Use a truth table to determine whether the statement is a tautology or contradiction: P⇒−Q⇔P∧Q.

The derivative of the function y = x²e^(-9x) is y' = 2xe^(-9x) - 9x²e^(-9x).

To find the derivative of the function y = x²e^(-9x), we can use the product rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by:

(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)

In this case, u(x) = x² and v(x) = e^(-9x). Let's find the derivatives of u(x) and v(x) first:

u'(x) = d/dx(x²) = 2x

v'(x) = d/dx(e^(-9x)) = -9e^(-9x) (using the chain rule)

Now, we can apply the product rule:

y' = u'(x)v(x) + u(x)v'(x)

= (2x)(e^(-9x)) + (x²)(-9e^(-9x))

= 2xe^(-9x) - 9x²e^(-9x)

Therefore,

To find the derivative of the function y = x²e^(-9x), we can use the product rule.

The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by:

(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)

In this case, u(x) = x² and v(x) = e^(-9x). Let's find the derivatives of u(x) and v(x) first:

u'(x) = d/dx(x²) = 2x

v'(x) = d/dx(e^(-9x)) = -9e^(-9x) (using the chain rule)

Now, we can apply the product rule:

y' = u'(x)v(x) + u(x)v'(x)

= (2x)(e^(-9x)) + (x²)(-9e^(-9x))

= 2xe^(-9x) - 9x²e^(-9x)

Therefore, the derivative of the function y = x²e^(-9x) is y' = 2xe^(-9x) - 9x²e^(-9x).

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A shipping company must design a closed rectangular shipping crate with a square base. The material for the top and sides costs $3 per square foot and the material for the bottom costs $13 per square foot. Find the dimensions of the crate that will minimize the total cost of material.

Let’s take the following dimensions: Length, width, and height are L, W, and H respectively. Then we have the following volume:LWH = 4608Hence, L²W = 4608We need to minimize the cost, which is given as:3(2LH + WH) + 13L²/WTo do this, we find the partial derivatives of the cost with respect to L and W (keeping H constant).

Hence,∂C/∂L = 6H + 26L/W²= 0∂C/∂W = 3H + 13L²/W²= 0Solving these equations gives,L = W/2, and L³ = 4608 or L = 16. Therefore,W = 32 and H = 9. Hence, the dimensions that minimize the cost are 16 feet by 32 feet by 9 feet.

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Since the value of g''(1) is negative (-2sin(1) < 0), we can conclude that the graph of g(x) is concave downwards at x = 1.

a. To find g(-1), we substitute the lower limit of integration into the integral:

g(-1) = ∫[-1, -1] cos[tex](t^2)[/tex] dt

Since the upper and lower limits are the same, the integral evaluates to zero:

g(-1) = 0

To approximate g(1) using a graphing calculator, you can follow these steps:

1. Enter the function f(x) = cos([tex]t^2[/tex]) into your graphing calculator.

2. Find the definite integral of f(x) from -1 to 1. This can usually be done by accessing the integral feature on your calculator.

3. The result of the definite integral will give you an approximation of g(1).

b. To find g'(x), we differentiate g(x) with respect to x using the Fundamental Theorem of Calculus:

g'(x) = d/dx ∫[-1, x] cos[tex](t^2)[/tex] dt

By the Second Fundamental Theorem of Calculus, we can find g'(x) by evaluating the integrand at the upper limit and multiplying by the derivative of the upper limit:

g'(x) = cos([tex]x^2)[/tex] * d/dx(x)

= cos[tex](x^2[/tex])

Therefore, g'(x) = cos([tex]x^2[/tex]).

c. To determine if the graph of g(x) is increasing, decreasing, or neither at x = 0, we need to analyze the sign of g'(x) around that point.

Let's consider the values of x slightly greater and slightly smaller than 0:

For x < 0: If we substitute a negative value into cos(x^2), we get positive values. Therefore, g'(x) > 0 for x < 0.

For x > 0: If we substitute a positive value into cos(x^2), we also get positive values. Therefore, g'(x) > 0 for x > 0.

Since g'(x) > 0 for x < 0 and g'(x) > 0 for x > 0, we can conclude that the graph of g(x) is increasing on the entire interval around x = 0.

d. To determine the concavity of the graph of g(x) at x = 1, we need to analyze the concavity of g(x) by examining the second derivative, g''(x).

Taking the derivative of g'(x) = cos([tex]x^2)[/tex], we get:

g''(x) = d/dx [cos(x^2)] = -2x * sin([tex]x^2[/tex])

To determine the concavity, we need to evaluate g''(x) at x = 1:

g''(1) = -2 * 1 * sin(1^2) = -2sin(1)

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The determinants for the given matrices:

(a) The determinant of matrix (a) is 2.

(b) The determinant of matrix (b) is 0.

(c) The determinant of matrix (c) is 3.

(d) The determinant of matrix (d) is 0.

(e) The determinant of matrix (e) is 2.

The determinant of a matrix is a scalar value that can be computed from the elements of the matrix. It provides important information about the matrix, such as whether it is invertible or singular. Let's calculate the determinants for the given matrices:

(a) The matrix is:

[ 1  -1 ]

[ 0   2 ]

To calculate the determinant of a 2x2 matrix, we use the formula: det(A) = ad - bc, where A is the matrix [a b; c d].

So, in this case, the determinant is: (1*2) - (-1*0) = 2.

Therefore, the determinant of matrix (a) is 2.

(b) The matrix is:

[ 1  1 ]

[ 1  1 ]

Using the same formula as above, we have: (1*1) - (1*1) = 0.

Hence, the determinant of matrix (b) is 0.

(c) The matrix is:

[ 1  2  0 ]

[ 0  1  0 ]

[ 2 -1  3 ]

To calculate the determinant of a 3x3 matrix, we can expand along any row or column using the cofactor expansion formula. Let's expand along the first row:

det(C) = 1 * det([1 0; -1 3]) - 2 * det([0 0; 2 3])

Calculating the determinants of the 2x2 matrices:

det([1 0; -1 3]) = (1*3) - (0*-1) = 3

det([0 0; 2 3]) = (0*3) - (0*2) = 0

Substituting back into the expansion formula:

det(C) = 1 * 3 - 2 * 0 = 3

Therefore, the determinant of matrix (c) is 3.

(d) The matrix is:

[ 1  2  0 ]

[ 0  1  1 ]

[ 1  0 -2 ]

Expanding along the first row:

det(D) = 1 * det([1 1; 0 -2]) - 2 * det([0 1; 1 -2])

Calculating the determinants of the 2x2 matrices:

det([1 1; 0 -2]) = (1*-2) - (1*0) = -2

det([0 1; 1 -2]) = (0*-2) - (1*1) = -1

Substituting back into the expansion formula:

det(D) = 1 * (-2) - 2 * (-1) = -2 + 2 = 0

Hence, the determinant of matrix (d) is 0.

(e) The matrix is:

[ 1  2  0  0 ]

[ 0  1  1  1 ]

[ 1  0 -2  0 ]

[ 0  1  0  0 ]

Expanding along the first row:

det(E) = 1 * det([1 1 1; 0 -2 0; 1 0 0]) - 2 * det([0 1 1; 1 -2 0; 0 0 0])

Calculating the determinants of the 3x3 matrices:

det([1 1 1; 0 -2 0; 1 0 0]) = (1 * (-2 *

0) - 1 * 0) - (0 - 1 * 1) + (1 - 0 * (-2)) = 0 - 1 + 1 = 0

det([0 1 1; 1 -2 0; 0 0 0]) = (0 - (-2 * 0)) - (0 - 1 * 1) + (0 * (-2) - 1 * 0) = 0 - 1 + 0 = -1

Substituting back into the expansion formula:

det(E) = 1 * 0 - 2 * (-1) = 0 + 2 = 2

Therefore, the determinant of matrix (e) is 2.

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The given function is f(x) = 2x³ − 96x + 2To find the x-coordinates of the points, we will find the derivative of the given function.

f(x) = 2x³ − 96x + 2f'(x) = 6x² - 96Now, the horizontal tangent lines have slope zero or m = 0.f'(x) = 0 => 6x² - 96 = 0 => x² - 16 = 0 => x = ±√16 => x = ±4

Hence, the x-coordinates of the points at which the graph of each function has a horizontal tangent line are -4 and 4, respectively.

Now, we need to find the equation of the tangent line at x = -4 and x = 4 respectively.  Equation of the tangent line at x = 4:Slope of the tangent line at x = 4, m = f'(4) = 6(4)² - 96 = 0y = f(4) = 2(4)³ - 96(4) + 2 = -190

Equation of the tangent line with slope m passing through (x₁, y₁) is given by (y - y₁) = m(x - x₁)⇒ (y - (-190)) = 0(x - 4)⇒ y = -190Hence, the equation of the tangent line at x = 4 is y = -190.

Equation of the tangent line at x = -4:Slope of the tangent line at x = -4, m = f'(-4) = 6(-4)² - 96 = 0y = f(-4) = 2(-4)³ - 96(-4) + 2 = 190

Equation of the tangent line with slope m passing through (x₁, y₁) is given by

(y - y₁) = m(x - x₁)

⇒ (y - 190) = 0(x + 4)

⇒ y = 190

Hence, the equation of the tangent line at x = -4 is y = 190.Solving f'(x) > 0f'(x) > 0 => 6x² - 96 > 0=> 6(x² - 16) > 0 => 6(x + 4)(x - 4) > 0For f'(x) > 0, the solution is (-infinity, -4)U(4, infinity).Solving f'(x) < 0f'(x) < 0 => 6x² - 96 < 0=> 6(x² - 16) < 0 => 6(x + 4)(x - 4) < 0For f'(x) < 0, the solution is (-4, 4).

Therefore, the solutions of the inequality f'(x) > 0 and f'(x) < 0 are (-infinity, -4)U(4, infinity) and (-4, 4), respectively.

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The concentration of an inhibitor directly affects the inhibition of an enzyme. As the concentration of the inhibitor increases, the inhibition of the enzyme becomes more pronounced.

When an inhibitor binds to an enzyme, it can either compete with the substrate for the active site (competitive inhibition) or bind to a different site on the enzyme, causing a conformational change that affects the enzyme's activity (non-competitive inhibition). In both cases, the concentration of the inhibitor plays a crucial role in determining the extent of enzyme inhibition.

In competitive inhibition, increasing the concentration of the inhibitor increases the likelihood of it occupying the active site, effectively preventing the substrate from binding and reducing the enzyme's activity. This inhibition can be overcome by increasing the concentration of the substrate to outcompete the inhibitor.

In non-competitive inhibition, the inhibitor binds to a different site on the enzyme, causing a change in the enzyme's structure or function. Increasing the concentration of the inhibitor increases the probability of inhibitor binding and leads to a greater inhibition of enzyme activity. Unlike competitive inhibition, increasing the substrate concentration does not alleviate the inhibition in non-competitive inhibition.

In summary, the concentration of an inhibitor directly impacts enzyme inhibition. Higher inhibitor concentrations lead to stronger inhibition, reducing the enzyme's activity and potentially interfering with its biological function.

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The differential is exact is 15. Hence, the correct answer is C.

Given that, [tex]\[ \int_{(1,1,1)}^{(2,2,2)}-\frac{2 z^{8}}{x^{3} y^{2}} d x-\frac{2 z^{8}}{x^{2} y^{3}} d y+\frac{8 z^{7}}{x^{2} y^{2}} d z \][/tex].

The given differential is exact; that is, it is a grouping of differentials that represent a single function, namely:

[tex]\[f(x,y,z)=-\frac{2z^8}{x^3y^2}+\frac{2z^8}{x^2y^3}-\frac{8z^7}{x^2y^2}\][/tex]

To find the definite integral, we need to evaluate the limits:

[tex]\[I=\int_{(1,1,1)}^{(2,2,2)}f(x,y,z)dxdy dz\][/tex]

Substituting in the limits and evaluating the integral using the Fundamental Theorem of Calculus, we have:

[tex]\begin{aligned}I &= \left[-\frac{2z^8}{x^3y^2}+\frac{2z^8}{x^2y^3}-\frac{8z^7}{x^2y^2}\right]_{(1,1,1)}^{(2,2,2)} \\&= 2^8+2^8-8\times 2^7 \\&= 15\end{aligned}[/tex]

Hence, the correct answer is C.

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"Your question is incomplete, probably the complete question/missing part is:"

Evaluate. The differential is exact. [tex]\[ \int_{(1,1,1)}^{(2,2,2)}-\frac{2 z^{8}}{x^{3} y^{2}} d x-\frac{2 z^{8}}{x^{2} y^{3}} d y+\frac{8 z^{7}}{x^{2} y^{2}} d z \][/tex].

A. 13 B. 16 C. 15 D. 1

1. dy/dx = 3x / √(1 + 3x²)  2. dh(r)/dr = e^(-5) - 5re^(-5)  4. dy/dx = cos(x¹) are the derivatives of every function

derivative of each function  are shown below

Given the following functions:

1) y = √(1 + 3x²)

Using the chain rule, we get

dy/dx = (1/2) (1 + 3x²)^(-1/2) * 6x

dy/dx = 3x / √(1 + 3x²)

2) h(r) = re^(-5)

Using the product rule, we get

dh(r)/dr = d/dr (r) * e^(-5) + r * d/dr(e^(-5))

dh(r)/dr = e^(-5) - 5re^(-5)

3)  f(x) = e^(sinz cos(ar))

Using the chain rule, we get

df/dx = e^(sinz cos(ar)) * d/dx(sinz cos(ar))

df/dx  = e^(sinz cos(ar)) * (cosz cos(ar) * dz/dx - sinz sin(ar) * dar/dx)

4) y = sin(x¹)

Using the power rule, we get

dy/dx = (x¹)' * cos(x¹ - π/2)

dy/dx = cos(x¹)

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Pump cavitation refers to the formation and collapse of vapor bubbles in a pump due to low pressure conditions. It can cause damage to the pump and decrease its efficiency. Preventing pump cavitation is crucial for maintaining optimal pump performance and avoiding potential issues.

Pump cavitation occurs when the pressure at the pump inlet drops below the vapor pressure of the liquid being pumped, causing the formation of vapor bubbles.

When these bubbles move to regions of higher pressure within the pump, they collapse, creating tiny shockwaves that can erode the pump impeller and other components.

This erosion can lead to reduced pump efficiency, increased vibration, and even mechanical failure.

To prevent pump cavitation, it is important to ensure an adequate Net Positive Suction Head (NPSH). NPSH is a measure of the available pressure at the pump inlet above the vapor pressure of the fluid. It determines the margin of safety against cavitation.

Maintaining a sufficient NPSH value is crucial to prevent cavitation. Preventing pump cavitation involves several measures. Firstly, selecting a pump with appropriate specifications and operating it within its recommended range can help ensure sufficient NPSH.

Additionally, proper system design, including adequate pipe sizing, minimizing pressure losses, and avoiding sudden changes in flow velocity, can contribute to preventing cavitation.

Proper maintenance, such as regular inspection of impellers and suction piping, can also help identify and address any issues that may lead to cavitation.

Overall, preventing pump cavitation involves maintaining a sufficient NPSH through proper pump selection, system design, and regular maintenance to ensure smooth and efficient pump operation.

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The bank should sample 429 customers.

To determine the number of customers that should be sampled, we need to use the formula for sample size calculation in estimating a population mean. The formula is given by:

n = (Z * σ / E)^2

Where:

n = sample size

Z =  corresponding to the desired level of confidence (98% confidence corresponds to a z-score of approximately 2.33)

σ = population standard deviation

E = desired margin of error

Plugging in the given values, we have:

n = (2.33 * 300 / 89)^2

n ≈ 429

Therefore, the bank should sample approximately 429 customers.

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The sum of the given infinite series is 15.

The given example of a series is  [tex]∑ k=0 [infinity] 5 3k 2k.[/tex]

Let's understand the following phrases with the help of the given series.

a. Sequence of un-summed terms

The un-summed terms of the series are 5, 15, 45, 135, and so on.

These terms are known as a sequence of un-summed terms.

b. Sequence of partial sums

Partial sum is the sum of a finite number of terms of an infinite series.

So, the sequence of partial sums of the given series is {5, 20, 65, 200, ...}.

c. Summable vs. not summable infinite series

The infinite series is summable if its partial sum has a finite limit as n approaches infinity, otherwise not summable.

In the given example, the infinite series is summable because it satisfies the condition of convergence, which is [tex]3/2 < r < 1[/tex], where r is the common ratio.

d. Sum of an infinite series

The sum of an infinite series is the limit of the sequence of partial sums as n approaches infinity.

The sum of the given series is: [tex]S = a / (1 - r)[/tex]

where a = first term of the series = 5

r = common ratio = 2/3

Putting these values in the formula, we get:

[tex]S = 5 / (1 - 2/3)\\= 5 / 1/3\\= 15[/tex]

So, the sum of the given infinite series is 15.

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Answer:

The connections securing the roof of a two story house to the exterior walls are critical for the building's overall lateral stability. These connections provide a structural link between the roof and walls, allowing them to act as a single unit and resist lateral forces such as wind or seismic activity. Without these connections, the roof and walls would be unable to transfer forces between each other, leading to an increased risk of collapse in the event of an earthquake or high winds. Additionally, these connections help ensure that the roof is properly supported by the walls, preventing it from sagging or collapsing due to its own weight.

The connections securing the roof of a two-story house to the exterior walls play a crucial role in the building's overall lateral stability. Lateral stability refers to the ability of a structure to resist horizontal forces such as wind or seismic activity. Here's why these connections are important:

1. Structural integrity: The roof connections provide a strong link between the roof and the walls, ensuring that the entire structure acts as a unified system. This helps distribute the loads imposed by lateral forces evenly throughout the building.

2. Wind resistance: During high winds, lateral forces can exert pressure on the exterior walls. The roof connections help transfer these forces to the walls, preventing the roof from lifting or separating from the rest of the structure. This is particularly important for two-story houses, as the height increases the vulnerability to wind loads.

3. Seismic protection: In earthquake-prone areas, lateral stability is crucial for minimizing structural damage. The connections between the roof and walls help resist the lateral forces generated during an earthquake. This prevents the roof from shifting or collapsing, reducing the risk of injuries and property damage.

4. Load distribution: The roof connections ensure that the weight of the roof is evenly distributed to the supporting walls. This helps prevent excessive stress on any single point, maintaining the structural integrity of the building over time.

To ensure the effectiveness of these connections, proper construction techniques and materials are essential. For example, hurricane ties or metal connectors may be used to reinforce the roof-to-wall connections. These components are designed to withstand the forces exerted on the structure and provide additional strength and stability.

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The annual increase in the value of the comic book will be 28.61% if the first appearance of Superman was sold at auction for $1,920,000 and was originally sold in 1942 for $.09.

Here is the detailed solution:

Given,In 2012, an Action Comics No. 1, featuring the first appearance of Superman, was sold at auction for $1,920,000.

The comic book was originally sold in 1942 for $.09.

The annual increase in the value of the comic book is to be found.

To find the percentage annual increase, we can use the compound interest formula as follows:

Present value (P) = $0.09

Future value (F) = $1,920,000

Time (n) = 2012 - 1942

= 70 years

Interest rate (r) = ?

We can use the formula for compound interest to solve for r:

F = P(1 + r)n

$1,920,000 = $0.09(1 + r)70

Taking the natural logarithm of both sides:

ln($1,920,000) = ln($0.09) + 70 ln(1 + r)

Simplifying:

70 ln(1 + r) = ln($1,920,000) - ln($0.09)

70 ln(1 + r) = 20.309

Dividing both sides by 70:

ln(1 + r) = 0.29013

Taking the antilogarithm of both sides:

1 + r = e0.29013

Simplifying:

1 + r = 1.3365

Subtracting 1 from both sides:

r = 0.3365 or 33.65%

Therefore, the annual increase in the value of the comic book is 33.65%.

Rounding this off to two decimal places gives 28.61%.

Thus, the annual increase in the value of the comic book will be 28.61%.

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The annual increase in the value of the comic book is 18.67%.

The annual increase in the value of the comic book can be calculated as follows:

We have, the original price of the Action Comics No. 1 was sold in 1942 for $.09.In 2012, it was sold at auction for $1,920,000.

Thus, the annual increase in the value of the comic book is as follows:

Annual increase = (Final value / Initial value)^(1/Number of years) - 1

Here, the final value = $1,920,000;

initial value = $0.09;

Number of years = 2012 - 1942

= 70 years.

Substituting the values in the above equation, we get

Annual increase % = (1920000/0.09)^(1/70) - 1

= 18.67% (rounded to two decimal places)

Therefore, the annual increase in the value of the comic book is 18.67%.

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We need to find the volume V of the parallelepiped such that the following four points A = (2, 1, 2), B = (3, 1, −2), C = (1, 3, 3), D = (2, 0, −1) are vertices and the vertices B, C, D are all adjacent to the vertex A.

The volume of a parallelepiped is given by the scalar triple product of the vectors representing three adjacent sides. The vectors representing three adjacent sides are:`

AB = (3 - 2)i + (1 - 1)j + (-2 - 2)k = i - 4k``AC = (1 - 2)i + (3 - 1)j + (3 - 2)k = -i + 2j + k``AD = (2 - 2)i + (0 - 1)j + (-1 - 2)k = -j - 3k`.

Let's find the scalar triple product of `Notice that `j² = 1` and `k² = 1`. So:`V = 8ij + j² - 42ik - 3kj - 24k²``  = 8i - j - 42k - 3k``  = (8, -1, -45)`Hence, the volume of the parallelepiped is 8 cubic units.

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Borrowing at low interest rates is a significant factor in the rising standard of living in the United States.

The correct answer is c) low interest rates; rising.

Borrowing at low interest rates is a major reason for the rising standard of living in the United States. When interest rates are low, it becomes more affordable for individuals, businesses, and the government to borrow money for various purposes such as purchasing homes, starting businesses, or investing in infrastructure.

Low interest rates mean that the cost of borrowing is lower, allowing people to access credit more easily and at a lower cost. This enables individuals and businesses to make large purchases or investments that they might not be able to afford otherwise. For example, low mortgage interest rates make homeownership more affordable, and low business loan rates facilitate entrepreneurship and business expansion.

Moreover, low interest rates can stimulate economic activity and boost consumer spending, which further contributes to a rising standard of living. When people can borrow money at lower costs, they have more disposable income, which can be spent on goods and services, driving economic growth and job creation.

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According to the question corresponding to an increase in production and sales from 10,000 to 20,000 iPhone 6's per hour is [tex]\$4,200,150.[/tex]

To determine the total additional hourly profit corresponding to an increase in production and sales from 10,000 to 20,000 iPhone 6's per hour, we can calculate the difference in profit between these two production levels.

The profit function is given by the difference between revenue and cost. The revenue is the product of the number of units sold and the selling price, while the cost is the integral of the marginal cost function.

For 10,000 iPhone 6's per hour:

[tex]\text{Revenue} &= \text{Number of units sold} \times \text{Selling price} \\&= 10,000 \times \$580 \\&= \$5,800,000[/tex]

The cost can be calculated by integrating the marginal cost function:

[tex]\text{Cost} &= \int_{0}^{10,000} C'(x) \, dx \\[/tex]

[tex]&= \int_{0}^{10,000} (160 - 0.002x) \, dx \\[/tex]

[tex]&= \left[160x - \frac{0.002}{2}x^2\right]_{0}^{10,000} \\[/tex]

[tex]&= 1,600,000 - 50 \\[/tex]

[tex]&= \$1,599,950[/tex]

The profit is given by the difference between revenue and cost:

[tex]\text{Profit} &= \text{Revenue} - \text{Cost} \\[/tex]

[tex]&= \$5,800,000 - \$1,599,950 \\[/tex]

[tex]&= \$4,200,050[/tex]

For 20,000 iPhone 6's per hour:

[tex]\text{Revenue} &= \text{Number of units sold} \times \text{Selling price} \\&= 20,000 \times \$580 \\&= \$11,600,000[/tex]

The cost can be calculated by integrating the marginal cost function:

[tex]\text{Cost} &= \int_{0}^{20,000} C'(x) \, dx \\\\&= \int_{0}^{20,000} (160 - 0.002x) \, dx \\\\&= \left[160x - \frac{0.002}{2}x^2\right]_{0}^{20,000} \\\\&= 3,200,000 - 200 \\&= \$3,199,800[/tex]

The profit is given by the difference between revenue and cost:

[tex]\text{Profit} &= \text{Revenue} - \text{Cost} \\[/tex]

[tex]&= \$11,600,000 - \$3,199,800 \\[/tex]

[tex]&= \$8,400,200[/tex]

To calculate the total additional hourly profit, we subtract the profit at the lower production level from the profit at the higher production level:

[tex]\text{Total additional hourly profit} &= \text{Profit at 20,000 units}[/tex] [tex]- \text{Profit at 10,000 units} \\[/tex]

[tex]&= \$8,400,200 - \$4,200,050 \\[/tex]

[tex]&= \$4,200,150[/tex]

corresponding to an increase in production and sales from 10,000 to 20,000 iPhone 6's per hour is [tex]\$4,200,150.[/tex]

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The figures that have rotational symmetry are:

Option D: Hexagon

Option E: Cross

Rotational symmetry, is also known as radial symmetry is defined as the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation.

Hexagon: The six sides of the hexagon are congruent, as they are all equal and this tells us that no mater the rotation it will always be the same.

The Cross: The four corners of the cross are equal and this tells us that no matter the rotation it will always be the same.

All the other figures have sides that are NOT congruent as they have different side lengths

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After considering the given data we conclude that the answers to the sub questions are
a) the machine will not be stopped and adjusted.
b) the calculated value of t (3.08) is greater than the critical value of t (1.734), we can reject the null hypothesis. Therefore, the machine needs to be stopped and adjusted.

a) Calculate the p-value for this test of hypothesis. Based on this p-value, will the quality control inspector decide to stop the machine and adjust it if he chooses the maximum probability of a Type I error to be .02? What if the maximum probability of a Type I error is .10?
To calculate the p-value for this test of hypothesis, we can use a t-test since the population standard deviation is unknown and the sample size is small (n = 20). The test statistic is given by:
[tex]t = (\bar{X} - \mu) / (s \sqrt(n))[/tex]
where [tex]\bar_{X}[/tex] is the sample mean, [tex]\mu[/tex] is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. Substituting the given values, we get:
[tex]t = (36.015 - 36) / (0.035 \sqrt(20)) = 3.08[/tex]
The degrees of freedom for this test is n - 1 = 19. Using a t-distribution table or a calculator, we can find that the two-tailed p-value for this test is approximately 0.006.
If the maximum probability of a Type I error is 0.02, then the quality control inspector will reject the null hypothesis since the p-value is less than 0.02. Therefore, the machine will be stopped and adjusted.
If the maximum probability of a Type I error is 0.10, then the quality control inspector will not reject the null hypothesis since the p-value is greater than 0.10. Therefore, the machine will not be stopped and adjusted.

b) Test the hypothesis of part a using the critical-value approach and a = .02. Does the machine need to be adjusted? What if a = .10?
To test the hypothesis of part a using the critical-value approach, we can find the critical values of t for a two-tailed test with 19 degrees of freedom and a significance level of 0.02. Using a t-distribution table or a calculator, we can find that the critical values are approximately ±2.539. Since the calculated value of t (3.08) is greater than the critical value of t (2.539), we can reject the null hypothesis. Therefore, the machine needs to be stopped and adjusted.
If a = 0.10, then the critical values of t are approximately ±1.734. Since the calculated value of t (3.08) is greater than the critical value of t (1.734), we can reject the null hypothesis. Therefore, the machine needs to be stopped and adjusted.
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a. The p-value for this test of hypothesis is approximately 0.5081. Based on this p-value, the quality control inspector would not stop the machine and adjust it, whether the maximum probability of a Type I error is 0.02 or 0.10.

b. Using the critical-value approach with a = 0.02, we fail to reject the null hypothesis. The machine does not need to be adjusted. The same conclusion holds true if a = 0.10.

To calculate the p-value for this test of hypothesis, we can use the standard normal distribution and the sample information provided.

- Sample mean ([tex]\bar{x}[/tex]) = 36.015 inches

- Population standard deviation (σ) = 0.035 inch

- Sample size (n) = 20

a. Calculating the p-value:

The test statistic for this hypothesis test is calculated using the formula:

t = ([tex]\bar{x}[/tex] - μ) / (σ / √n)

Substituting the values:

t = (36.015 - 36) / (0.035 / √20)

t = 0.015 / (0.035 / √20)

t ≈ 0.6745

To find the p-value, we need to determine the probability of observing a test statistic as extreme as the calculated t-value, assuming the null hypothesis is true. Since the alternative hypothesis is "μ ≠ 36 inches," we are conducting a two-tailed test.

Using a t-distribution table or a statistical calculator, we find the p-value associated with a t-value of 0.6745 (degrees of freedom = n - 1 = 19) to be approximately 0.5081.

b. Decision based on p-value:

If the maximum probability of a Type I error (α) is 0.02, and since the p-value of 0.5081 is greater than α, we fail to reject the null hypothesis. Therefore, the quality control inspector would not decide to stop the machine and adjust it.

If the maximum probability of a Type I error is 0.10, and since the p-value of 0.5081 is still greater than α, we again fail to reject the null hypothesis. The machine would not need to be adjusted in this case as well.

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A line through A and X, then extending it by a factor of 2 to reach A'. The coordinates of vertex A' are (8,6). The coordinates of vertex A' are (8,6).

Trapezium ABCD is being enlarged using a scale factor of 2 and center X to give trapezium A'B'C'D'

To find: The coordinates of vertex A'As we know that the trapezium is being enlarged by a scale factor of 2 with center X.

We use the following steps to find the coordinates of vertex A'.

Draw a diagonal through the trapezium, from vertex A to vertex C.2.

The center of enlargement, X, is the midpoint of AC.

Mark it clearly.3. From X, draw lines that pass through each vertex of the original trapezium, ABCD.

These lines will intersect the opposite side of the enlarged trapezium, A'B'C'D', at the corresponding vertices.

Label these vertices clearly.

In our case, we need to find the coordinates of vertex A' which is the image of vertex A.

So, we construct a diagram as shown in the figure attached below.

Here, the center of enlargement, X, is the midpoint of AC. The lines through vertices A, B, C, and D meet the opposite side of A'B'C'D' at vertices A', B', C', and D', respectively.

Therefore, we can find the coordinates of A' by drawing a line through A and X, then extending it by a factor of 2 to reach A'.

Hence, the coordinates of vertex A' are (8,6).

The coordinates of vertex A' are (8,6).

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The coordinates of vertex A' are given as follows:

A'(4, 12).

A dilation is defined as a non-rigid transformation that multiplies the distances between every point in a polygon or even a function graph, called the center of dilation, by a constant factor called the scale factor.

The coordinates of the vertex A in this problem are given as follows:

A(2, 6).

The scale factor is given as follows:

k = 2.

Hence the coordinates of vertex A' are given as follows:

A'(4, 12).

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Depending on the specific integral in question, one of these methods may be more appropriate. It's important to carefully analyze the integral and choose the appropriate test based on the characteristics and available information.

To test the convergence of an integral, we can utilize integration, the direct comparison test, or the limit comparison test. Let's explore each method briefly:

1. Integration Test:

By directly evaluating the integral, we can determine if it converges or diverges. If the integral yields a finite value, then it converges; otherwise, if the integral diverges to infinity or negative infinity, then it diverges. This method is straightforward but may not be applicable to all integrals.

2. Direct Comparison Test:

The direct comparison test is useful when we have a non-negative function that may not be easy to integrate. By comparing the given function with a known function whose convergence is already established, we can determine the convergence of the given function. If the known function converges and is greater than or equal to the given function, then the given function also converges. If the known function diverges and is less than or equal to the given function, then the given function diverges.

3. Limit Comparison Test:

The limit comparison test involves comparing the given function to a known function and taking the limit of their ratio as x approaches infinity. If the limit is a positive, finite value, then both functions either both converge or both diverge. If the limit is zero or infinity, the test is inconclusive, and another method may be needed.

Depending on the specific integral in question, one of these methods may be more appropriate. It's important to carefully analyze the integral and choose the appropriate test based on the characteristics and available information.

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The function f: R² -> R² given by f(x, y) = (2xy, 4x² + y²) satisfies the Lipschitz condition on the set Q = {(x, y): |x| ≤ 1, |y| ≤ 2} with a Lipschitz constant L = 16.

To show that f satisfies the Lipschitz condition, we need to find a constant L such that for any two points (x₁, y₁) and (x₂, y₂) in Q, the difference between the function values of f at these points is bounded by L times the difference in the corresponding coordinates.

The difference in the function values:

|f₁(x₁, y₁) - f₁(x₂, y₂)| + |f₂(x₁, y₁) - f₂(x₂, y₂)| = |2x₁y₁ - 2x₂y₂| + |4x₁² + y₁² - 4x₂² - y₂²|

The properties of absolute values, simplify this expression:

|2x₁y₁ - 2x₂y₂| + |4x₁² + y₁² - 4x₂² - y₂²| ≤ 2|x₁ - x₂||y₁| + |4x₁² - 4x₂²| + |y₁² - y₂²|

The ranges of x and y in Q: |x| ≤ 1 and |y| ≤ 2. Using these bounds, we can further simplify the expression:

2|x₁ - x₂||y₁| + |4x₁² - 4x₂²| + |y₁² - y₂²| ≤ 2|x₁ - x₂|2 + 4|x₁ - x₂| + 4|y₁ - y₂|

Choose L = 16 as a Lipschitz constant that bounds the expression above. Thus, for any two points (x₁, y₁) and (x₂, y₂) in Q, we have:

|f₁(x₁, y₁) - f₁(x₂, y₂)| + |f₂(x₁, y₁) - f₂(x₂, y₂)| ≤ L(|x₁ - x₂| + |y₁ - y₂|)

Therefore, f satisfies the Lipschitz condition on Q with L = 16.

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To factorize -3x² + 2x + 1 using completing the square, we follow the steps given below.

Step 1: Rewrite the quadratic equation in the form ax² + bx + c. This gives a = -3, b = 2, and c = 1.

Thus, the equation becomes -3x² + 2x + 1 = 0.

Step 2: Divide the equation throughout by -3 to get x² - (2/3)x - (1/3) = 0.

Step 3: To make the left-hand side of the equation a perfect square, we add and subtract `1/9` as shown below: x² - [tex](2/3)x + 1/9 - 1/9 - 1/3[/tex]= 0.

Step 4: Rearrange the terms to get x²[tex]- (2/3)x + 1/9 = 1/3[/tex].

Step 5: Factorize the left-hand side of the equation as (x - 1/3)² = 1/3 + 1/9. This gives (x - 1/3)² = 4/9.

Step 6: Take the square root of both sides of the equation to get x - 1/3 = ± 2/3.Thus, x = 1/3 ± 2/3.Simplifying the solution, we get x = 1 or x = -1/3.

We can use completing the square to factorize -3x² + 2x + 1. The steps involved in factorizing the equation are given above.

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The dimension of the column space of the matrix is 2. The dimension of the null space of a matrix is also known as the nullity.

In this case, if the nullity of the \(4 \times 6\) matrix is 4, it means that there are 4 linearly independent vectors that satisfy the equation \(Ax = 0\), where \(A\) is the matrix and \(x\) is a vector.

The null space consists of all the vectors that get mapped to the zero vector when multiplied by the matrix. Geometrically, it represents the set of solutions to a homogeneous system of linear equations.

Now, the rank-nullity theorem states that for any matrix \(A\), the sum of the rank and nullity of \(A\) is equal to the number of columns of \(A\). In this case, we have a \(4 \times 6\) matrix, so it has 6 columns.

Using the rank-nullity theorem, we can find the dimension of the column space (also known as the rank) of the matrix:

\[ \text{{rank}}(A) + \text{{nullity}}(A) = \text{{number of columns of }} A\]

\[ \text{{rank}}(A) + 4 = 6\]

\[ \text{{rank}}(A) = 6 - 4\]

\[ \text{{rank}}(A) = 2\]

Therefore, the dimension of the column space of the matrix is 2.

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The given triple integral represents the volume of the region E, which lies below the plane z = 1 + x + y and above the region in the xy-plane bounded by the curves y = x and y = 0.

To evaluate the triple integral ∭E 6xy dV, we first need to determine the limits of integration for each variable. The region E is defined as the space below the plane z = 1 + x + y and above the region in the xy-plane bounded by the curves y = x and y = 0.

In the xy-plane, the lower boundary is y = 0, and the upper boundary is y = x. Therefore, the limits of integration for y are 0 to x. For x, the region is not explicitly bounded, so we need to consider the intersection of the plane z = 1 + x + y with the xy-plane. Setting z = 0, we can solve for x and y to find the boundaries of x. We have x = -1 - y.

Now, for z, the region E lies below the plane z = 1 + x + y, so the upper boundary for z is given by the equation of the plane itself, z = 1 + x + y.

To evaluate the integral, we integrate 6xy with respect to z, from 0 to 1 + x + y, then integrate the result with respect to y, from 0 to x, and finally integrate the resulting expression with respect to x, considering the limits of x as -1 - y to 0.

Evaluating this triple integral will yield the volume of the region E described by the given conditions.

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The average value of f(x) = sin(2x) on the interval [4π, 2π] is 0.

To find the average value of the function f(x) = sin(2x) on the interval [4π, 2π], we need to calculate the definite integral of f(x) over that interval and then divide it by the length of the interval.

The average value of f(x) on the interval [a, b] is given by:

Average value = (1 / (b - a)) ∫[a to b] f(x) dx

In this case, a = 4π and b = 2π. Let's calculate the definite integral of f(x) over [4π, 2π]:

∫[4π to 2π] sin(2x) dx

Using the integral rules for sin(2x), we have:

= (-1/2) * [cos(2x)] [4π to 2π]

= (-1/2) * (cos(4π) - cos(2π))

cos(4π) and cos(2π) both evaluate to 1, so we have:

= (-1/2) * (1 - 1)

= (-1/2) * 0

= 0

Now, let's calculate the length of the interval [4π, 2π]:

Length of interval = b - a

                  = 2π - 4π

                  = -2π

The length of the interval is negative because the upper bound (2π) is smaller than the lower bound (4π). However, the length itself should be positive, so we take the absolute value:

|Length of interval| = |-2π|

                    = 2π

Finally, we can calculate the average value of f(x):

Average value = (1 / |Length of interval|) * ∫[4π to 2π] sin(2x) dx

            = (1 / (2π)) * 0

            = 0

Therefore, the average value of f(x) = sin(2x) on the interval [4π, 2π] is 0.

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Given that the probability distribution function of a discrete variable X is as follows:f(x) = {1/7kx, x = 3, 4, 5, 6{0, Otherwise where k is a constant. a) Find the value of k.

Probability distribution function is given byf(x) = {1/7kx, x = 3, 4, 5, 6{0, OtherwiseAs we know that the sum of probabilities of all possible outcomes is equal to 1, thus the sum of the probability for x = 3, 4, 5, and 6 is 1.So,1/7k(3) + 1/7k(4) + 1/7k(5) + 1/7k(6) = 11/7kThus, we havek = 44.

b) Construct the probability distribution table. The probability distribution table is as follows:x 3 4 5 6 f(x) 1/11 4/11 5/11 6/11 c) Calculate the mean and variance of X. The mean of X is given byμ = ∑xf(x) = 3(1/11) + 4(4/11) + 5(5/11) + 6(6/11) = 94/11The variance of X is given byσ² = ∑(x-μ)²f(x) = [(3-94/11)²(1/11)] + [(4-94/11)²(4/11)] + [(5-94/11)²(5/11)] + [(6-94/11)²(6/11)] = 1972/121d) Find P(1

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(a) About 7.0% of all applicants had scores higher than 13 on the MCAT biological sciences part.

(b) About 89% of those who entered medical school had scores between 9 and 12 on the MCAT biological sciences part.

(a) Approximately 15.87% of children aged 13 to 15 years old have scores above 101 on the intelligence test.

(b) The score marking the lowest 30 percent of the distribution is approximately 98.78 on the intelligence test.

(c) The score marking the highest 15 percent of the distribution is approximately 135.86 on the intelligence test.

(a) About 2.5% of men are taller than 74 inches in height.

(b) Approximately 2.5% of all men are shorter than 4 inches in height.

(c) About 13.59% of men are between 64 and 66.5 inches in height.

(a) For the scores on the MCAT biological sciences part, which are normally distributed with a mean of 9.9 and a standard deviation of 2.1, we want to find the percentage of all applicants who had scores higher than 13.

To calculate this, we need to find the area under the normal curve to the right of the score 13.

Using the standard normal distribution table or a calculator, we can find that the area to the right of 13 is approximately 0.070, or 7.0%. Therefore, about 7.0% of all applicants had scores higher than 13.

(b) For those applicants who entered medical school, we want to find the percentage of them who had scores between 9 and 12 on the MCAT biological sciences part.

To calculate this, we need to find the area under the normal curve between the scores 9 and 12.

Using the standard normal distribution table or a calculator, we can find that the area between 9 and 12 is approximately 0.89, or 89%.

Therefore, about 89% of those who entered medical school had scores between 9 and 12.

(a) For the scores on the intelligence test for children aged 13 to 15 years, which are normally distributed with a mean of 111 and a standard deviation of 24, we want to find the proportion of children who have scores above 101.

To calculate this, we need to find the area under the normal curve to the right of the score 101.

Using the standard normal distribution table or a calculator, we can find that the area to the right of 101 is approximately 0.8413.

Therefore, the proportion of children aged 13 to 15 years with scores above 101 is approximately 1 - 0.8413 = 0.1587, or 15.87%.

(b) The score which marks the lowest 30 percent of the distribution can be found by finding the z-score corresponding to the lower 30th percentile and then converting it back to the original scale using the mean and standard deviation.

The z-score corresponding to the lower 30th percentile is approximately -0.524. Converting this back to the original scale using the mean of 111 and the standard deviation of 24, we have:

Score = Mean + (Z-score * Standard deviation) = 111 + (-0.524 * 24) ≈ 98.78.

Therefore, the score which marks the lowest 30 percent of the distribution is approximately 98.78.

(c) To find the score which marks the highest 15 percent of the distribution, we need to find the z-score corresponding to the upper 15th percentile and then convert it back to the original scale.

The z-score corresponding to the upper 15th percentile is approximately 1.036. Converting this back to the original scale using the mean of 111 and the standard deviation of 24, we have:

Score = Mean + (Z-score * Standard deviation) = 111 + (1.036 * 24) ≈ 135.86.

Therefore, the score which marks the highest 15 percent of the distribution is approximately 135.86.

(a) For the heights of adult men in the U.S., which are normally distributed with a mean of 69 inches and a standard deviation of 2.5 inches, we want to find the percentage of men who are taller than 74 inches.

Using the 68-95-99.7 rule (also known as the empirical rule), we know that approximately 2.5% of the data falls beyond two standard deviations above the mean.

Since 74 inches is two standard deviations above the mean (69 + 2 * 2.5 = 74), we can conclude that about 2.5% of men are taller than 74 inches.

(b) According to the same 68-95-99.7 rule, approximately 2.5% of the data falls beyond two standard deviations below the mean.

Since 4 inches is more than two standard deviations below the mean (69 - 2 * 2.5 = 64), we can conclude that about 2.5% of all men are shorter than 4 inches.

(c) To find the percentage of men who are between 64 and 66.5 inches, we need to calculate the area under the normal curve between these two values.

First, we need to standardize the values by calculating the z-scores:

Z1 = (64 - 69) / 2.5 = -2

and Z2 = (66.5 - 69) / 2.5 = -1.

This corresponds to the area between -2 and -1 under the standard normal distribution.

Using the standard normal distribution table or a calculator, we can find that the area between -2 and -1 is approximately 0.1359.

Therefore, about 0.1359 or 13.59% of men are between 64 and 66.5 inches in height.

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The required partial derivative is -54.29 (approximate value).

We are given a function as z = xcos(xy).

We are also given that x = e^t and y = st, we need to find ∂z/∂t at x = 2 and y = 1.

The partial derivative of z with respect to x is obtained by considering y as a constant.

z = xcos(xy)

Taking partial derivative of z with respect to x,

∂z/∂x = cos(xy) + (-ysin(xy))(by applying chain rule)

= cos(st e^(t2)) - s e^(t2) sin(st e^(t2))

Now, we need to find partial derivative of z with respect to t.

For this, we consider y as a function of t.

z = xcos(xy)

= e^t cos(st e^(t2))

Now, taking partial derivative of z with respect to t, we get

∂z/∂t = (-sin(st e^(t2)))(s e^(t2)) (by applying chain rule)

= -s^2 e^(3t2) sin(st e^(t2))

Hence, the required partial derivative is -s^2 e^(3t2) sin(st e^(t2)).

As given, x = e^t and y = st

We need to find ∂z/∂t at x = 2 and y = 1

Substituting the given values, we get

[tex]x = e^t \\= 2[/tex]

=> [tex]t = ln(2)\\y = st \\= 1[/tex]

=> [tex]s = 1/t \\= 1/ln(2)[/tex]

Now, substituting these values in the expression of partial derivative obtained above, we get

∂z/∂t = -s^2 e^(3t2) sin(st e^(t2))= -1/ln^2(2) * e^(3ln^2(2)) sin(2 e^(ln^2(2)))≈ -54.29

Therefore, the required partial derivative is -54.29 (approximate value).

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The work done by the monkey is 450 lb·ft. Firstly, let's find the weight of the rope that hangs above the monkey.

Weight of the rope that hangs above the monkey = (weight per foot) × (length of the rope above the monkey).

Given that weight per foot = 0.2 lb/ft Length of the rope above the monkey = 30 ft

Weight of the rope that hangs above the monkey = (0.2 lb/ft) × (30 ft) = 6 lb

Now, let's set up the integral for the work done. We know that work done = force × distance

Since the monkey is climbing up, the force is against gravity, and it is equal to the weight of the monkey and the rope above it.

Let's assume that the monkey has climbed up a distance of x feet.

Then, the weight of the monkey and the rope above it is given by

(10 + 6) lb = 16 lb

The distance moved by the monkey is (30 - x) feet.

Work done = force × distance = (16 lb) × (30 - x) ft

Since the monkey starts at the bottom and climbs to the top, we need to integrate the work done over the distance x from 0 to 30.

Therefore, the integral for the work done is given by

∫(0 to 30) (16 lb) × (30 - x) dx

To evaluate the integral, we need to first multiply the integrand:

∫(0 to 30) (16 lb) × (30 - x) dx= 16 lb

∫(0 to 30) (30 - x) dx

Now, we can integrate:

∫(0 to 30) (30 - x) dx= [30x - (x²/2)]

evaluated from 0 to 30= [(30 × 30) - (30²/2)] - [(0 × 0) - (0²/2)]= 450 lb·ft

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The matrix A is orthogonally diagonalized as A = PDP^T where P is the orthogonal matrix and D is the diagonal matrix.

To orthogonally diagonalize a matrix A using spectral decomposition, we need to find its eigenvalues and eigenvectors.

Given matrix A:

A = [-7 24]

     [24 7]

First, let's find the eigenvalues λ of matrix A by solving the characteristic equation det(A - λI) = 0, where I is the identity matrix.

A - λI = [-7-λ 24]

                 [24 - λ]

Determinant of A - λI:

det(A - λI) = (-7-λ)(7-λ) - 24*24

                 = λ^2 - 14λ + 49 - 576

                 = λ^2 - 14λ - 527

Setting the determinant equal to zero:

λ^2 - 14λ - 527 = 0

Solving this quadratic equation gives us the eigenvalues of matrix A. Let's solve it:

Using the quadratic formula, we have:

λ = (-(-14) ± √((-14)^2 - 4*1*(-527))) / (2*1)

λ = (14 ± √(196 + 2108)) / 2

λ = (14 ± √(2304)) / 2

λ = (14 ± 48) / 2

λ = 31 or λ = -17

Now, let's find the eigenvectors corresponding to these eigenvalues.

For λ = 31:

A - 31I = [-7-31 24]

                  [24 - 31]

Row reducing (A - 31I):

[A - 31I | 0] => [1  -1]

                          [0  0]

The solution to (A - 31I)v = 0 is v = [1 1].

For λ = -17:

A + 17I = [-7+17 24]

                 [24 - 17]

Row reducing (A + 17I):

[A + 17I | 0] => [1  0]

                          [0  1]

The solution to (A + 17I)v = 0 is v = [0 1].

Now, let's normalize the eigenvectors to obtain orthonormal eigenvectors.

For λ = 31:

v1 = [1 1]

||v1|| = sqrt(1^2 + 1^2) = sqrt(2)

Normalized eigenvector v1: u1 = (1/sqrt(2))[1 1]

For λ = -17:

v2 = [0 1]

||v2|| = sqrt(0^2 + 1^2) = 1

Normalized eigenvector v2: u2 = [0 1]

Finally, let's form the orthogonal matrix P using the normalized eigenvectors as columns:

P = [u1 u2]

     [(1/sqrt(2)) 0]

     [(1/sqrt(2)) 1]

Now, let's compute the diagonal matrix D:

D = P^T * A * P

D = [u1^T u2^T] * [-7 24] * [u1 u2]

       [(-7/sqrt(2))  (24/sqrt(2))]

       [(24/sqrt(2)) (7/sqrt(2))]    

D = [-7/sqrt(2) 24/sqrt(2)] * [(-7/sqrt(2))  (24/sqrt(2))]

                                     [(24/sqrt(2)) (7/sqrt(2))] * [1/sqrt(2) 0]

[1/sqrt(2) 1]

D = [-7/sqrt(2) 24/sqrt(2)] * [(-7/sqrt(2))*(1/sqrt(2))  (24/sqrt(2))*0]

                                     [(24/sqrt(2))*(1/sqrt(2)) (7/sqrt(2))*1]

D = [-7/sqrt(2) 24/sqrt(2)] * [(-7/2) 0]

                                     [12 7/2]

D = [(-7/2)(-7/sqrt(2)) (24/2)(-7/sqrt(2))]

       [(-7/2)(12) (24/2)(7/2)]

D = [49/2 - 49/2]

     [-42 147/4]

Therefore, the matrix A is orthogonally diagonalized as:

A = PDP^T

where P is the orthogonal matrix and D is the diagonal matrix.

Please note that the above calculations involve complex computations, and it's important to ensure accuracy when performing these calculations.

Learn more about matrix here

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