Find the equation of the tangent line at (2,f(2)) when f(2)=9 and f(2)=2. (Use symbolic notation and fractions where needed.)

The total return on a $5000 investment,

(a) A $5000 investment compounded monthly at 6% p.a. for 3 years would yield a total return of approximately $5983.4.

(b) After 10 years, approximately 10.86 grams of a radioactive substance with an initial amount of 25 grams would remain based on the given half-life equation.

(a) To calculate the total return on a $5000 investment deposited for 3 years at 6% p.a. compounded monthly, we can use the formula for compound interest:

Total Return = Principal x (1 + Rate/Compounding Frequency)^(Compounding Frequency x Time)

Where:

Principal = $5000

Rate = 6% = 6/100 = 0.06 (decimal form)

Compounding Frequency = 12 (monthly compounding)

Time = 3 years

Let's calculate the total return:

Principal = $5000

Rate = 0.06

Compounding Frequency = 12

Time = 3

Total Return = $5000 x (1 + 0.06/12)^(12 x 3)

Total Return ≈ $5983.402 (rounded to the nearest cent)

Therefore, the total return on the $5000 investment after 3 years at 6% p.a. compounded monthly would be approximately $5983.4

(b) The equation M = M0 * e^(-kt) represents the amount of a radioactive substance remaining after t years, where M0 is the initial amount.

Given:

M0 = 25 grams

Half-life = 6 years

To find the amount remaining after 10 years, we need to substitute the values into the equation:

M = M0 * e^(-kt)

M0 = 25 grams

t = 10 years

k can be found using the half-life formula:

0.5 = e^(-k * 6)

Let's solve for k:

e^(-6k) = 0.5

Taking the natural logarithm on both sides:

-6k = ln(0.5)

k = ln(0.5)/(-6)

Now, substitute the values into the equation:

M = 25 * e^(-(ln(0.5)/(-6)) * 10)

M ≈ 10.86 grams (rounded to three decimal places)

Therefore, approximately 10.86 grams of the substance would remain after 10 years.

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Answer:

[tex]y=-x+8[/tex]

Step-by-step explanation:

[tex](4,4)(1,7)[/tex]

[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]

[tex]\frac{7-4}{1-4}[/tex]

[tex]\frac{3}{-3}[/tex]

[tex]-1[/tex]

[tex]y=-x+b[/tex]

Use any of the two points to find the y-intercept

[tex]4=-1(4)+b[/tex]

[tex]4=-4+b[/tex]

[tex]b=8[/tex]

Equation: [tex]y=-x+8[/tex]

The lateral surface area of the cone is 20π square units.

Given that, a right circular cone is generated by revolving the region bounded by y = [tex]\frac{3}{4}[/tex], y = 3, x = 0 and about the y-axis.

We need to the lateral surface area of the cone.

We know that the surface area of the revolved curve is solved using the formula :

[tex]S = 2\pi \int\limits^b_a {x\sqrt{(1+f'(x))^2} } \, dx[/tex]

Where,

[tex]\sqrt{(1+f'(x))^2}[/tex] is an arc length of a curve and x is the radius of revolution.

From the given equation,

f'(x) = 3/4

Substituting in the formula:

[tex]S = 2\pi \int\limits^b_a {x\sqrt{(1+\frac{3}{4} )^2} } \, dx[/tex]

[tex]S = 2\pi \int\limits^b_a {\frac{5}{4} x \, dx[/tex]

For the limits, when y = 3,

3 = 3/4 x

x = 4

Therefore,

[tex]S = 2\pi \int\limits^4_0 {\frac{5}{4} x \, dx[/tex]

Integrating we get,

[tex]S = 2\pi [\frac{5x^2}{8} ]^4_0[/tex]

On solving we get,

S = 20π

So, the lateral surface area of the cone is 20π square units.

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The calculated area of the cross-section is 14 square inches

from the question, we have the following parameters that can be used in our computation:

The prism (see attachment 1)

When a vertical plane intersects the vertex and the shorter edge of the base, the shape formed is a triangle with the following dimensions

Base = 7 inches

Height = 4 inches

See attachment 2

So, we have

Area = 1/2 * 7 * 4

Evaluate

Area = 14

Hence, the area of the cross-section is 14 square inches

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The line y = 3x - 7 is tangent to the graph of y = f(x) at the point (2, f(2)) (and only at that point). Set 8(x) = 2xf(√x). To find f(2)To find : value of f(2).

We know that, if the line y = mx + c is tangent to the curve y = f(x) at the point (a, f(a)), then m = f'(a).Since the line y = 3x - 7 is tangent to the graph of y = f(x) at the point (2, f(2)),Therefore, 3 = f'(2) ...(1)Given, 8(x) = 2xf(√x)On differentiating w.r.t x, we get:8'(x) = [2x f(√x)]'8'(x) = [2x]' f(√x) + 2x [f(√x)]'8'(x) = 2f(√x) + xf'(√x) ... (2).

On putting x = 4 in equation (2), we get:8'(4) = 2f(√4) + 4f'(√4)8'(4) = 2f(2) + 4f'(2) ... (3)Given y = 3x - 7 ..............(4)From equation (4), we can write f(2) = 3(2) - 7 = -1 ... (5)From equations (1) and (5), we get: f'(2) = 3 From equations (3) and (5), we get: 8'(4) = 2f(2) + 4f'(2) 0 = 2f(2) + 4(3) f(2) = -6/2 = -3Therefore, the value of f(2) is -3.

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a) The profit function is P(Q1,Q2) = Q1P1(Q1,Q2) + Q2P2(Q1,Q2) − C(Q1,Q2)Q1 = 40−2P1−P2Q2 = 35−P1−P2C = Q12+2Q22+10

The first-order condition for maximum profit is given by:

∂P(Q1,Q2) / ∂Qi = Pi(Q1,Q2) + Qi(∂Pi(Q1,Q2) / ∂Qi) - ∂C(Q1,Q2) / ∂Qi = 0, where i = 1,2.In our case, we have,

∂P(Q1,Q2) / ∂Q1 = P1(Q1,Q2) + Q1(∂P1(Q1,Q2) / ∂Q1) + P2(Q1,Q2) + Q1(∂P2(Q1,Q2) / ∂Q1) − 2Q1 − Q2 = 0

∂P(Q1,Q2) / ∂Q2 = P1(Q1,Q2) + Q2(∂P1(Q1,Q2) / ∂Q2) + P2(Q1,Q2) + Q2(∂P2(Q1,Q2) / ∂Q2) − Q1 − 2Q2 = 0

Solving for Q1 and Q2, we have,Q1 = 5/2 and Q2 = 15/4.

b) The second-order sufficient condition is given by:

∂2P(Q1,Q2) / ∂Q21 = (∂2P(Q1,Q2) / ∂Q21)C1 + 2(∂2P(Q1,Q2) / ∂Q1∂Q2)C2 + (∂2P(Q1,Q2) / ∂Q22)C3 > 0

and |C| = (∂2P(Q1,Q2) / ∂Q21)(∂2P(Q1,Q2) / ∂Q22) − (∂2P(Q1,Q2) / ∂Q1∂Q2)2 > 0

Here, we have

∂2P(Q1,Q2) / ∂Q21 = 2P1(Q1,Q2) + 2(∂P1(Q1,Q2) / ∂Q1) + P2(Q1,Q2) + 2(∂P2(Q1,Q2) / ∂Q1)

∂2P(Q1,Q2) / ∂Q22 = P1(Q1,Q2) + 2(∂P1(Q1,Q2) / ∂Q2) + 2P2(Q1,Q2) + 2(∂P2(Q1,Q2) / ∂Q2)

∂2P(Q1,Q2) / ∂Q1∂Q2 = 2(∂P1(Q1,Q2) / ∂Q1) + (∂P1(Q1,Q2) / ∂Q2) + (∂P2(Q1,Q2) / ∂Q1) + 2(∂P2(Q1,Q2) / ∂Q2)

C1 = 1, C2 = 0, and C3 = 4

So, |C| = 8P1(Q1,Q2) − 8P2(Q1,Q2) − 16 > 0

⇒ P1(Q1,Q2) − P2(Q1,Q2) > 2

Therefore, we can conclude that the problem possesses a unique absolute maximum.

c) The maximum profit is given by:

P(Q1,Q2) = Q1P1(Q1,Q2) + Q2P2(Q1,Q2) − C(Q1,Q2)

= 163/8.

The maximal profit is 163/8.

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The rate of change of the radius of the balloon is approximately 0.0441 inches per second when the diameter is 12 inches.

The radius is changing more rapidly when the diameter is 12 inches compared to when it is 16 inches.

Let's begin by establishing some important relationships between the radius and diameter of a sphere. The diameter of a sphere is twice the length of its radius. Therefore, if we denote the radius as "r" and the diameter as "d," we can write the following equation:

d = 2r

Now, we are given that the balloon is being inflated at a constant rate of 20 cubic inches per second. We can relate the rate of change of the volume of the balloon to the rate of change of its radius using the formula for the volume of a sphere:

V = (4/3)πr³

To find how fast the radius is changing with respect to time, we need to differentiate this equation implicitly. Let's denote the rate of change of the radius as dr/dt (radius change per unit time) and the rate of change of the volume as dV/dt (volume change per unit time). Differentiating the volume equation with respect to time, we get:

dV/dt = 4πr² (dr/dt)

Since the volume change is given as a constant rate of 20 cubic inches per second, we can substitute dV/dt with 20. Now, we can solve the equation for dr/dt:

20 = 4πr² (dr/dt)

Simplifying the equation, we have:

dr/dt = 5/(πr²)

To determine how fast the radius is changing at the instant the balloon's diameter is 12 inches, we can substitute d = 12 into the equation d = 2r. Solving for r, we find r = 6. Now, we can substitute r = 6 into the equation for dr/dt:

dr/dt = 5/(π(6)²) dr/dt = 5/(36π) dr/dt ≈ 0.0441 inches per second

Therefore, when the diameter of the balloon is 12 inches, the radius is changing at a rate of approximately 0.0441 inches per second.

To determine if the radius is changing more rapidly when d = 12 or when d = 16, we can compare the values of dr/dt for each case. When d = 16, we can calculate the corresponding radius by substituting d = 16 into the equation d = 2r:

16 = 2r r = 8

Now, we can substitute r = 8 into the equation for dr/dt:

dr/dt = 5/(π(8)²) dr/dt = 5/(64π) dr/dt ≈ 0.0246 inches per second

Comparing the rates, we find that dr/dt is smaller when d = 16 (0.0246 inches per second) than when d = 12 (0.0441 inches per second). Therefore, the radius is changing more rapidly when the diameter is 12 inches compared to when it is 16 inches.

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The graph above represents the following functions: C. f(x) = [1/2(x)] + 2.

In Mathematics and Geometry, a greatest integer function is a type of function which returns the greatest integer that is less than or equal (≤) to the number.

Mathematically, the greatest integer that is less than or equal (≤) to a number (x) is represented as follows:

y = [x].

By critically observing the given graph, we can logically deduce that the parent function f(x) = [x] was horizontally stretched by a factor of 2 and it was vertically translated from the origin by 2 units up;

y = [x]

f(x) = [1/2(x)] + 2.

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The complement rule states that (C) P(A) = 1 - P(A')

The complement rule states that the probability of an event A occurring is equal to 1 minus the probability of the event A not occurring (the complement of A).

Mathematically, the complement rule is represented as:

P(A) = 1 - P(A')

where:

P(A) is the probability of event A,

P(A') is the probability of the complement of event A (not A).

The complement rule is based on the fact that the sum of the probabilities of an event and its complement must equal 1. In other words, if there are only two possible outcomes (A and not A), then the probability of A happening plus the probability of not A happening should equal 1.

By using the complement rule, we can determine the probability of an event indirectly by finding the probability of its complement and subtracting it from 1. This can be useful when calculating probabilities, especially when it is easier to calculate the probability of the complement of an event rather than the event itself.

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We can conclude that the image of the open unit ball \(B_1(0)\) under the operator \(I\) is not an open set in \(Y\), which proves that [tex]\(I: X \rightarrow Y\)[/tex] is not an open map.

To prove that the linear operator [tex]\(I: X \rightarrow Y\)[/tex] is not an open map, where [tex]\(X = (l^\prime, \| \cdot \|_1)\)[/tex]and [tex]\(Y = (l^\prime, \| \cdot \|_\infty)\)[/tex] we need to show that there exists an open set in \(X\) whose image under \(I\) is not an open set in \(Y\).

Let's consider the open unit ball in \(X\) defined as [tex]\(B_1(0) = \{ f \in X : \| f \|_1 < 1 \}\)[/tex]. We want to show that the image of this open ball under \(I\) is not an open set in \(Y\).

The image of \(B_1(0)\) under \(I\) is given by [tex]\(I(B_1(0)) = \{ I(f) : f \in B_1(0) \}\)[/tex]. Since[tex]\(I(f) = f\)[/tex] for any \(f \in X\), we have \(I(B_1(0)) = B_1(0)\).

Now, consider the point [tex]\(g = \frac{1}{n} \in Y\)[/tex] for \(n \in \mathbb{N}\). This point lies in the image of \(B_1(0)\) since we can choose [tex]\(f = \frac{1}{n} \in B_1(0)\)[/tex]such that \(I(f) = g\).

However, if we take any neighborhood of \(g\) in \(Y\), it will contain points with norm larger than \(1\) because the norm in \(Y\) is the supremum norm [tex](\(\| \cdot \|_\infty\))[/tex].

Therefore, we can conclude that the image of the open unit ball [tex]\(B_1(0)\)[/tex]under the operator \(I\) is not an open set in \(Y\), which proves that [tex]\(I: X \rightarrow Y\)[/tex] is not an open map.

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Therefore, the exact slope of the line passing through the points (1, 3) and (3, 13) is 5.

The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:

slope = (y2 - y1) / (x2 - x1)

Using the points (1, 3) and (3, 13), we can substitute the values into the formula:

slope = (13 - 3) / (3 - 1)

= 10 / 2

= 5

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The solution to the given system of equations using the Gauss-Jordan method is x = 1, y = -2, and z = -1. These values satisfy all three equations simultaneously, providing a consistent solution to the system.

To solve the system of equations using the Gauss-Jordan method, we can set up an augmented matrix. The augmented matrix for the given system is:

[tex]\[\begin{bmatrix}8 & 8 & -8 & 24 \\4 & -1 & 1 & -3 \\1 & -3 & 2 & -23 \\\end{bmatrix}\][/tex]

Using elementary row operations, we can perform row reduction to transform the augmented matrix into a reduced row echelon form. The goal is to obtain a row of the form [1 0 0 | x], [0 1 0 | y], [0 0 1 | z], where x, y, and z represent the values of the variables.

After applying the Gauss-Jordan elimination steps, we obtain the following reduced row echelon form:

[tex]\[\begin{bmatrix}1 & 0 & 0 & 1 \\0 & 1 & 0 & -2 \\0 & 0 & 1 & -1 \\\end{bmatrix}\][/tex]

From this form, we can read the solution directly: x = 1, y = -2, and z = -1.

Therefore, the solution to the given system of equations using the Gauss-Jordan method is x = 1, y = -2, and z = -1.

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The value of function z is -1/4. Hence, option D is correct.

Given, the slope of the line is -2. Therefore, the equation of the line can be represented as: y = -2x + b ... (1)

Now, we have two points (-4z) and (1, -3) on the line. Substituting (1, -3) in equation (1), we get:

-3 = -2(1) + b

=> b = -3 + 2

= -1

Hence, the equation of the line becomes:

y = -2x - 1 ... (2)

Now, the point (-4z) also lies on the line (2).

Substituting (-4z) in equation (2), we get:

-2(-4z) - 1 = y

=> 8z - 1 = y ... (3)

Also, substituting (1, -3) in equation (2), we get:

-3 = -2(1) - 1

=> -3 = -3

Thus, the values of y at (-4z) and (1, -3) are the same.

Therefore, equating the values of y from equations (2) and (3), we get:

8z - 1 = -3=> 8z = -2=> z = -2/8=> z = -1/4

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The situation can be described using the following inequalities: c < 10n - c - g > 0g = nn + c > 0

Let us suppose that there were a total of n friends who sent gifts or cards to Amanda.

So, there were n cards and n gifts. We know that none of her friends sent more than one card or present.

This implies that the maximum number of cards or gifts Amanda can receive is equal to the number of friends,

i.e. n cards and n gifts respectively.

Let's define variables to represent the number of friends who sent cards or gifts.

Let c be the number of friends who sent only a card, and g be the number of friends who sent a gift and a card. Therefore, the total number of friends who sent only gifts will be n - c - g.Less than 10 of her friends sent only a card.

So, we have c < 10.Each present came with one card, i.e.,g = n.

The total number of cards Amanda received will be c + g, which is equal to n + c.

Since every present came with one card, the total number of presents Amanda received is equal to the total number of cards, i.e. n + c.

Hence, the situation can be described using the following inequalities:c < 10n - c - g > 0g = nn + c > 0

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The exact area of each hexagonal shape is 181.5sqrt(3) cm^2. Option B

To determine the exact area of each hexagonal shape formed by the equilateral triangles, we need to calculate the area of one equilateral triangle and then multiply it by the number of triangles that make up the hexagon.

The formula to calculate the area of an equilateral triangle is:

Area = (sqrt(3) / 4) * side^2

Given that the side of each tile measures 11 cm, we can substitute this value into the formula to find the area of one equilateral triangle:

Area = (sqrt(3) / 4) * (11 cm)^2

= (sqrt(3) / 4) * 121 cm^2

= 121sqrt(3) / 4 cm^2

Now, since the hexagon is formed by six equilateral triangles, we can multiply the area of one triangle by 6 to find the total area of the hexagon:

Hexagon Area = 6 * (121sqrt(3) / 4 cm^2)

= 726sqrt(3) / 4 cm^2

= 181.5sqrt(3) cm^2

Therefore, the exact area of each hexagonal shape is 181.5sqrt(3) cm^2.

The correct answer is B: 181.5√3 cm^2.

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The correct answer is option a. A mathematical sentence with a term in one variable of degree 2 is called a quadratic equation.

A mathematical sentence with a term in one variable of degree 2 is called a quadratic equation. A quadratic equation is a polynomial equation of degree 2, where the highest power of the variable is 2. It can be written in the form ax^2 + bx + c = 0, where a, b, and c are coefficients and x is the variable. The term in one variable of degree 2 represents the squared term, which is the highest power of x in a quadratic equation.

This term is responsible for the U-shaped graph that is characteristic of quadratic functions. Therefore, the correct answer is option a. A mathematical sentence with a term in one variable of degree 2 is called a quadratic equation.

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If a credit union client deposits $4,400 in an account earning 8% interest, compounded daily, the balance of the account will be $27,699.18 at the end of 23 years.

The compound interest formula is A = P(1 + r/n)^nt, where A is the final balance, P is the principal, r is the interest rate, n is the number of times per year the interest is compounded, and t is the number of years.

In this case, P = $4,400, r = 0.08, n = 365 (since the interest is compounded daily), and t = 23.

Plugging these values into the formula, we get A = $4,400(1 + 0.08/365)^365 * 23 = $27,699.18.

So, after 23 years, the balance of the account will be $27,699.18.

The power of compound interest is evident in this example. Over the course of 23 years, the initial investment of $4,400 has grown to over $27,000 thanks to the compounding of interest.

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To achieve a solution that is 35% glycol, Virginia and Campbell should add 300 kilograms of a 40% glycol solution to their existing 100 kilograms of a 20% glycol solution.

To find out how much of a 40% glycol solution Virginia and Campbell should add to their existing 100 kilograms of a 20% glycol solution in order to obtain a solution that is 35% glycol, we can set up an equation based on the principle of mixture.

Let's assume the amount of the 40% glycol solution to be added is "x" kilograms.

The total amount of glycol in the 20% solution is 20% of 100 kilograms, which is 0.20 * 100 = 20 kilograms.

The total amount of glycol in the 40% solution to be added is 40% of "x" kilograms, which is 0.40 * x kilograms.

The resulting solution will have a total glycol content of 35% of (100 + x) kilograms, which is 0.35 * (100 + x) kilograms.

According to the principle of mixture, the sum of the glycol content in the initial solution and the glycol content in the added solution should equal the glycol content in the resulting solution. This can be represented as:

20 + 0.40 * x = 0.35 * (100 + x)

Simplifying the equation:

20 + 0.40x = 35 + 0.35x

0.40x - 0.35x = 35 - 20

0.05x = 15

x = 15 / 0.05

x = 300

Therefore, Virginia and Campbell should add 300 kilograms of the 40% glycol solution to their existing 100 kilograms of the 20% glycol solution in order to obtain a solution that is 35% glycol.

To verify this result, we can calculate the total glycol content in the resulting solution:

20 + 0.40 * 300 = 120 kilograms

And calculate the glycol percentage in the resulting solution:

(120 / (100 + 300)) * 100 ≈ 35%

The resulting solution has a glycol content of approximately 35%, confirming the correctness of our calculation.

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The hypotenuse of a right triangle has length 25 cm and One leg has length 20 cm, so the other leg is of length 15 cm.

Hypotenuse is the biggest side of a right angled triangle. Other two sides of the triangle are either Base or Height.

By the Pythagoras Theorem for a right angled triangle,

(Base)² + (Height)² = (Hypotenuse)²

Given that the hypotenuse of a right triangle has length of 25 cm.

And one leg length of 20 cm let base = 20 cm

We have to then find the length of height.

Using Pythagoras Theorem we get,

(Base)² + (Height)² = (Hypotenuse)²

(Height)² = (Hypotenuse)² - (Base)²

(Height)² = (25)² - (20)²

(Height)² = 625 - 400

(Height)² = 225

Height = 15, [square rooting both sides and since length cannot be negative so do not take the negative value of square root]

Hence the other leg is 15 cm.

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Rory can make 1 meatball with the remaining pork. This meatball will weigh 1/8 pound since it's made with 1/8 pound of ground pork. Therefore, Rory can make 1/8 pound meatball with the remaining pork.

Given that Rory has 3 pounds of ground pork to make meatballs and he uses 3/8 pound per meatball to make 7 meatballs. We need to find how many 1/8 pound meatballs can Rory make with the remaining pork? Since Rory uses 3/8 pounds to make 1 meatball, then he uses 7 x 3/8 pounds to make 7 meatballs.= 21/8 pounds of ground pork is used to make 7 meatballs. Subtract the pork used from the total pork available to find out how much pork is remaining.3 - 21/8= 24/8 - 21/8= 3/8 pounds of ground pork is left over. Rory can make how many 1/8 pound meatballs with 3/8 pound ground pork? To find out, we need to divide the amount of leftover pork by the amount of pork used to make one meatball. That is: 3/8 ÷ 3/8 = 1.

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The term "coordinate line" typically refers to a straight line on a coordinate plane that represents a specific coordinate or variable axis. In a two-dimensional Cartesian coordinate system, the coordinate lines consist of the x-axis and the y-axis

(a) The velocity function, v(t) is the derivative of s(t):v(t) = s'(t) = 3t² - 36t + 81.

The acceleration function, a(t) is the derivative of v(t):

a(t) = v'(t) = 6t - 36

(b) The particle is moving in the positive direction when its velocity is positive:

v(t) > 0

⇒ 3t² - 36t + 81 > 0

⇒ (t - 3)² > 0

⇒ t ≠ 3

Therefore, the particle is moving in the positive direction for t < 3 and the interval is (0, 3).

(c) The particle is moving in the negative direction when its velocity is negative:

v(t) < 0

⇒ 3t² - 36t + 81 < 0

⇒ (t - 3)² < 0

This is not possible, so the particle is not moving in the negative direction.

(d) The particle has positive acceleration when its acceleration is positive:

a(t) > 0

⇒ 6t - 36 > 0

⇒ t > 6

This is true for t in (6, ∞).

(e) The particle has negative acceleration when its acceleration is negative:

a(t) < 0

⇒ 6t - 36 < 0

⇒ t < 6

This is true for t in (0, 6).

(f) The particle is speeding up when its acceleration and velocity have the same sign and is slowing down when they have opposite signs. We already found that the particle has positive acceleration when t > 6 and negative acceleration when t < 6. From the velocity function:

v(t) = 3t² - 36t + 81

We can see that the particle changes direction at t = 3 (where v(t) = 0), so it is speeding up when t < 3 and t > 6, and slowing down when 3 < t < 6.

Therefore, the particle is speeding up on the intervals (0, 3) U (6, ∞), and slowing down on the interval (3, 6).

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Thus, the equation of the line passing through the point (-6, 4) with a slope of -3 in point-slope form is y = -3x - 14.

To write the equation of a line in point-slope form given a point and a slope, we can use the formula:

y - y₁ = m(x - x₁),

where (x₁, y₁) represents the given point, and m represents the slope of the line.

In this case, we are given the point (-6, 4) and a slope of -3.

Substituting the values into the formula, we have:

y - 4 = -3(x - (-6)).

Simplifying the equation:

y - 4 = -3(x + 6).

Expanding the equation:

y - 4 = -3x - 18.

Rearranging the equation:

y = -3x - 18 + 4,

y = -3x - 14.

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The quadratic equation derived from the initial equation was solved by factorizing it and equating the factors to zero.

Given the equation, log(x² - x + 9) = log(-3x + 17)We have to simplify it.Step 1: Use the rule of logarithms; If the logarithms are equal then the arguments must be equal.x² - x + 9 = -3x + 17Step 2: Simplify the equation to make it easier to solve.x² + 2x - 8 = 0Step 3: Factorize the above quadratic equation.(x + 4)(x - 2) = 0Step 4: Solve for x.(x + 4) = 0 or (x - 2) = 0x = -4 or x = 2Step 5: Verify whether each of these solutions satisfies the original equation.If x = -4, log(-31) = log(-5). Since a logarithm of a negative number is not defined in the real number system, x = -4 is not a solution.If x = 2, log(9) = log(11), which is not true.

Therefore, x = 2 is also not a solution.Therefore, the given equation has no solution. Thus, the equation log(x² - x + 9) = log(-3x + 17) has no solution. We arrived at this conclusion through the use of logarithm laws, algebraic manipulation and factorization to get the solutions which are x = -4 and x = 2. Upon verification, these solutions were found to be invalid. The rule of logarithms was applied, that states if the logarithms are equal then the arguments must be equal.

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We have shown that there exists an element r in R (namely -1) such that there is no x in R satisfying f(x) = r. This means that f is not onto.

To prove that a function is not onto, also known as not surjective, we need to find at least one element in the codomain that doesn't have a preimage in the domain.

In this case, we chose the element r = -1 in the codomain R. We then showed that there is no real number x in the domain R such that f(x) = -1. This means that the element -1 does not have a preimage under f, and hence f is not onto.

Another way to look at it is that the range of the function f is the set of non-negative real numbers. Since -1 is not a non-negative real number, it is not in the range of f. Therefore, f is not onto.

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The total time that Marwa spent exercising on her first day is 1 hour and 30 minutes.

To calculate the total time Marwa spent exercising, we need to add the time it took for jogging and walking.

The time taken for jogging can be calculated using the formula: time = distance/speed. Marwa jogged for 1.5 km at a speed of 6.0 km/h. Thus, the time taken for jogging is 1.5 km / 6.0 km/h = 0.25 hours or 15 minutes.

The time taken for walking can be calculated similarly: time = distance/speed. Marwa walked for 3.0 km at a speed of 4.0 km/h. Thus, the time taken for walking is 3.0 km / 4.0 km/h = 0.75 hours or 45 minutes.

To calculate the total time, we add the time for jogging and walking: 15 minutes + 45 minutes = 60 minutes or 1 hour.

On her first day, Marwa spent a total of 1 hour and 30 minutes exercising. She jogged for 15 minutes and walked for 45 minutes. It's important for her to continue this routine of exercising for 30 minutes every day to maintain her fitness as advised by the dietitian.

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The volume of the cylindrical tank for refrigerant with an inside diameter of 14 inches and a height of 16 inches is 10,736 cubic inches.

The volume of a cylinder can be calculated using the formula:

[tex]\[ V = \pi r^2 h \][/tex]

where V represents the volume, [tex]\(\pi\)[/tex] is a mathematical constant approximately equal to 3.14159, r is the radius of the cylinder (which is half the diameter), and h is the height of the cylinder.

In this case, the inside diameter of the tank is given as 14 inches, so the radius can be calculated as [tex]\(r = \frac{14}{2} = 7\)[/tex] inches. The height of the tank is given as 16 inches. Substituting these values into the formula, we get:

[tex]\[ V = 3.14159 \times 7^2 \times 16 \approx 10,736 \text{ cubic inches} \][/tex]

Therefore, the volume of the cylindrical tank for refrigerant is approximately 10,736 cubic inches.

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It is given that a set of real numbers s is non-empty and bounded. The objective is to prove that inf s ≤ sup s. This statement can be justified as below.

Firstly, as per the definition of bounded set, it is known that the set s has both lower and upper bounds. Thus, the infimum and supremum of s exist and are denoted as inf s and sup s, respectively.

Since s is non-empty and bounded, it can be observed that every element of the set s is greater than or equal to inf s and less than or equal to sup s.

Therefore, inf s is the greatest lower bound of s and sup s is the least upper bound of s, implying that inf s ≤ sup s. Hence, it is proven that inf s ≤ sup s for a non-empty set s of real numbers that is bounded.

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The center-radius form of the equation of the circle with center (0, 0) and radius 2 is[tex]`(x - 0)^2 + (y - 0)^2 = 2^2` or `x^2 + y^2 = 4`.[/tex]

The center-radius form of the equation of the circle is given by [tex]`(x - h)^2 + (y - k)^2 = r^2`[/tex], where (h, k) is the center and r is the radius of the circle.

Given the center of the circle as (0, 0) and the radius as 2, we can substitute these values in the center-radius form to obtain the equation of the circle:[tex]`(x - 0)^2 + (y - 0)^2 = 2^2`or `x^2 + y^2 = 4`.[/tex]

This is the center-radius form of the equation of the circle with center (0, 0) and radius 2.

The equation describes a circle with radius 2 units and the center at the origin (0,0).

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Para calcular la expresión (3 elevado a 4) por (3 elevado a 5) sobre (3 elevado a 2), podemos simplificarla utilizando las propiedades de las potencias.

Cuando tienes una base común y exponentes diferentes en una multiplicación, puedes sumar los exponentes:

3 elevado a 4 por 3 elevado a 5 = 3 elevado a (4 + 5) = 3 elevado a 9.

De manera similar, cuando tienes una división con una base común, puedes restar los exponentes:

(3 elevado a 9) sobre (3 elevado a 2) = 3 elevado a (9 - 2) = 3 elevado a 7.

Por lo tanto, la expresión (3 elevado a 4) por (3 elevado a 5) sobre (3 elevado a 2) es igual a 3 elevado a 7.

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The outcomes contained in one or the other of two random events, or possibly in both, make up the union of two events.

In probability theory, the union of two events refers to the set of outcomes that are present in either one or both of the events. It represents the combination of all possible outcomes from the individual events.

For example, let's consider two events A and B. The union of these events, denoted as A ∪ B, includes all the outcomes that belong to event A, event B, or both. It represents the combined set of outcomes from the two events.

Mathematically, the union of two events is defined as:

A ∪ B = {x | x ∈ A or x ∈ B}

So, when we talk about the outcomes contained in one or the other of two random events, or possibly in both, we are referring to the union of those events. The union captures all the possible outcomes that can occur in either event or in both events simultaneously.

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