Factorise: (x-y/xy)^3

The rate of change of the surface area of a beach ball is -20/pie cm^2 per second when the volume of the ball is 256/3pie cubic centimeters, assuming the ball retains its spherical shape as it deflates at a constant rate of 10 cubic centimeters per second.

The formula for the volume of a sphere is V = 4/3 * pie * r^3, and the formula for the surface area of a sphere is A = 4 * pie * r^2. If the volume of the beach ball is 256/3pie cubic centimeters, we can solve for the radius by setting the volume equation equal to this value and solving for r:

256/3pie = 4/3 * pie * r^3

r^3 = (256/3) / (4/3 * pie)

r^3 = 16/3pie

r = (16/3pie)^(1/3)

Since the beach ball is deflating at a constant rate of 10 cubic centimeters per second, the rate of change of the volume can be expressed as dV/dt = -10. Taking the derivative of the surface area formula with respect to time gives us:

dA/dt = 8 * pie * r * dr/dt

We can substitute in the values we calculated for r and dV/dt to find:

dA/dt = 8 * pie * (16/3pie)^(1/3) * (-10)

dA/dt = -20/pie cm^2 per second

Therefore, the rate of change of the surface area of the beach ball is -20/pie cm^2 per second when the volume of the ball is 256/3pie cubic centimeters.

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The distance from Y to the subspace spanned by V is approximately 4.05 units.

To compute the distance from a point to a subspace, we can use the formula:

d = ||Y - proj(V)Y||

where Y is the given point and V is a set of vectors spanning the subspace.

In this case, Y = [4, -6, 0, 0] and V = {[2, 2, -2, -1], [4, -1, 1, 4]}.

Step 1: Find the projection of Y onto the subspace spanned by V.

To find the projection, we can use the formula:

proj(V)Y = ((Y⋅V1)/(V1⋅V1)) * V1 + ((Y⋅V2)/(V2⋅V2)) * V2

where ⋅ denotes the dot product.

Calculating the dot products:

Y⋅V1 = [4, -6, 0, 0] ⋅ [2, 2, -2, -1] = 8 + (-12) + 0 + 0 = -4

V1⋅V1 = [2, 2, -2, -1] ⋅ [2, 2, -2, -1] = 4 + 4 + 4 + 1 = 13

Y⋅V2 = [4, -6, 0, 0] ⋅ [4, -1, 1, 4] = 16 + 6 + 0 + 0 = 22

V2⋅V2 = [4, -1, 1, 4] ⋅ [4, -1, 1, 4] = 16 + 1 + 1 + 16 = 34

Calculating the projection:

proj(V)Y = ((-4/13) * [2, 2, -2, -1]) + ((22/34) * [4, -1, 1, 4])

= [-8/13, -8/13, 8/13, 4/13] + [44/34, -11/34, 11/34, 44/34]

= [108/34, -108/34, 108/34, 156/34]

= [54/17, -54/17, 54/17, 78/17]

Step 2: Calculate the distance.

To calculate the distance, we use the formula:

d = ||Y - proj(V)Y|| = ||[4, -6, 0, 0] - [54/17, -54/17, 54/17, 78/17]||

Calculating the vector difference:

Y - proj(V)Y = [4, -6, 0, 0] - [54/17, -54/17, 54/17, 78/17]

= [68/17, -102/17, -54/17, -78/17]

Calculating the norm (magnitude):

||Y - proj(V)Y|| = √((68/17)^2 + (-102/17)^2 + (-54/17)^2 + (-78/17)^2)

= √(4624/289 + 10404/289 + 2916/289 + 6084/289)

= √(23428/289)

≈ 4.05

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Answer:

  -3

Step-by-step explanation:

You want the value of p(2), given the piecewise definition of function p.

The function definition divides its domain into three (3) parts. The first step in evaluating the function for a particular value of x is to find the applicable domain.

Here, you want the value of p(x) for x = 2.

The third domain expression (2 ≤ x < 5) includes the value x = 2, so the third function definition applies:

  p(x) = x -5

  p(2) = 2 -5 = -3

The value of p(2) is -3.

__

Additional comment

A graph of the function is attached. The point of interest is circled in green. (2, p(2)) = (2, -3).

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Applying the properties of a parallelogram, we have:

A. x = 3; AB = 17 units; CD = 17 units

B. y = 18; m<A = 86 degrees; m<D = 94 degrees.

Recall that. the opposite sides of a parallelogram are both parallel to each other and also congruent to each other, while its adjacent angles are supplementary.

Therefore, we have:

A. 5x + 2 = 8x - 7

Combine like terms:

5x - 8x = -2 - 7

-3x = -9

x = 3

AB = 5x + 2 = 5(3) + 2 = 17 units

CD = 17 units

B. 2y + 50 + 3y + 40 = 180 [supplementary angles]

Combine like terms:

5y = + 90 = 180

5y = 180 - 90

5y = 90

y = 18

m<A = 2y + 50 = 2(18) + 50 = 86 degrees

m<D = 3y + 40 = 3(18) + 40 = 94 degrees.

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Step-by-step explanation:

To multiply the negative of a fraction by the inverse of another fraction, follow these steps:

1. Convert the first fraction to its negative equivalent.

- The negative of 2/3 is -2/3.

2. Find the inverse (or reciprocal) of the second fraction.

- The inverse of 9/7 is 7/9.

3. Multiply the two fractions.

- Multiply the numerators: (-2) * (7) = -14

- Multiply the denominators: (3) * (9) = 27

The result is -14/27.

Answer:

Prolly bout a hunnit miles per hour

Step-by-step explanation:

The complete R code to generate a random number x with probability 0.2 is;

R is a programming language for statistical computation and graphics that is backed by the R Core Team and the R Foundation for Statistical Computing.

In R programming, the Poisson distribution is used to describe the probability that a certain number of events will take place in a given amount of time or space if they occur at a known constant mean rate.

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The equation 6csc(0)-7=5 can be solved as follows:

Add 7 to both sides of the equation to get:

6csc(0) = 12

Divide both sides of the equation by 6 to get:

csc(0) = 2

Take the reciprocal of both sides of the equation to get:

sin(0) = 1/2

Therefore, the solution to the equation is 0 = 30 degrees (or pi/6 radians).

To make the frame for her favorite poster, Monica needs to calculate the total amount of wood required. Since the poster is 3 feet by 2 feet, the frame needs to be slightly larger to fit around the edges. Assuming that Monica wants to add 1 foot of wood around each edge of the poster, she needs to add 2 feet to both the length and the width of the poster. Therefore, the total dimensions of the poster and the frame will be 5 feet by 4 feet.

  To calculate the total amount of wood required, Monica needs to find the perimeter of the frame, which is the sum of the lengths of all four sides. The length of each of the two longer sides is 5 feet, and the length of each of the two shorter sides is 4 feet. Therefore, the total amount of wood required to make the frame is 5+5+4+4 = 18 feet. Hence, Monica needs 18 feet of wood to make the frame for her favorite poster.

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To make the frame for her favorite poster, Monica needs to calculate the total amount of wood required. Since the poster is 3 feet by 2 feet, the frame needs to be slightly larger to fit around the edges. Assuming that Monica wants to add 1 foot of wood around each edge of the poster, she needs to add 2 feet to both the length and the width of the poster. Therefore, the total dimensions of the poster and the frame will be 5 feet by 4 feet.

      To calculate the total amount of wood required, Monica needs to find the perimeter of the frame, which is the sum of the lengths of all four sides. The length of each of the two longer sides is 5 feet, and the length of each of the two shorter sides is 4 feet. Therefore, the total amount of wood required to make the frame is 5+5+4+4 = 18 feet. Hence, Monica needs 18 feet of wood to make the frame for her favorite poster.

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The Amplitude of a function is the maximum distance a point on the graph is from the function's midline, usually measured in the vertical direction.The amplitude of the function graphed will be 2.

A graph,

The highest value = 3,

Lowest value = -1

So,

From the definition mentioned above,

Amplitude

We get,

Amplitude = 2

Hence, we can say that the amplitude of the function graphed will be 2.

For a sine or cosine function, the amplitude is the absolute value of the coefficient in front of the sine/cosine term.
The period of a function is the length of one complete cycle of the graph. In other words, it's the horizontal distance required for the graph to repeat its pattern. For a sine or cosine function, the period can be calculated as (2π) / |B|, where B is the coefficient of the angle in the sine/cosine term.

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Answer:

y=4x-6

Step-by-step explanation:

y=4x+5

-30=-24-6

y=4x-6

The upgraded model costs $22.42 more than the basic model.

Given that a basic stereo system costs $189.67.

An upgraded model costs $212.09.

We have to find the how much more does the upgraded model cost

To find this we have to find the difference of costs between two models.

The upgraded model costs $212.09 and the basic model costs $189.67

So the difference in cost is:

Two hundred twelve minus one hundred eighty nine point six seven

$212.09 - $189.67

We get twenty two point four two

$22.42

Therefore, the upgraded model costs $22.42 more than the basic model.

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The coefficient of determination is 0.49. This means that approximately 49% of the variation in the dependent variable can be explained by the variation in the independent variable.

What is coefficient of determination?

The coefficient of determination, also known as R-squared (R²), quantifies the extent to which the variation in the dependent variable can be accounted for or attributed to the independent variable(s) in a regression model.

On the other hand, the coefficient of correlation, represented as r, measures the strength and direction of the linear relationship between two variables. It ranges from -1 to 1, where a value of 1 indicates a perfect positive linear relationship, -1 indicates a perfect negative linear relationship, and 0 suggests no linear relationship.

In order to assess the percentage of variability in the dependent variable (y) that can be explained by the independent variable (x), we calculate the coefficient of determination (r²). This value indicates the proportion of the variance in the dependent variable that can be understood based on the variation in the independent variable.

To calculate [tex]r^2[/tex], we square the coefficient of correlation (r):

[tex]r^2 = (0.7)^2 = 0.49[/tex]

So, the coefficient of determination is 0.49. This means that approximately 49% of the variation in the dependent variable can be explained by the variation in the independent variable.

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The surface area of the figure is equal to 641.1 square inches.

The figure comprises of a smaller and a bigger cone, the sum of their surface area is the surface area of the figure.

Surface area of cone = πr[r + √(h² + r²)]

surface area of smaller cone = 22/7 × 6in[6in + √(8² + 6²)]

surface area of smaller cone = 301.7 square inches

Height of the bigger cone = √(12² - 6²) = 10.4in

surface area of bigger cone = 22/7 × 6in[6in + √(10.4² + 6²)]

surface area of bigger cone = 339.4 square inches

surface area of the shape = 301.7 + 339.4

surface area of the shape = 641.1 square inches.

Therefore, the surface area of the figure is equal to 641.1 square inches.

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The value of volume of the cone​ is,

⇒ V = 53.9π cm³

We have to given that;

The slant side of a cone is,

⇒ 15 cm

And, the radius of the base is

⇒ 7 cm.

Since, We know that;

Volume of cone is,

⇒ V = 1/3 πr² (√l² - r²)

Here, r = 7 cm

l = 15 cm

Hence, We get;

⇒ V = 1/3 πr² (√l² - r²)

⇒ V = 1/3 π (7)² (√15² - 7²)

⇒ V = 1/3π × 49 (√176)

⇒ V = 1/3π × 49 × 13.3

⇒ V = 53.9π cm³

Thus, The volume of the cone​ is,

⇒ V = 53.9π cm³

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Answer:

5x^4y(6x^3y^3 - 2x + 15y^x)

Step-by-step explanation:

the factor the expression 30x^7 y^4 - 10x^5 y +75x^4 y^3 we can first factor out the greatest common factor, which is 5x^4y:

30x^7 y^4 - 10x^5 y +75x^4 y^3

= 5x^4y(6x^3y^3 - 2x + 15y^2)

the expression 6x^3y^3 - 2x + 15y^2 cannot be factored further, so the final factored form is:

Therefore, the integral evaluates to 2tan^2(x) + c.

Let u = tan(x), then du/dx = sec^2(x) dx

Using du/dx, we can write the integral as:

∫4 tan(x) sec^3(x) dx = ∫4u du = 2u^2 + c = 2tan^2(x) + c

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A normal distribution has a mean of 120 and a standard deviation of 10 then P(x > 110) is 0.84

Given that mean is 120 and a standard deviation of 10

The z-score for x=110 can be calculated as:

z = (x - μ) / σ = (110 - 120) / 10 = -1

Using a standard normal distribution table

we can find that the probability of a z-score being greater than -1 is 0.8413.

Therefore, P(x > 110) = 1 - P(x ≤ 110)

= 1 - P(z ≤ -1)

= 1 - 0.1587

= 0.8413.

Hence,  a normal distribution has a mean of 120 and a standard deviation of 10 then P(x > 110) is 0.84

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Answer:1250

Step-by-step explanation:

The required length RU is 16 units for the given triangle.

As we know that Pythagoras's theorem states that in a right-angled triangle, the square of one side is equal to the sum of the squares of the other two sides.

Using Pythagoras's theorem in ΔSUT,

ST² = SU² + UT²

37² = SU² + 35²

SU² = 37² - 35²

SU² = 1369 - 1225

SU² = 144

SU = 12 units

Now, using Pythagoras's theorem in ΔRSU,

SR² = SU² + RU²

20² = 12² + RU²

400 = 144 + RU²

RU² = 400 - 144

RU² = 256

RU = 16 units

Therefore, the required length RU is 16 units.

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Rolling an even number (2, 4, or 6) has a 50% probability because there are 3 even numbers out of the 6 possible outcomes. Similarly, rolling an odd number (1, 3, or 5) also has a 50% probability.

To determine an outcome with a probability of 50% when Ushi throws a fair 6-sided dice.
A fair 6-sided dice has 6 equally likely outcomes: 1, 2, 3, 4, 5, and 6. Since there are 6 possible outcomes, no single outcome (such as rolling a 1 or a 6) can have a 50% probability. However, we can group outcomes to achieve a 50% probability.
For instance, rolling an even number (2, 4, or 6) has a 50% probability because there are 3 even numbers out of the 6 possible outcomes. Similarly, rolling an odd number (1, 3, or 5) also has a 50% probability.

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At pH 12, the predominant form of methionine (pKa = 2.28 and 9.21) would be the deprotonated form, which is represented as -CH2-S-CH2-CH(NH3+)COO-.

At this high pH, both the carboxylic acid (-COOH) and the amino group (-NH3+) on the side chain of methionine will be deprotonated, resulting in a negatively charged side chain. This form of methionine has a net charge of -1. At this pH, the amino acid will exist predominantly in its anionic form since the pH is higher than both of its pKa values. Methionine's isoelectric point (pI) is 5.74, which is the pH at which its net charge is zero.

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The discount given on the aquarium is 75%.

Given that an aquarium has an original price of $52 which is now being sold at $13, we need to find the discount given on it,

So, the discount is calculated by =

original price - selling price / original price × 100%

So,

Discount = 52 - 13 / 52 × 100%

= 39 / 52 × 100%

= 0.75 × 100%

= 75%

Hence the discount given on the aquarium is 75%.

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Therefore, the largest possible volume of the box is approximately 6,821.05 cubic centimeters.

Let the side length of the square base be x. Then, the height of the box will also be x, since it has an open top. The surface area of the box will be:

Area of the base: x^2

Area of the sides: 4xh = 4x^2, since h = x

Total surface area: x^2 + 4x^2 = 5x^2

We know that the available material is 1,800 square centimeters, so:

5x^2 = 1800

Solving for x, we get:

x^2 = 360

x ≈ 18.97

The largest possible volume of the box will be when x = 18.97, which gives:

V = x^2h = 18.97^2 * 18.97 ≈ 6,821.05 cubic centimeters

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The possible largest number of times that the student can eat at the restaurant without getting the same fortune four times is 10.

To solve this problem, we need to use the pigeonhole principle. This principle states that if n items are placed into m containers where n > m, then at least one container must contain more than one item.

1. The company makes 213 different fortunes.
2. The student wants to avoid getting the same fortune four times.
3. To calculate the largest possible number of times the student can eat at the restaurant without getting the same fortune four times, we need to multiply the number of different fortunes by 3 (since they can get each fortune up to 3 times).
4. So, 213 different fortunes multiplied by 3 visits per fortune equals 639 total visits (213 * 3 = 639).

Therefore, the student can eat at the restaurant 639 times without getting the same fortune four times.

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Answer:

[tex] \sqrt{ {5}^{2} + {1}^{2} } = \sqrt{25 + 1} = \sqrt{26} = 5.1[/tex]

The distance between diagonally opposite corners of this wall is about 5.1 meters.

To decrease the width of a confidence interval for a proportion when taking a random sample of 600 adults from a population of over one million, researchers can either increase the sample size or decrease the desired level of confidence.

These actions reduce the margin of error, resulting in a narrower confidence interval.

When constructing a confidence interval for a proportion, the width of the interval is determined by the sample size, the level of confidence, and the estimated proportion. To decrease the width of the confidence interval, researchers can focus on adjusting the sample size or the level of confidence.

Firstly, increasing the sample size will lead to a narrower confidence interval. A larger sample size provides more data points, which improves the precision of the estimated proportion. As the sample size increases, the margin of error decreases, resulting in a narrower interval. However, it is important to note that there are practical limitations to the sample size, such as time, cost, and feasibility.

A higher level of confidence, such as 95% or 99%, results in a wider interval because it requires a higher level of certainty. By decreasing the level of confidence, such as using a 90% confidence level instead of 95%, the margin of error decreases, resulting in a narrower confidence interval.

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The translation of angles and parallel lines is discussed in the following query. Below is a detailed response.

When you translate an item in geometry, you are essentially turning it in a different direction. Consequently, an angle that has been translated is one that has been turned in a new direction.

The photograph from the first selection in section A makes it obvious that the angle remained the same. From the positive to the negative side of the x-axis, it moves seven units.

The angles also don't alter from Part B. It should be noted that a translation or rotation of an angle has no effect on the angles.

What we got is the reflection of the angles from Part C. Due to the fact that they are reflections of one another, this indicates that both angles are equal.

The two parallel sets of lines from Part D will continue to be parallel as long as they are both translated at the same time.

The parallel line that is at an angle to the Y-axis will not meet the parallel line that is at an angle to the x-axis if they are extended infinitely, despite the fact that in each set of parallel lines, the two sets of lines remain equally distant from one another.

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Your question is incomplete but most probably your full question was,  

The volume of the ball (sphere) that Jerry needs to determine can be computed with a diameter of 4 inches, rounded to the nearest hundredth, as 33.52 inches³.

The volume refers to the capacity of an object, cylinder, sphere, or box.

Diameter of the ball = 4 inches

Radius = d/2

= 2 inches

The volume can be determined using the radius, which is half of the diameter, as follows:

Volume = ⁴/₃πr³

Volume = ⁴/₃ x ²²/₇ x 2³

Volume = ⁴/₃ x ²²/₇ x 8

Volume = 33.52 inches³

Thus, we can conclude that Jerry's ball has a volume of 33.52 inches³.

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