What is the value of m in the following equation? m 3= 1 125 m=

The average value of the function f(x) = x^2 on the interval [-2, 2] is f = 2/3.

To find the average value of a function on a given interval, we need to calculate the definite integral of the function over that interval and divide it by the length of the interval. In this case, the function f(x) = x^2 is a simple quadratic function. We can integrate it using the power rule, which states that the integral of x^n is (1/(n+1)) * x^(n+1).

Integrating f(x) = x^2, we get F(x) = (1/3) * x^3. To find the definite integral over the interval [-2, 2], we evaluate F(x) at the endpoints and subtract the values: F(2) - F(-2).

F(2) = (1/3) * (2)^3 = 8/3

F(-2) = (1/3) * (-2)^3 = -8/3

Therefore, the definite integral of f(x) on the interval [-2, 2] is F(2) - F(-2) = (8/3) - (-8/3) = 16/3. To calculate the average value, we divide the definite integral by the length of the interval, which is 2 - (-2) = 4. So, the average value of the function f(x) = x^2 on the interval [-2, 2] is f = (16/3) / 4 = 2/3.

Graphically, the average value corresponds to the height of the horizontal line that cuts the area under the curve in half. In this case, the average value of 2/3 can be represented by a horizontal line at y = 2/3, intersecting the curve of f(x) = x^2 at some point within the interval [-2, 2].

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a) AB = (6, 8), ||AB|| = 10 and c) a unit vector in the same direction as AB is (0.6, 0.8).

To find the values requested, we can follow these steps:

a) AB: The vector AB is the difference between the coordinates of point B and point A.

AB = (x2 - x1, y2 - y1)

= (8 - 2, 5 - (-3))

= (6, 8)

Therefore, AB = (6, 8).

b) ||AB||: To find the length or magnitude of the vector AB, we can use the formula:

||AB|| = √(x² + y²)

||AB|| = √(6² + 8²)

= √(36 + 64)

= √100

= 10

Therefore, ||AB|| = 10.

c) Unit vector in the same direction as AB:

To find a unit vector in the same direction as AB, we can divide the vector AB by its magnitude.

Unit vector AB = AB / ||AB||

Unit vector AB = (6, 8) / 10

= (0.6, 0.8)

Therefore, a unit vector in the same direction as AB is (0.6, 0.8).

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Answer:

C. (-2, -5) and (3,0)

Step-by-step explanation:

the solutions to the system of equations is the points where both graphs meet and cross over each other

Answer:

I don't remember this math all too well, however, I think it's asking where both lines intersect with each other. If that is the question, the answer is C.

Step-by-step explanation:

The lines intersect with each other first at (-2,-5) and then at (3,0).

Hope this helps.

The equation of the tangent line to the graph of x' + x + y = 1 at the point (0, 1) is y = -x + 1. To determine the equation of the tangent line, we need to find the slope of the line and a point on the line.

The equation x' + x + y = 1 represents a curve. To determine the slope of the tangent line, we differentiate the equation with respect to x, treating y as a function of x. Differentiating x' + x + y = 1 yields 1 + 1 + dy/dx = 0, which simplifies to dy/dx = -2. Hence, tangent line has a slope of -2.

To determine a point on the tangent line, we consider that the curve passes through the point (0, 1). Thus, this point must also lie on the tangent line. Consequently, the equation of the tangent line can be expressed as y = mx + b, where m represents the slope (-2) and b denotes the y-intercept. Substituting the values, we obtain 1 = -2(0) + b, which leads to b = 1. Thus, y = -x + 1 is equation of the tangent line.

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The joint probability density function of Y1 and Y2, where Y1 = X1 - X2 and Y2 = e^X2, can be determined as follows:

To find the joint probability density function of Y1 and Y2, we need to determine the transformation between the variables X1, X2 and Y1, Y2.

First, let's find the relationship between Y1 and X1, X2. We have Y1 = X1 - X2.

Next, let's find the relationship between Y2 and X1, X2. We have Y2 = e^X2.

To determine the joint probability density function of Y1 and Y2, we can use the method of transformation of variables. We need to find the joint probability density function of X1 and X2, and then apply the appropriate transformation to obtain the joint probability density function of Y1 and Y2.

Since X1 and X2 are independent exponential random variables with parameter λ, their joint probability density function is given by f(x1, x2) = λ^2 * e^(-λ(x1+x2)) for x1 > 0 and x2 > 0, and 0 otherwise.

To find the joint probability density function of Y1 and Y2, we need to determine the corresponding region in the Y1-Y2 space and the Jacobian of the transformation.

The region in the Y1-Y2 space is determined by the inequalities Y1 > 0 and Y2 > 0.

The transformation from X1, X2 to Y1, Y2 can be represented as Y1 = X1 - X2 and Y2 = e^X2.

To find the joint probability density function of Y1 and Y2, we need to find the joint probability density function of X1 and X2 and then apply the appropriate transformation.

Applying the transformation, we have X1 = Y1 + X2 and X2 = ln(Y2).

To find the Jacobian of the transformation, we calculate the determinant of the Jacobian matrix:

|d(X1, X2)/d(Y1, Y2)| = |1 1|

                        |0 1| = 1.

The joint probability density function of Y1 and Y2 is given by f(y1, y2) = f(x1, x2) / |d(X1, X2)/d(Y1, Y2)| = λ^2 * e^(-λ(y1+ln(y2))) / 1 = λ^2 * y2 * e^(-λy1-λln(y2)).

Therefore, the joint probability density function of Y1 and Y2 is f(y1, y2) = λ^2 * y2 * e^(-λy1-λln(y2)) for y1 > 0 and y2 > 0, and 0 otherwise.

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Matthew's estimate will have the smaller margin of error because the sample size is larger and the level of confidence is higher.

The margin of error in an estimate is influenced by two factors: sample size and level of confidence. A larger sample size tends to reduce the margin of error because it provides a more representative and reliable sample of the population. Additionally, a higher level of confidence, typically expressed as a percentage (e.g., 95% confidence level), means that there is a greater certainty in the estimate falling within the specified range. Therefore, when comparing Matthew and Katrina's estimates, where Matthew has a larger sample size and a higher level of confidence, it is reasonable to conclude that Matthew's estimate will have the smaller margin of error.

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The derivative of the function y = 6x^2 - 7x + 8x + 5 is y' = 12x + 1.

To find the derivative of the function y = 6x^2 - 7x + 8x + 5, we differentiate each term of the function separately using the power rule of differentiation.

The power rule states that if we have a term of the form ax^n, the derivative with respect to x is given by nx^(n-1).

Differentiating each term:

d/dx (6x^2) = 12x^(2-1) = 12x

d/dx (-7x) = -7

d/dx (8x) = 8

d/dx (5) = 0 (the derivative of a constant is zero)

Now, combining the derivatives, we get:

y' = 12x - 7 + 8

Simplifying, we have:

y' = 12x + 1

Therefore, the derivative of the function y = 6x^2 - 7x + 8x + 5 is y' = 12x + 1.

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To locate the absolute extreme points for the given function over the given interval, we need to take the derivative of the function and set it equal to zero.

Then we can find the critical points and determine whether they correspond to maximum or minimum values.Let's differentiate g(x) = -2 -2x2 + 14.6x – 16.5:$$g'(x)=-4x+14.6$$Now, let's find the critical points by setting g'(x) equal to zero:$$g'(x)=-4x+14.6=0$$$$-4x=-14.6$$$$x=\frac{14.6}{4}=3.65$$So the only critical point over the given interval is x = 3.65. We can now determine whether this critical point corresponds to a maximum or minimum value by examining the sign of the second derivative. Let's take the second derivative of the function:$$g''(x)=-4$$Since g''(x) is negative for all x, we know that the critical point x = 3.65 corresponds to a maximum value. Therefore, the absolute extreme point for the given function over the given interval is (3.65, g(3.65)). Let's evaluate g(3.65) to find the y-coordinate of the absolute extreme point:$$g(3.65)=-2 -2(3.65)^2 + 14.6(3.65) – 16.5=6.452$$Therefore, the absolute extreme point for the given function over the given interval is approximately (3.65, 6.452), rounded to three decimal places.

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The amount (net earnings) that Amy will have after giving her parents $200 a month for room and board is $565.12.

The difference (net earnings) between Amy's monthly earnings and the amount she spends on her parents shows the amount that Amy will have.

The difference is the result of a subtraction operation, which is one of the four basic mathematical operations.

The hourly rate that Amy earns = $7.97

The number of hours per week that Amy works = 24 hours

4 weeks = 1 month

The monthly earnings = $765.12 ($7.97 x 24 x 4)

Amy's monthly expenses on parents' rooom and board = $200

The net earnings (ignoring taxes and other lawful deductions) = $565.12 ($765.12 - $200)

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How much is left for her at the end of the month, ignoring taxes and other lawful deductions?

We need to find the partial derivatives of f(x, y, z) with respect to x, y, and z.To find the partial derivative of f(x, y, z) with respect to x (fx), we differentiate the function with respect to x while treating y and z as constants.

fx(x, y, z) = d/dx(3x^2 + 6y^2 + x^2)

Differentiating each term separately:

fx(x, y, z) = d/dx(3x^2) + d/dx(6y^2) + d/dx(x^2)

Applying the power rule of differentiation, where
d/dx(x^n) = nx^(n-1):

fx(x, y, z) = 6x + 0 + 2x

Simplifying:

fx(x, y, z) = 8x

Similarly, to find the partial derivatives fy(x, y, z) and fz(x, y, z), we differentiate the function with respect to y and z, respectively, while treating the other variables as constants.

fy(x, y, z) = d/dy(3x^2 + 6y^2 + x^2)

fy(x, y, z) = 0 + 12y + 0

fy(x, y, z) = 12y

fz(x, y, z) = d/dz(3x^2 + 6y^2 + x^2)

fz(x, y, z) = 0 + 0 + 0

fz(x, y, z) = 0

Therefore, the partial derivatives are:

fx(x, y, z) = 8x

fy(x, y, z) = 12y

fz(x, y, z) = 0

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We will solve the ordinary differential equation (ODE) y' + 0.5y = 0 using the Euler method with the initial condition y(0) = 1.

The Euler method is a numerical technique used to approximate the solution of an ODE. It involves discretizing the interval of interest and using iterative steps to approximate the solution at each point.

For the given ODE y' + 0.5y = 0, we can rewrite it as y' = -0.5y. Applying the Euler method, we divide the interval into smaller steps, let's say h, and approximate the solution at each step.

Let's choose a step size of h = 0.1 for this example. Starting with the initial condition y(0) = 1, we can use the Euler method to approximate the solution at the next step as follows:

y(0.1) ≈ y(0) + h * y'(0)

≈ 1 + 0.1 * (-0.5 * 1)

≈ 0.95

Similarly, we can continue this process for subsequent steps. For example:

y(0.2) ≈ y(0.1) + h * y'(0.1)

≈ 0.95 + 0.1 * (-0.5 * 0.95)

≈ 0.9025

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The expanded forms are:(a) log7 y(b) 2 ln x + ln (2 + x)(c) ln 81 + 8 ln x + ln y.

(a) expand log7 y:using the logarithmic property logb(xⁿ) = n logb(x), we have:log7 y = log7 (y¹) = 1 log7 y = log7 y.

(b) expand ln (x²(2 + x)):using the logarithmic property ln (ab) = ln a + ln b, we have:ln (x²(2 + x)) = ln (x²) + ln (2 + x) = 2 ln x + ln (2 + x).

(c) expand ln 81x⁸y:using the logarithmic property ln (aⁿ) = n ln a, we have:ln 81x⁸y = ln 81 + ln (x⁸y) = ln 81 + ln (x⁸) + ln y = ln 81 + 8 ln x + ln y.

logarithmic expressions: (49.23 (a) log7 y (b) In (x2(2 + x)) (c) In 81x8y

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The solution to the given differential equation with the initial condition y = 70 when x = 0 is y = 70e^(5x^2).

The given differential equation is:

dy/dx = 10xy

To solve this, we'll separate the variables and integrate both sides.

First, let's separate the variables:

dy/y = 10x dx

Now, we'll integrate both sides:

∫ (1/y) dy = ∫ 10x dx

Integrating, we get:

ln|y| = 5x^2 + C1

Where C1 is the constant of integration.

To find the particular solution, we'll use the initial condition y = 70 when x = 0.

Substituting these values into the equation, we get:

ln|70| = 5(0)^2 + C1

ln|70| = C1

So, the equation becomes:

ln|y| = 5x^2 + ln|70|

Combining the logarithms:

ln|y| = ln|70e^(5x^2)|

We can remove the absolute value by taking the exponential of both sides:

y = 70e^(5x^2)

Therefore, the solution to the given differential equation with the initial condition y = 70 when x = 0 is y = 70e^(5x^2).

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The equation f(x, y) = x + 4y + 7 has no critical points. We cannot categorize them as local minimum, local maximum, or saddle points because there are no critical points.

To find the critical points of the function f(x, y) = x + 4y + 7, we need to find the points where the partial derivatives with respect to x and y are equal to zero.

The partial derivatives of f(x, y) are:

∂f/∂x = 1

∂f/∂y = 4

Setting these partial derivatives equal to zero, we have:

1 = 0 (for ∂f/∂x)

4 = 0 (for ∂f/∂y)

However, there are no values of x and y that satisfy these equations simultaneously. Therefore, there are no critical points for the function f(x, y) = x + 4y + 7.

Since there are no critical points, we cannot classify them as local minimum, local maximum, or saddle points.

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The distance of the race is 300 meters.

Tom's average speed is 10 meters per minute.

To solve this problem, we'll first calculate the time it took Tom to complete half of the race and then use that information to find the distance of the entire race.

Let's denote the distance of the race as "d."

Since Tom had only finished running half of the race when Kelly completed it in 15 minutes, we can find the time it took Tom to run half the distance. We know that Tom's speed is 10 m/min less than Kelly's speed. Let's denote Kelly's speed as "v" m/min. Tom's speed would then be "v - 10" m/min.

The time it took Tom to run half the distance can be calculated using the formula:

time = distance / speed

For Tom, the time is 15 minutes (the time Kelly took to complete the race) and the distance is half of the total distance, which is "d/2." The speed is "v - 10" m/min.

So, we have the equation:

15 = (d/2) / (v - 10)

To find the distance of the race (d), we need to eliminate the fraction. We can do this by multiplying both sides of the equation by 2(v - 10):

15 * 2(v - 10) = d

30(v - 10) = d

Expanding the equation:

30v - 300 = d

Now we have an expression for the distance of the race (d) in terms of Kelly's speed (v).

To find Tom's average speed in meters per minute, we need to find Kelly's speed (v). We know that Kelly completed the race in 15 minutes, so her average speed is:

v = distance / time

v = d / 15

Substituting the expression for d:

v = (30v - 300) / 15

Multiplying both sides by 15:

15v = 30v - 300

Subtracting 30v from both sides:

-15v = -300

Dividing by -15:

v = 20

Now that we know Kelly's speed (v = 20 m/min), we can find the distance of the race (d):

d = 30v - 300

d = 30 * 20 - 300

d = 600 - 300

d = 300

Therefore, the distance of the race is 300 meters.

To find Tom's average speed in meters per minute, we can subtract 10 m/min from Kelly's speed:

Tom's speed = Kelly's speed - 10

Tom's speed = 20 - 10

Tom's speed = 10 m/min

Therefore, Tom's average speed is 10 meters per minute.

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To find the value of sec(-θ) given sec(θ), we can use the reciprocal property of trigonometric functions. In this case, since sec(θ) is known to be -0.37, we can determine sec(-θ) by taking the reciprocal of -0.37.

The secant function is the reciprocal of the cosine function. Therefore, if sec(θ) = -0.37, we can find sec(-θ) by taking the reciprocal of -0.37. The reciprocal of a number is obtained by dividing 1 by that number.

Reciprocal of -0.37:

sec(-θ) = 1 / sec(θ)

sec(-θ) = 1 / (-0.37)

sec(-θ) = -2.7027

Therefore, sec(-θ) is equal to -2.7027. By applying the reciprocal property of trigonometric functions, we can find the value of sec(-θ) using the known value of sec(θ).

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The triple integral evaluates to zero because the given solid R lies entirely within the plane z = 0, so the integral of z over that region is zero.

The given solid R is bounded by the paraboloid z = 1 – x^2 - y^2 and the plane z = 0. Cylindrical coordinates are well-suited to represent this solid. In cylindrical coordinates, the equation of the paraboloid becomes z = 1 - r^2, where r represents the radial distance from the z-axis. Since the solid lies entirely below the z = 0 plane, the limits of integration for z are 0 to 1 - r^2. The integral of z over the region will be zero because the limits of integration are symmetric around z = 0, resulting in equal positive and negative contributions that cancel each other out. Therefore, the triple integral evaluates to zero.

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The axis of rotation is the y-axis, and the solid is a cylinder with a cylindrical hole in the center.

To describe the solid, we first need to find the bounds for y. From the integral, we see that y ranges from 0 to the value that makes 2-y=0 or y=2, whichever is smaller. Thus, the bounds for y are 0 to 2.

Next, we need to determine the axis of rotation. The integral is set up using cylindrical shells, which means the axis of rotation is perpendicular to the y-axis.

To find the axis of rotation, we look at the bounds for x. We are given two options: x=??, x=0, and y=0 OR x=??, x=7, and y=0. We need to choose the one that makes sense for the given integral.

If we look at the integrand, we see that it contains factors of (2-y) and (7-y^2), which suggests that the region being rotated is bounded by the curves y=2-x and y=sqrt(7-x^2).

This region lies between the y-axis and the curve y=2-x, so rotating it about the y-axis would give us a solid with a hole in the center.

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To find the y-intercept of the function k(x) = 3x^4 + 4x^3 - 36x^2 - 10, we evaluate the function at x = 0. The y-intercept is the point where the graph of the function intersects the y-axis. In this case, the y-intercept is -10.

The y-intercept of a function is the value of the function when x = 0. To find the y-intercept of the function k(x) = 3x^4 + 4x^3 - 36x^2 - 10, we substitute x = 0 into the function:

k(0) = 3(0)^4 + 4(0)^3 - 36(0)^2 - 10

= 0 + 0 - 0 - 10

= -10

Therefore, the y-intercept of the function is -10. This means that the graph of the function k(x) intersects the y-axis at the point (0, -10).

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To find the trigonometric values and quadrant of an angle whose terminal side lies on the line 24x + 7y = 0, we need to determine the values of sin(theta), cos(theta), tan(theta), csc(theta), sec(theta), and cot(theta).

The equation of the line is 24x + 7y = 0. To find the slope of the line, we can rearrange the equation in slope-intercept form:

y = (-24/7)xFrom this equation, we can see that the slope of the line is -24/7. Since the slope is negative, the angle formed by the line and the positive x-axis will be in the second quadrant (Quadrant II).

Now, let's find the values of the trigonometric functions:

sin(theta) = y/r = (-24/7) / sqrt((-24/7)^2 + 1^2)

cos(theta) = x/r = 1 / sqrt((-24/7)^2 + 1^2)

tan(theta) = sin(theta) / cos(theta)

csc(theta) = 1 / sin(theta)

sec(theta) = 1 / cos(theta)

cot(theta) = 1 / tan(theta)After evaluating these expressions, we can find the values of the trigonometric functions for the angle theta whose terminal side lies on the given line in the second quadrant.Please note that since the specific angle theta is not provided, we can only calculate the values of the trigonometric functions based on the given information about the line.

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Answer:

12q⁹s⁸

Step-by-step explanation:

In mathematics, the brackets () means that you have to multiply, and this is an algebraic expression, so:

[tex]6 \times 2 = 12 \\ {q}^{7} \times {q}^{2} = {q}^{9} [/tex]

The reason we do this I in mathematics, when me multiply expression with exponents, add the exponents together

Eg:

[tex] {p}^{2} \times {p}^{3} = {p}^{5} [/tex]

So we continue:

[tex] {s}^{5} \times {s}^{3} = {s}^{8} [/tex]

Therefore, we add them and it becomes

[tex]12 {q}^{9} {s}^{8}[/tex]

Hope this helps

By integrating the marginal cost function and adding the fixed costs, we can find the total cost to produce 40 items.

The total cost to produce 40 items can be determined by integrating the marginal cost function and adding the fixed costs. By evaluating the integral and adding the fixed costs, we can find the total cost to produce 40 items, rounding the answer to the nearest integer.

The marginal cost function is given by C′(q) = q² - 40q + 700, where q represents the quantity of items produced. To find the total cost, we need to integrate the marginal cost function to obtain the cost function, and then evaluate it at the quantity of interest, which is 40.

Integrating the marginal cost function C′(q) with respect to q, we obtain the cost function C(q) = (1/3)q³ - 20q² + 700q + C, where C is the constant of integration.

To determine the constant of integration, we use the given information that fixed costs are $500. Since fixed costs do not depend on the quantity of items produced, we have C(0) = 500, which gives us the value of C.

Now, substituting q = 40 into the cost function C(q), we can calculate the total cost to produce 40 items. Rounding the answer to the nearest integer gives us the final result.

Therefore, by integrating the marginal cost function and adding the fixed costs, we can find the total cost to produce 40 items.

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Answer: To find the area of the side of the cylinder that needs to be painted, we need to calculate the lateral surface area.

The formula for the lateral surface area of a right cylinder is:

Lateral Surface Area = 2πrh

where r is the radius and h is the height of the cylinder.

Plugging in the values:

Lateral Surface Area = 2π(5 feet)(20 feet)

Now we can calculate the lateral surface area:

Lateral Surface Area = 2π(5 feet)(20 feet)

Since we know that one gallon of paint covers 325 square feet, we can calculate the number of gallons needed:

Number of gallons = Lateral Surface Area / Coverage per gallon

Therefore, the farmer will need to buy approximately 1.932 gallons of paint to complete the job.

The correct answer is 1728 in terms of p and q: k²3 supposing logk p = 11 and logk q = -7, where k, p, q. We will use the laws of logarithms.

a) The value of log (p²q-8) is -6.

To solve for log (p²q-8), we can use the laws of logarithms:

p²q-8 as (pq²)/2^3

log (p²q-8) = log [(pq²)/2^3]

= log (pq²) - log 2^3

= log p + 2log q - 3

log (p²q-8) = 11 + 2(-7) - 3 (Substituting the values)

= -6

b) The value of logk (wp-5q³) is (1/11) * log w + (1/-7) * log (p-5q³).

To solve for logk (wp-5q³),

Using the property that log ab = log a + log b:

logk (wp-5q³) = logk w + logk (p-5q³)

logk w = (1/logp k) * log w  (first equation)

logk (p-5q³) = (1/logp k) * log (p-5q³) (second equation)

Substituting the given values of logk p and logk q, we get:

logk w = (1/11) * log w

logk (p-5q³) = (1/-7) * log (p-5q³)

logk (wp-5q³) = (1/11) * log w + (1/-7) * log (p-5q³)

c) To express k²3 in terms of p and q, we need to eliminate k from the given expression. Using the property that (loga b)^c = loga (b^c), we can write:

k²3 = (k^2)^3

= (logp kp)^3

= (logp k + logp p)^3

= (logp k + 1)^3

k²3 = (11 + 1)^3 (Substitution)

= 12^3

= 1728

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We can compare these values to find the absolute maximum and minimum values of F(x, y).

To find the absolute maximum and minimum values of the function[tex]F(x, y) = r + y + ry + 3[/tex] on the domain[tex]D = {(x, y) | x^2 + y^2 ≤ 51}[/tex], we need to evaluate the function at critical points and boundary points of the domain. First, let's find the critical points by taking the partial derivatives of F(x, y) with respect to x and y:

[tex]∂F/∂x = r∂F/∂y = 1 + r[/tex]

To find critical points, we set both partial derivatives equal to zero:

[tex]r = 0 ...(1)1 + r = 0 ...(2)[/tex]

From equation (2), we can solve for r:

[tex]r = -1[/tex]

Now, let's evaluate the function at the critical point (r, y) = (-1, y):

[tex]F(-1, y) = -1 + y + (-1)y + 3F(-1, y) = 2y + 2[/tex]

Next, let's consider the boundary of the domain, which is the circle defined by [tex]x^2 + y^2 = 51.[/tex]To find the extreme values on the boundary, we can use the method of Lagrange multipliers.

Let's define the function [tex]g(x, y) = x^2 + y^2.[/tex] The constraint is [tex]g(x, y) = 51.[/tex]

Now, we set up the Lagrange equation:

[tex]∇F = λ∇g[/tex]

Taking the partial derivatives:

[tex]∂F/∂x = r∂F/∂y = 1 + r∂g/∂x = 2x∂g/∂y = 2y[/tex]

The Lagrange equation becomes:

[tex]r = λ(2x)1 + r = λ(2y)x^2 + y^2 = 51[/tex]

From the first equation, we can solve for λ in terms of r and x:

[tex]λ = r / (2x) ...(3)[/tex]

Substituting equation (3) into the second equation, we get:

[tex]1 + r = (r / (2x))(2y)1 + r = ry / xx + xr = ry ...(4)[/tex]

Next, we square both sides of equation (4) and substitute [tex]x^2 + y^2 = 51:(x + xr)^2 = r^2y^2x^2 + 2x^2r + x^2r^2 = r^2y^251 + 2(51)r + 51r^2 = r^2y^251(1 + 2r + r^2) = r^2y^251 + 102r + 51r^2 = r^2y^251(1 + 2r + r^2) = r^2(51 - y^2)1 + 2r + r^2 = r^2(1 - y^2 / 51)[/tex]

Simplifying further:

[tex]1 + 2r + r^2 = r^2 - (r^2y^2) / 51(r^2y^2) / 51 = 2rr^2y^2 = 102ry^2 = 102[/tex]

Taking the square root of both sides, we get:

[tex]y = ±√102[/tex]

Since the square root of 102 is approximately 10.0995, we have two values for [tex]y: y = √102 and y = -√102[/tex].

Substituting y = √102 into equation (4), we can solve for x:

[tex]x + xr = r(√102)x + x(-1) = -√102x(1 - r) = -√102x = -√102 / (1 - r)[/tex]

Similarly, substituting y = -√102 into equation (4), we can solve for x:

[tex]x + xr = r(-√102)x + x(-1) = -r√102x(1 - r) = r√102x = r√102 / (1 - r)[/tex]

Now, we have the following points on the boundary of the domain:

[tex](x, y) = (-√102 / (1 - r), √102)(x, y) = (r√102 / (1 - r), -√102)[/tex]

Let's evaluate the function F(x, y) at these points:

[tex]F(-√102 / (1 - r), √102) = -√102 / (1 - r) + √102 + (-√102 / (1 - r))√102 + 3F(r√102 / (1 - r), -√102) = r√102 / (1 - r) + (-√102) + (r√102 / (1 - r))(-√102) + 3[/tex]

To find the absolute maximum and minimum values of F(x, y), we need to compare the values obtained at the critical points and the points on the boundary.

Let's summarize the values obtained:

[tex]F(-1, y) = 2y + 2F(-√102 / (1 - r), √102)F(r√102 / (1 - r), -√102)[/tex]

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To rewrite the function (18x - 1)e^(9x^2 - x) in terms of u using the substitution method, we let u = 9x^2 - x. By finding the derivative of u with respect to x, we can express the integrand in terms of u.

To rewrite the function (18x - 1)e^(9x^2 - x) in terms of u, we let u = 9x^2 - x. Differentiating both sides of this equation with respect to x, we get du/dx = 18x - 1. Solving for dx, we have dx = (1/(18x - 1)) du.

Substituting the expression for dx into the original function, we have:

(18x - 1)e^(9x^2 - x) dx = (18x - 1)e^(u) (1/(18x - 1)) du.

Simplifying, we cancel out the (18x - 1) terms:

(18x - 1)e^(u) (1/(18x - 1)) du = e^u du.

We have successfully rewritten the integrand in terms of u. The function (18x - 1)e^(9x^2 - x) is now expressed as e^u. We can now proceed with the integration using the new expression.

In conclusion, by letting u = 9x^2 - x and finding the derivative du/dx, we can rewrite the function (18x - 1)e^(9x^2 - x) in terms of u as e^u. This substitution simplifies the integration process.

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After the given code is executed, the value of y will still be 12.

The code starts by assigning the value 10 to the variable x. Then, the variable y is assigned the value of x + 2, which is 12 (10 + 2). Next, the value of x is changed to 12. However, this change does not affect the value of y, which was already assigned as 12.

Therefore, the correct answer is c. 12.

what is variable?

In the context of mathematics and programming, a variable is a symbol or name that represents a value that can change. It is used to store and manipulate data within a program or equation.

A variable can hold different types of data, such as numbers, text, or boolean values, and its value can be modified during the execution of a program or when solving equations. Variables provide a way to store and retrieve data, perform calculations, and control the flow of a program.

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The answer explains how to find the average value of a function on a given interval and evaluates the definite integral of a given expression.

To find the average value of the function f(x) on the interval [2, 2e], we need to evaluate the definite integral of f(x) over that interval and divide it by the length of the interval.

The definite integral of f(x) over the interval [2, 2e] can be written as:

∫[2,2e] f(x) dx

To evaluate the definite integral, we need the expression for f(x). However, the function f(x) is not provided in the question. Please provide the function expression, and I will be able to calculate the average value.

Regarding the given definite integral, ∫ (16 + p^2) dp, we can evaluate it by integrating the expression:

∫ (16 + p^2) dp = 16p + (p^3)/3 + C,

where C is the constant of integration. If you have specific limits for the integral, please provide them so that we can calculate the definite integral.

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The length of the curve defined by the parametric equations x = 1 + 3t^2 and y = 4 + 2t^3 can be found using the arc length formula. The formula involves integrating the square root of the sum of the squares of the derivatives of x and y with respect to t.

To find the length of the curve, we can use the arc length formula. Let's denote the derivatives of x and y with respect to t as dx/dt and dy/dt, respectively.

The derivatives are:

dx/dt = 6t,

dy/dt = 6t^2.

The arc length formula is given by:

L = ∫[a, b] √((dx/dt)^2 + (dy/dt)^2) dt.

Substituting the derivatives into the formula, we have:

L = ∫[a, b] √((6t)^2 + (6t^2)^2) dt.

Simplifying the expression inside the square root:

L = ∫[a, b] √(36t^2 + 36t^4) dt.

Factoring out 36t^2 from the square root:

L = ∫[a, b] 6t √(1 + t^2) dt.

To solve this integral, a specific range for t needs to be provided. Without that information, we cannot proceed further with the calculations. However, this is the general process for finding the length of a curve defined by parametric equations using the arc length formula.

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The scale factor ratio between the smaller and larger rectangular prisms is 2:3, and the surface area ratio is 2:3.

To find the scale factor ratio, we can take the cube root of the volume ratio. The cube root of 162 is approximately 5.08, and the cube root of 384 is approximately 7.87. Therefore, the scale factor ratio is approximately 5.08:7.87, which can be simplified to 2:3.

The surface area of a rectangular prism is proportional to the square of the scale factor. Since the scale factor ratio is 2:3, the surface area ratio would be the square of that ratio, which is 4:9.

Therefore, the scale factor ratio between the smaller and larger rectangular prisms is 2:3, and the surface area ratio is 4:9.

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