(q20) Is the function f(x) = (x - 9)(x + 9)x even, odd, or neither?

Answer:

c = 52 m

Step-by-step explanation:

using Pythagoras' identity in the right triangle

the square on the hypotenuse is equal to the sum of the squares on the other 2 sides , that is

c² = 48² + 20² = 2304 + 400 = 2704 ( take square root of both sides )

c = [tex]\sqrt{2704}[/tex] = 52 m

The correct statement regarding David's claim is given as follows:

David’s claim is incorrect because the interquartile range for his scores is greater than the interquartile range for Elise’s scores.

The interquartile range of a data-set is given by the difference of the third quartile by the first quartile of the data-set.

The interquartile range is a metric of consistency, and the lower the interquartile range, the more consistent the data-set is.

The interquartile range for David is greater than for Elise, hence his claim is incorrect.

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Dakota can fill the 1-liter measuring cup with water and pour it into the fish tank multiple times until the tank is full, then multiply the number of times she filled the cup by 1 liter to determine the total volume of water used.

What  is Measuring cup.?

A measuring cup is a kitchen tool used to measure the volume of liquid or bulk solid ingredients, typically made of glass or plastic and marked with graduated lines to indicate different measurements, such as milliliters, fluid ounces, and cups.

Dakota has a 1-liter measuring cup. How could she use the measuring cup to measure the volume of water that could fill a fish tank?

Dakota could use the 1-liter measuring cup to measure the volume of water that could fill a fish tank by filling the cup with water and pouring it into the fish tank, repeating the process until the fish tank is filled to the desired volume. She could keep track of the number of times she fills the measuring cup and multiply that by 1 liter to determine the total volume of water used.

Let's say Dakota wants to measure the volume of water in a fish tank that has a capacity of 5 liters. She can use the 1-liter measuring cup to do this.

She can start by filling the measuring cup with water from a tap or a water source.

Then, she can carefully pour the water from the measuring cup into the fish tank.

She can repeat this process four more times until the fish tank is filled to the desired volume.

Each time she fills the measuring cup, she can keep track of how many cups she has used.

In this example, she would have used the measuring cup five times, and therefore the total volume of water used would be 5 liters (1 liter per cup x 5 cups).

So, by using the 1-liter measuring cup, Dakota could measure the volume of water in the fish tank by filling and pouring the cup multiple times until the tank is full, then multiplying the number of cups used by 1 liter to determine the total volume of water used.

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Answer:

  f(-8) = 14

Step-by-step explanation:

You want f(-8) when f(x) = x² +8x +14.

The function is evaluated for x = -8 by putting -8 where you see x, then doing the arithmetic.

  f(-8) = (-8)² +8(-8) +14

  f(-8) = 64 -64 +14 = 14

The value of f(-8) is 14.

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Answer:

Mandy scored a total of 22 points in the basketball game. She made 9 field points, which can be worth either 2 or 3 points. Let's assume that she made x three-point goals and y two-point goals.Then, we can set up the following system of equations:x + y = 9 (because she made a total of 9 field points)3x + 2y = 22 (because the total point value of her field goals was 22).

Solving this system of equations, we can first multiply the first equation by 2 to get:2x + 2y = 18Then, we can subtract this equation from the second equation to eliminate y:3x + 2y - (2x + 2y) = 22 - 18Simplifying this gives:x = 4

Therefore, Mandy made a total of 4 three-point goals and 5 two-point goals in the game.

To test the hypothesis that the average time to deliver goods, once the order is placed over the phone, is more than 30 minutes in the population, we can use a one-sample t-test.

The one-sample t-test is used to compare the mean of a sample to a known or hypothesized population mean. In this case, the null hypothesis would be that the population mean delivery time is equal to 30 minutes, and the alternative hypothesis would be that the population mean delivery time is greater than 30 minutes. We would collect a sample of delivery times and calculate the sample mean and standard deviation. We would then use the t-test to determine whether the sample mean is significantly different from the hypothesized population mean of 30 minutes.

Therefore, a one-sample t-test would be the appropriate statistical test to use to test the hypothesis that the average time to deliver goods.

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If the eigenvectors of a are the columns of the identity matrix (i), then a is a diagonal matrix. If the eigenvector matrix p is triangular, then a is a triangular matrix.

If the eigenvectors of a are the columns of the identity matrix (i), then a is a diagonal matrix. This is because the eigenvectors of a diagonal matrix are simply the columns of the identity matrix, and the eigenvectors of a matrix do not change under similarity transformations.

If the eigenvector matrix p is triangular, then a is a triangular matrix. This is because the eigenvector matrix p is related to the matrix a through the equation:

A = PDP⁻¹

where D is a diagonal matrix whose diagonal entries are the eigenvalues of a, and P is the matrix whose columns are the eigenvectors of a. If the matrix P is triangular, then the matrix A is also triangular. This can be seen by noting that the inverse of a triangular matrix is also triangular, and the product of two triangular matrices is also triangular.

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-- The given question is incomplete, the complete question is

"If the eigenvectors of A are the columns of I, then A is what sort of matrix? If the eigenvector matrix P is triangular, what sort of matrix is A?"

Dr. Toming ran a regression analysis to settle the debate between L. Wood and H. Wood about whether to invest in large houses or large lots. She included all variables collected by the Woods and controlled for house size and lot size, which tend to correlate highly with each other. The output showed that the mean sale price is over $200,000. To determine whether this is a valid model, additional information is needed, such as the significance level and the R-squared value. Without this information, it is impossible to determine the validity of the model.

Dr. Toming's regression analysis controlled for both house size and lot size, which is important because they tend to correlate highly with each other. This means that the analysis accounted for the fact that larger houses tend to be on larger lots, and vice versa. However, the mean sale price alone does not provide enough information to determine the validity of the model. Additional information such as the significance level and the R-squared value would be necessary to make a determination.

Without additional information about the significance level and R-squared value, it is impossible to determine the validity of Dr. Toming's regression analysis. While controlling for house size and lot size is important in this case, more information is needed to evaluate the overall effectiveness of the model.

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Answer:

168=18x +24x

168=42x

168÷42=X

X=4

The three additional points on the line are (9, 4), (12, 3) and (15, 2)

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

(3, 6) and (6, 5)

From the above, we can see that

As x increases by 3, the value of y decreases by 1

This means that the slope of the line is -1/3

Also, we can use the following transformation rule to generate the other points

(x + 3, y - 1)

When used, we have

(9, 4), (12, 3) and (15, 2)

Hence, the three additional points on the line are (9, 4), (12, 3) and (15, 2)

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The  inverse Laplace transform of L{f(t)} is:

f(t) = L^-1{2/s} + L^-1{3/s^2} + L^-1{4} + L^-1{13/s^2}
    = 2 + 3t + 4δ(t) + 13t

Thus, f(t) = 2 + 16t for t > 0, and f(t) = 2 for t = 0.

We are given the Laplace transform of a function f(t) as:

L{f(t)} = 2s/(s^2) + 3/(s^2) + 4s/(s^2) + 13/(s^2)

We can simplify this expression as:

L{f(t)} = 2/s + 3/s^2 + 4 + 13/s^2

To find f(t), we need to take the inverse Laplace transform of each term in this expression. We can use the following formulas:

L{t^n} = n!/s^(n+1)
L{e^at} = 1/(s-a)

Using these formulas, we can find that the inverse Laplace transform of each term is:

L^-1{2/s} = 2
L^-1{3/s^2} = 3t
L^-1{4} = 4δ(t)
L^-1{13/s^2} = 13t

where δ(t) is the Dirac delta function.

Therefore, the inverse Laplace transform of L{f(t)} is:

f(t) = L^-1{2/s} + L^-1{3/s^2} + L^-1{4} + L^-1{13/s^2}
    = 2 + 3t + 4δ(t) + 13t

Thus, f(t) = 2 + 16t for t > 0, and f(t) = 2 for t = 0.

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The volume of the solid is (11π/3) cubic units.

We have,

To find the volume of the solid obtained by rotating the region bounded by y = x and y = x^2 about the line x = -3, we can use the method of cylindrical shells.

The formula for the volume using cylindrical shells is given by:

V = 2π ∫ [a, b] x h(x) dx,

where [a, b] is the interval of integration, x represents the variable of integration, and h(x) represents the height of the shell at each value of x.

In this case, we want to rotate the region bounded by y = x and y = x² about the line x = -3.

Since we are rotating about a vertical line, the height of the shell at each value of x will be given by the difference between the x-coordinate of the curve and the line of rotation:

h(x) = (x - (-3)) = x + 3.

To find the interval of integration, we need to determine the x-values where the two curves intersect.

Setting x = x², we have:

x = x²,

x² - x = 0,

x (x - 1) = 0.

This gives us two intersection points: x = 0 and x = 1.

Therefore, the interval of integration is [0, 1].

Now we can set up the integral to find the volume:

V = 2π ∫ [0, 1] x (x + 3) dx.

Evaluating this integral, we have:

V = 2π ∫ [0, 1] (x² + 3x) dx

= 2π [x³/3 + (3/2)x²] evaluated from 0 to 1

= 2π [(1/3 + 3/2) - (0/3 + 0/2)]

= 2π [(2/6 + 9/6) - 0]

= 2π (11/6)

= (22π/6)

= (11π/3).

Therefore,

The volume of the solid is (11π/3) cubic units.

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The property to be used is vertical angle theorem and the value of x is 20/3.

Given is a figure in which two lines are intersecting at a point, making two angles,

The angles are = 3x and 20°,

We need to determine the value of x and the property involved.

So, according to figure we can say, the property involved is vertical angle theorem.

Therefore,

3x = 20

x = 20/3

Hence the property to be used is vertical angle theorem and the value of x is 20/3.

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Since the p-value (0.06) is greater than the alpha level (0.05), you would fail to reject the null hypothesis.

The alpha level, or significance level, is the threshold below which you would reject the null hypothesis in favor of the alternative hypothesis. The p-value is the probability of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true.

In this case, you set the alpha level at 0.05, meaning that there is a 5% chance of incorrectly rejecting the null hypothesis if it is true. The p-value of 0.06 indicates that there is a 6% chance of observing a test statistic as extreme or more extreme than the one observed, assuming the null hypothesis is true. Since 6% is greater than 5%, you do not have enough evidence to reject the null hypothesis.

Based on the alpha level of 0.05 and the p-value of 0.06, you would conclude that there is not enough evidence to reject the null hypothesis, and you should fail to reject it.

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Step-by-step explanation:

log(5x) - log(2) = log(5x/2)

therefore,

log(4x - 1) = log(5x/2)

4x - 1 = 5x/2

8x - 2 = 5x

3x - 2 = 0

3x = 2

x = 2

since this is basically a linear equation in x, there is only one solution, and that is x = 2.

for x = 2 all arguments of the log functions are positive.

4x - 1 = 4×2 - 1 = 8 - 1 = 7

5x = 5×2 = 10

these are all valid arguments for the log function.

so, x = 2 is a valid and not extraneous solution.

The "transient-solution" for X'' + 8X' + 12X = 15,  X(O) = 2, X'(0) = 2 is X(t) = (-7/8) × [tex]e^{-6t}[/tex] + (13/8) × [tex]e^{-2t}[/tex] + 5/4.

In order to find the transient solution of given second-order system, we solve the homogeneous equation associated with it and then find the particular solution for non-homogeneous term.

The homogeneous equation is obtained by setting the right-hand side (RHS) of the equation to zero:

X'' + 8X' + 12X = 0

The characteristic-equation is obtained by assuming a solution of the form X(t) = [tex]e^{rt}[/tex]:

r² + 8r + 12 = 0

(r + 2)(r + 6) = 0

So, the two roots are : r = -2 and r = -6,

The general solution of homogeneous equation is given by:

[tex]X_{h(t)}[/tex] = C₁ × [tex]e^{-6t}[/tex] + C₂ × [tex]e^{-2t}[/tex]

Now, we find the particular-solution for the non-homogeneous term, which is 15. Since 15 is a constant, we assume a constant solution for [tex]X_{p(t)[/tex]:

[tex]X_{p(t)[/tex] = k

Substituting this into original equation,

We get,

0 + 8 × 0 + 12 × k = 15,

12k = 15

k = 15/12 = 5/4

So, particular solution is [tex]X_{p(t)[/tex] = 5/4.

The "transient-solution" is sum of homogeneous and particular solutions:

X(t) = [tex]X_{h(t)[/tex] + [tex]X_{p(t)[/tex]

X(t) = C₁ × [tex]e^{-6t}[/tex] + C₂ × [tex]e^{-2t}[/tex] + 5/4, and

X'(t) = -6C₁ × [tex]e^{-6t}[/tex] -2C₂ × [tex]e^{-2t}[/tex] ,

To find the values of C₁ and C₂, we use initial-conditions: X(0) = 2 and X'(0) = 2.

X(0) = C₁ × [tex]e^{-6\times 0}[/tex] + C₂ × [tex]e^{-2\times 0}[/tex] + 5/4,
X(0) = C₁ + C₂ + 5/4,

Since X(0) = 2, We have:

C₁ + C₂ + 5/4 = 2      ...Equation(1)

and Since X'(0) = 2, we have:

3C₁ + C₂ = -1     ....Equation(2)

On Solving equation(1) and equation(2),

We get,

C₁ = -7/8  and C₂ = 13/8,

Substituting the values, the transient-solution can be written as :

X(t) = (-7/8) × [tex]e^{-6t}[/tex] + (13/8) × [tex]e^{-2t}[/tex] + 5/4.

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The given question is incomplete, the complete question is

For the following second-order system and initial conditions, find the transient solution: X'' + 8X' + 12X = 15,  X(O) = 2, X'(0) = 2.

Thus, the results of this bivariate correlation analysis suggest that there is little to no relationship between students' love of cats and their love of school.

A bivariate correlation analysis is a statistical tool that is used to determine whether there is a relationship between two variables. In this case, the analysis tests the relationship between students' love of cats and their love of school.

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The multiplication of 8x³ by (x² + 5x - 6) using distribution is 8x⁵ + 40x⁴ - 48x³.

To multiply the polynomial 8x³ by the polynomial (x² + 5x - 6) using distribution, we will distribute each term of the first polynomial (8x³) to every term in the second polynomial (x² + 5x - 6).

Here's the step-by-step process:

Distribute 8x³ to each term of (x² + 5x - 6):

8x³ · x² + 8x³ · 5x + 8x³ · (-6)

Multiply each term:

8x³ · x² = 8x³ · x² = 8x⁵

8x³ · 5x = 40x³⁺¹ = 40x⁴

8x³ · (-6) = -48x³

Combine the resulting terms:

8x⁵ + 40x⁴ - 48x³

Therefore, the multiplication of 8x³ by (x² + 5x - 6) using distribution is 8x⁵ + 40x⁴ - 48x³.

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To classify the critical point (0,0) using the determinant test, we need to compute the Hessian matrix. The Hessian matrix is a matrix of second partial derivatives of the function with respect to x and y. The Hessian matrix for f(x,y) is given by:

H = [[12x + 2y, 2x], [2x, 2y]]

Evaluating the Hessian matrix at (0,0), we get:

H(0,0) = [[0, 0], [0, 0]]

The determinant of the Hessian matrix is zero, which indicates that the test is inconclusive. In this case, we need to use another method to classify the critical point (0,0). One possible method is to examine the signs of the second partial derivatives of f(x,y) at (0,0).

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

f(x)x = 12x + 2y = 0
fxy = 2x = 0
fyy = 2y = 0

Since all the second partial derivatives of f(x,y) are zero at (0,0), we cannot determine the nature of the critical point using this method either. We would need to use additional methods, such as the Taylor series expansion or graphing, to classify the critical point.

The critical point (0,0) is a local minimum.

To classify the critical point (0,0) of the function [tex]f(x, y) = 2x^3 + xy^2 + 5x^2 + y^2[/tex] using the determinant test, we need to compute the Hessian matrix and evaluate its determinant at the critical point.

The Hessian matrix of f(x, y) is given by:

[tex]H = | f_{xx} f_{xy} |[/tex]

       [tex]| f_{yx} f_{yy} |[/tex]

Where f_xx represents the second partial derivative of f with respect to x, [tex]f_{xy}[/tex] represents the mixed partial derivative of f with respect to x and y, [tex]f_{yx}[/tex] represents the mixed partial derivative of f with respect to y and x, and [tex]f_{yy}[/tex] represents the second partial derivative of f with respect to y.

Taking the partial derivatives of f(x, y), we have:

[tex]f_x = 6x^2 + y^2 + 10x\\f_y = 2xy + 2y[/tex]

Calculating the second partial derivatives:

[tex]f_{xx} = 12x + 10\\f_{xy} = 2y\\f_{yx} = 2y\\f_{yy} = 2x + 2[/tex]

Now, evaluating the Hessian matrix at the critical point (0,0):

[tex]H(0,0) = | f_{xx}(0,0) f_{xy}(0,0) |[/tex]              

              [tex]| f_{yx}(0,0) f_{yy}(0,0) |[/tex]

H(0,0) = | 10  0 |

              | 0    2 |

The determinant of the Hessian matrix at (0,0) is:

Det[H(0,0)] = det | 10  0 |

                          | 0    2 |

Det[H(0,0)] = (10)(2) - (0)(0) = 20

Therefore, the determinant (Det[H(0,0)]) is positive (20 > 0), we can conclude that the critical point (0,0) is a local minimum.

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The box spans from Q1 = 52 to Q3 = 62, and the spread of the middle 50% of the data is Q3 - Q1 = 62 - 52 = 10 miles per gallon.

To find the spread of the middle 50% of the data using the box plot, we need to first find the boundaries of the box, which represents the middle 50% of the data.

Looking at the box plot, we can see that the box spans from the lower quartile (Q1) to the upper quartile (Q3), with a line inside the box representing the median.

From the data given in the box plot, we can see that the minimum value is 48 and the maximum value is 64. The median is the middle value of the data, which is the average of the two middle values since we have an even number of values. Therefore, the median is (56 + 58) / 2 = 57.

To find Q1 and Q3, we can split the data into two halves at the median and find the medians of each half. The lower half of the data is {48, 50, 52, 54, 56} and the upper half is {58, 60, 62, 64}. The medians of these halves are 52 and 62, respectively.

Therefore, the box spans from Q1 = 52 to Q3 = 62, and the spread of the middle 50% of the data is Q3 - Q1 = 62 - 52 = 10 miles per gallon.

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If the marked price is $816 and the selling price is $800, the discount offered is $16, which is 1.96 percent off the marked price.

The discount refers to the percentage off the marked price of an item.

The discount amount is the dollar value that is taken off the marked price before arriving at the selling price, also known as the discounted price.

The marked price of the item = $816

The selling price (discounted price) = $800

The discount amount in dollars = $16 ($816 - $800)

The discount percentage = 1.96% ($16/$816 x 100)

Thus, the discount that the retailer offered the customer is $16, which translates to 1.96% off the marked price.

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To find a linear differential operator that annihilates the function 1 + 8e^(2x), we can start by differentiating the function.

d/dx (1 + 8e^(2x)) = 0 + 16e^(2x) = 16e^(2x)

Notice that the derivative of the function is a constant multiple of itself. This suggests that the linear differential operator we are looking for involves a constant coefficient multiplied by the derivative operator.

Let's try multiplying the derivative operator d/dx by a constant c and applying it to the function:

c(d/dx)(1 + 8e^(2x)) = c(0 + 16e^(2x)) = 16ce^(2x)

We want this result to be equal to zero, so we can solve for the constant c:

16ce^(2x) = 0

c = 0

Therefore, the linear differential operator that annihilates the function 1 + 8e^(2x) is simply d/dx. In other words, taking the derivative of the function will result in zero.

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The graph is given below.

The area of the square TUVW is 9 square units.

We have,

The square TUVW with vertices

T = (-2, -4)
U = (-5, -4)

V = (-5, -1)

W = (-2, -1)

Now,

To find the area of the square TUVW, we need to find the length of its sides first.

Using the distance formula, we can find the length of TU:

TU = √((Ux - Tx)² + (Uy - Ty)²)

= √((-5 - (-2))² + (-4 - (-4))²)

= √(9)

= 3

Since TUVW is a square, all of its sides have the same length,

So UV = VW = WT = 3 as well.

The area of the square is the length of one side squared.

Area = side²

= 3²

= 9

Therefore,

The area of the square TUVW is 9 square units.

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Answer 136

Step-by-step explanation:

Therefore, the ladder reaches a height of 12 feet up the wall.

To solve this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides (the distance from the base of the wall and the height of the ladder on the wall). In this case, we have a right triangle with a base of 9 feet, a hypotenuse of 15 feet, and an unknown height.
So, using the Pythagorean theorem, we can solve for the height:
15^2 = 9^2 + height^2
225 = 81 + height^2
144 = height^2
12 = height
Therefore, the ladder reaches a height of 12 feet up the wall.

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The value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = 54 sec^2 x, over the interval [-4, 4] is zero.

The Mean Value Theorem for Integrals states that if a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that the definite integral of f(x) from a to b is equal to f(c) times (b-a). In this case, the given function f(x) is continuous and differentiable over the interval [-4, 4]. Hence, by the Mean Value Theorem for Integrals, there exists a value c in (-4, 4) such that the integral of f(x) from -4 to 4 is equal to f(c) times (4-(-4)) = 8f(c).

As the function is periodic, its integral over the interval from 0 to π is equal to zero. Hence, the integral of the function over the interval [-4, 4] is also equal to zero. Therefore, the value(s) of c guaranteed by the Mean Value Theorem for Integrals is zero. Thus, the answer is 0.

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The equations of the lines with the given gradients and points are:

1. y = -3x + 16

2. y = -5x + 5

3. y = 4x - 20

To find the equation of a line given its gradient and a point it passes through, we can use the point-slope form of a linear equation:

y - y₁ = m(x - x₁),

where (x₁, y₁) represents the given point and m represents the gradient.

Let's calculate the equations for each given gradient and point:

1. Gradient = -3, Point Q(4,4):

Using the point-slope form:

y - 4 = -3(x - 4)

y - 4 = -3x + 12

y = -3x + 16

2. Gradient = -5, Point P(0,5):

Using the point-slope form:

y - 5 = -5(x - 0)

y - 5 = -5x

y = -5x + 5

3. Gradient = 4, Point A(6,4):

Using the point-slope form:

y - 4 = 4(x - 6)

y - 4 = 4x - 24

y = 4x - 20

Therefore, the equations of the lines with the given gradients and points are:

1. y = -3x + 16

2. y = -5x + 5

3. y = 4x - 20

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The volume of the region in the first octant bounded by the coordinate planes, the plane y z=12, and the cylinder x=144−y2 is 432 cubic units. To find the volume of the region in the first octant bounded by the coordinate planes, the plane y z=12, and the cylinder x=144−y2, we need to set up a triple integral.

Since the region is in the first octant, we have the following limits of integration:
0 ≤ x ≤ 144 - y^2
0 ≤ y ≤ √(12/z)
0 ≤ z ≤ 12
So the volume V of the region is given by the triple integral:
V = ∫∫∫ R dV
Where R is the region defined by the above limits of integration, and dV = dxdydz is the differential volume element. Substituting in the limits of integration, we have:
V = ∫0^12 ∫0^√(12/z) ∫0^(144-y^2) dxdydz
Evaluating the integral using the order dzdydx, we get:
V = ∫0^12 ∫0^√(12/z) (144-y^2)dydz
   = ∫0^12 [144y - (1/3)y^3]0^√(12/z) dz
   = ∫0^12 [144√(12/z) - (1/3)(12/z)^(3/2)]dz
   = 576∫0^1 (1 - u^3)du          (where u = √(12/z))
Evaluating the final integral, we get:
V = 576(1 - 1/4)
 = 432 cubic units.

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