(after 3.1) Assume T: R^m → R^n is a linear transformation. (a) Suppose there is a nonzero vector xERm such that T(x) = 0. Is it possible that T is one-to-one? Give an example, or explain why it's not possible. (b) Suppose there is a nonzero vector xe Rm such that T(x) = 0. Is it possible that T is onto? Give an example, or explain why it's not possible. (c) Suppose that u and v are linearly dependent vectors in Rm. Show that T(u) and T(v) are also linearly dependent. (d) Suppose that u and v are linearly independent vectors in R™ Is it guaranteed that Tu) and Tv) are also linearly independent? If yes, explain why. If no, give an example where this is not the case.

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.

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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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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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?"

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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Answer:

  -1572864

Step-by-step explanation:

You want the 19th term of the geometric sequence with first term -6 and common ratio -2.

The n-th term of a geometric sequence is ...

  an = a1·r^(n-1)

where a1 is the first term, and r is the common ratio.

Using the given values of a1 and r, the 19th term is ...

  a19 = (-6)·(-2)^(19-1) = -1572864

<95141404393>

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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Answer:

168=18x +24x

168=42x

168÷42=X

X=4

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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We can conclude that the left-hand side (AU (BoC) - (AUB) A (AUC)) and the right-hand side (∅) have no common elements, which proves the equality AU (BoC) - (AUB) A (AUC) = ∅.

To prove the equality AU (BoC) - (AUB) A (AUC) = ∅, we need to show that the left-hand side is an empty set.

First, let's break down the expression step by step:

AU (BoC) represents the union of A with the intersection of B and C. This implies that any element in A, or in both B and C, will be included.

(AUB) represents the union of A and B, which includes all elements present in either A or B.

(AUC) represents the union of A and C, which includes all elements present in either A or C.

Now, let's analyze the right-hand side:

(AUB) A (AUC) represents the intersection of (AUB) and (AUC), which includes elements that are common to both sets.

To prove the equality, we need to show that the left-hand side and the right-hand side have no common elements, i.e., their intersection is empty.

If an element belongs to the left-hand side (AU (BoC) - (AUB) A (AUC)), it must either belong to A and not belong to (AUB) A (AUC), or it must belong to (BoC) and not belong to (AUB) A (AUC).

However, if an element belongs to (BoC), it implies that it belongs to both B and C. Since it does not belong to (AUB) A (AUC), it means that it cannot belong to either A or B or C. Similarly, if an element belongs to A, it cannot belong to (AUB) A (AUC).

Therefore, we can conclude that the left-hand side (AU (BoC) - (AUB) A (AUC)) and the right-hand side (∅) have no common elements, which proves the equality AU (BoC) - (AUB) A (AUC) = ∅.

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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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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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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

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 expected frequency is 50 hence, the answer is (d) 50.

Expected frequency is the frequency we would expect to see in a particular category or group if the null hypothesis is true. The null hypothesis assumes a specific distribution or pattern in the data, and the expected frequency is calculated based on that assumption.

Here we have

Individuals in a random sample of 150 were asked whether they supported capital punishment.

If the opinions of the individuals are uniformly distributed, then the expected frequency for each group would be the same.

Since there are three groups (yes, no, and no opinion), the expected frequency for each group is:

Expected frequency = Total frequency / Number of groups

= 150/3 = 50

Therefore,

The expected frequency is 50 hence, the answer is (d) 50.

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Answer:

-35.375

Step-by-step explanation:

(-1.5+9.5)=8

5/8 =0.625

7+11=18

0.4*18=36/5

=7.2

7.2/-0.2=

-283/8=

-35.375

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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The car travels 1.40 meters in 0.187 seconds before coming to a stop. To answer this question, we need to use the equation of motion: v(t) = v0 + at where v(t) is the velocity at time t, v0 is the initial velocity, a is the acceleration, and t is the time.

(a) To find how much time it takes for the car to stop, we need to find the time when v(t) = 0. Using the given values, we have:

0 = 15 - 80.2t

Solving for t, we get:

t = 15/80.2 = 0.187 seconds

Therefore, it takes the car 0.187 seconds to stop.

(b) To find how far the car travels in this time, we can use the equation:

d(t) = v0t + 0.5at^2

Substituting the given values, we get:

d(t) = 15(0.187) + 0.5(-80.2)(0.187)^2

Simplifying, we get:

d(t) = 1.40 meters

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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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Answer:

11/6

Step-by-step explanation:

For the reciprocal just flip it

Answer 136

Step-by-step explanation:

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 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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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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Answer:

576 (square yards)

Step-by-step explanation:

length of slanted height of triangle = √(6² + 8²)

= √100

= 10.

surface area = area of 2 triangle faces + area of 3 lengths

= 2 (1/2 X 12 X 8) + 3 (10 X 16)

= 576 (square yards)

The quadratic function f(x) = x² + 10x + 37 from standard form to vertex form is f(x) = (x + 5)² + 12

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

f(x) = x² + 10x + 37

The above quadratic function is its standard form

f(x) = ax² + bx + c

Start by calculating the axis of symmetry using

h = -b/2a

So, we have

h = -10/2

h = -5

Next, we have

f(-5) = (-5)² + 10(-5) + 37

k = 12

The vertex form is then represented as

f(x) = a(x - h)² + k

So, we have

f(x) = (x + 5)² + 12

Hence, the vertex form is f(x) = (x + 5)² + 12

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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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