Henry will paint 5 birdhouses with 1 quart of paint. How many quarts of paint will Henry use to paint 42 birdhouses? ​

Consider the integral

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx[/tex]

which is the negative of yours. Bit strange to integrate over [tex](\infty,-\infty)[/tex], but if that's what you actually intended, just multiply the final result by -1. Of course, I've already canceled the superfluous factors of [tex]x^2[/tex].

Expand the integrand into partial fractions.

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = \int_{-\infty}^\infty \left(2 + \frac3{1+x^2}\right) e^{-x^2} \, dx[/tex]

Recall that for [tex]\alpha>0[/tex],

[tex]\displaystyle \int_{-\infty}^\infty e^{-\alpha x^2} \, dx = \sqrt{\frac\pi\alpha}[/tex]

Now let

[tex]\displaystyle I(a) = \int_{-\infty}^\infty \frac{e^{-ax^2}}{1+x^2} \, dx[/tex]

Together, these give

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = 2\sqrt\pi + 3I(1)[/tex]

Differentiate [tex]I(a)[/tex] under the integral sign with respect to [tex]a[/tex] to obtain a simple linear differential equation.

[tex]\displaystyle \frac{dI}{da} = -\int_{-\infty}^\infty \frac{x^2 e^{-ax^2}}{1+x^2} \, dx \\\\ ~~~~~~~~ = - \int_{-\infty}^\infty \left(1 - \frac1{1+x^2}\right) e^{-ax^2} \, dx \\\\ ~~~~~~~~ = -\sqrt{\frac\pi a} + I(a)[/tex]

Solve for [tex]I(a)[/tex] with the initial value [tex]I(1) = \sqrt\pi[/tex]. Using an integrating factor,

[tex]\displaystyle \frac{dI}{da} - I(a) = -\sqrt{\frac\pi a} \\\\ e^{-a} \frac{dI}{da} - e^{-a} I(a) = -\sqrt{\frac\pi a}\,e^{-a} \\\\ \frac{d}{da}\left[e^{-a} I(a)\right] = -\sqrt{\frac\pi a}\,e^{-a}[/tex]

By the fundamental theorem of calculus,

[tex]\displaystyle e^{-a} I(a) = e^{-a}I(a)\bigg|_{a=0} - \sqrt\pi \int_0^a \frac{e^{-\xi}}{\sqrt\xi} \, d\xi \\\\ I(a) = \pi e^a - \sqrt\pi \, e^a \int_0^a \frac{e^{-\xi}}{\sqrt\xi} \, d\xi[/tex]

so that

[tex]\displaystyle I(1) = \pi e - \sqrt\pi\,e \int_0^1 \frac{e^{-\xi}}{\sqrt\xi} \, d\xi[/tex]

Substitute [tex]t=\sqrt\xi[/tex].

[tex]\displaystyle I(1) = \pi e - 2\sqrt\pi\,e \int_0^1 e^{-t^2} \, dt[/tex]

Recall the error function,

[tex]\mathrm{erf}(x) = \displaystyle \frac2{\sqrt\pi} \int_0^x e^{-t^2} \, dt[/tex]

which we can use to write

[tex]I(1) = \pi e - 2\sqrt\pi e \cdot \dfrac{\sqrt\pi}2\,\mathrm{erf}(1) = \pi e - \pi e \,\mathrm{erf}(1)[/tex]

Finally, we arrive at

[tex]\displaystyle \int_{-\infty}^\infty \frac{5 + 2x^2}{1 + x^2} e^{-x^2} \, dx = \boxed{2\sqrt\pi + 3\pi e - 3\pi e \, \mathrm{erf}(1)}[/tex]

The height of the box that has a volume of 1250 cubic cm with a length of 20cm and a width of 5cm will be 12.5cm.

It should be noted that the formula for the volume will be:

= Length × Width × Height

Therefore the height will be:

Volume = LWH

where

L = length = 20cm

W = width = 5cm

H = height = unknown

1250 = (20 × 5 × H)

1250 = 100H

Height = 1250/100

Height = 12.5 cm

Therefore, the height is 12.5cm.

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Given that XZ is the perpendicular bisector of segment WY, the numerical value of k is 6.

A bisector divides a segment or angle into two equal halves.

Given the data in the question;

Since XZ is a perpendicular bisector of segment WY,

Segment WX is equal to Segment XY

Segment WX = Segment XY

Plug in the given values and solve for k.

3k + 17 = 7k - 7

Collect like terms

3k - 7k = -7 - 17

-4k = -24

Divide both sides by the coefficient of k

-4k/-4 = -24/-4

k = 6

Given that XZ is the perpendicular bisector of segment WY, the numerical value of k is 6.

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The image of triangle DEF after the dilation with a scale factor of -1/2  is given at the end of the answer.

Transformations in the graph of a function involve operations in the vertices of the image, such as addition, subtraction, division or multiplication.

The transformation used in this problem is a dilation, in which the coordinates of the vertices of the original figure are multiplied by the scale factor.

For this problem, the vertices of the triangle are given as follows:

D(3,6), E(3,2) and F(5,4).

The dilation has a scale factor of -0.5 = -1/2, hence the coordinates of the image are given as follows:

The dilated triangle is given by the graph at the end of this answer.

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The turning point of the function f(x) = |x-2| + 3 will be (3, 2).

Here,

We have to find the turning point of the graph of function,

f(x) = |x−2|+3

What is an absolute function?

The non-negative value of real number without regard to its sign is known as its absolute value.

Now,

The absolute function is;

f(x) = |x+3| -2.

Since, When we plot the graph of the absolute function f(x) = |x+3| -2,

it will give two straight lines which is shown in the graph.

Clearly, The graph of the function changes its position at the coordinate

(3, 2).

Therefore, The absolute function having its turning point at the coordinate ( 3, 2).

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Answer:

Step-by-step explanation:

The length of the pen is 2√3 feet

The formula for calculating the area of a rectangle is expressed according to the formula below:

Area of a right triangle = 1/2 * base * height

A = 1/2 bh

Given the following parameters

Area = 20 square feet

If the height is five times the length, then;

h = 5b

Substitute

A = 1/2 bh

A = 1/2(b)5b

30 = 1/2(5b²)

60 =5b²

b² = 12

b = 2√3

Hence the length of the pen is 2√3 feet

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Answer:

31x+0.5

Step-by-step explanation:

In this problem, we must combine like terms to simplify the expression. We can begin by writing out the expression:

13x + 5 + 17x - 4.5 + x

Now that we have rewritten the expression, we can combine like terms to simplify the expression. This would look like:

13x + 17x + x = 31x

5 - 4.5 = 0.5

The answer would be “31x + 0.5” when simplified.

Answer:

x=29 angle a=63 Ang b=46 and c=71

Step-by-step explanation:

2x+5+x+17+3x-16=180 (sum of angles in triangles)

value of x is 29 substitute it in angle a b c

a=2*29+5

=63

b=29+17 = 46

c=3*29-16 = 71

Answer:

1/8

Step-by-step explanation:

When worked out, all numbers are equal to 1.8 except 1/8. (1/8 is equal to .125)

1/8) 1 divided by 8 is .125

18/10) 18 divided by 10 is 1.8

1.8) 1.8=1.8

1 4/5) 4 divided by 5 is .8, and when you add the 1 it ends up as 1.8

The given system has zero solutions.

Lines that never cross one other are said to be parallel. Hence, a pair of parallel lines must have the same slope but distinct intercepts. (On the other hand, identical lines have same intercept).

In the general equation of a line, y = mx + b, m represents the slope of the line.

The general equations of parallel lines would be of the form:

(1) y = mx + b

(2) y = mx + c

where b and c are any constants.

Now, we observe that the given equations are of the above form with m=2, b=2 and c=0, therefore, they are parallel lines. Since, parallel lines never intersect each other, they have no solution.

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Answer:

  C.  Multiply by 5, then subtract 7.

Step-by-step explanation:

You want to know the two steps required to solve this 2-step linear equation.

The variable x has 7 added to it, and the sum is divided by 5. To find the value of x, we must undo these operations in reverse order.

To undo division by 5, we must multiply by 5.

Then, to undo the addition of 7, we must subtract 7.

Here is the "work":

  [tex]\dfrac{x+7}{5}=10\qquad\text{given}\\\\x+7 = 50\qquad\text{multiply both sides by 5}\\\\x=43\qquad\text{subtract 7 from both sides}[/tex]

The operations in order of use are ...

  Multiply both sides by 5 first, then subtract 7 from both sides.

__

Additional comment

The properties of equality tell you that you must do the same operation to both sides of the equation.

Answer:

7.5 : 3.5

Step-by-step explanation:

ratio of time spent studying in math to studying for English is ;

time spent studying in math. / time spent in studying English

[tex] \frac{7.5}{3.5} [/tex]

= 7.5 : 3.5

The limit dimensions of the part are 3.135 inches and 3.155 inches

The given parameters are

The limit dimensions of the part are calculated as

Limit = Thickness +/- Tolerance

So, we have

Limit = 3.145 inches +/- 0.010 inches

Expand the above expression

So, we have

Limit = (3.145 inches - 0.010 inches, 3.145 inches + 0.010 inches)

Evaluate the sum

Limit = (3.135 inches, 3.155 inches)

Hence, the limit dimensions of the part are 3.135 inches and 3.155 inches

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The  average for a student with 64 on attendance, 82 on the first paper, 61 on the second paper, 72 on test 1, 64 on test 2, 93 on test 3, and 82 on the final exam is 77.31% and the average for a student with the above scores on the papers, tests, and final exam, but with a score of only 29 on attendance is 73.23%.

a. Average:

Average: 82(0.07)+ 61(0.09)+ 72(0.14) +64(0.14) + 93(0.14)+ 82(0.33)÷0.07+0.09+3(0.14)+0.33

Average=5.74+5.49+10.08+8.96+13.02+27.06÷0.91

Average=70.35÷0.91

Average=77.31%

b. Average

Average= 29(0.07)+ 61(0.09)+ 72(0.14) +64(0.14) + 93(0.14)+ 82(0.33)÷0.07+0.09+3(0.14)+0.33

Average=2.03+5.49+10.08+8.96+13.02+27.06÷0.91

Average=66.64÷0.91

Average=73.23%

Therefore the average for both a and b is 77.31% and 73.23%.

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Answer:

Step-by-step explanation:

According to the graph, point A has a negative value, between - 1 and zero.

All the expressions could result in negative value apart from the sum of the squares, which is never negative.

Correct choice is C.

The unit tangent vector for the parametrized curve is [tex]\vec T(t) = - \frac{\sin t}{\sqrt{1 + \cos ^{2} t}}\,\hat{i} + \frac{\cos t}{\sqrt{1 + \cos ^{2} t}}\,\hat{j} + \frac{\cos t}{\sqrt{1 + \cos ^{2} t}}\,\hat{k}[/tex].

In this question we know a three-dimension parametrized curve, of which we must derive its unit tangent vector in accordance with the following formula:

[tex]\vec T(t) = \frac{\vec R'(t)}{\|\vec R'(t)\|}[/tex]

Where:

First, determine the first derivative of the parametrized curve:

[tex]\vec R'(t) = -\sin t \,\hat{i} + \cos t \,\hat{j} + \cos t \,\hat{k}[/tex]

Second, derive the magnitude of the parametrized curve:

[tex]\|\vec R'(t)\|[/tex] = √[(- sin t)² + (cos t)² + (cos t)²]

[tex]\|\vec R'(t)\|[/tex] = √(1 + cos² t)

Third, substitute every term in the unit tangent curve formula:

[tex]\vec T(t) = - \frac{\sin t}{\sqrt{1 + \cos ^{2} t}}\,\hat{i} + \frac{\cos t}{\sqrt{1 + \cos ^{2} t}}\,\hat{j} + \frac{\cos t}{\sqrt{1 + \cos ^{2} t}}\,\hat{k}[/tex]

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The distribution is a probability distribution, and the reasons are

The probability distribution is given as:

X     0   1    2

P(X) 0 1/3 2/3

From the above table of probability distribution, we can see that each probability value is between 0 and 1 (inclusive)

Also, the sum of probabilities is 1

i.e. 0 + 1/3 + 2/3 = 1

Hence, the distribution is a probability distribution, and the reasons are

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Answer:

See work below.

30 sold for full price and 68 sold for a discount.

Step-by-step explanation:

Therefore, the price elasticity of demand by percentage change method is -0.025.

Given the price of a first commodity (p₁) = 6 rs

the price of a second commodity (p₂) = 10 rs

Expenditure on first commodity (s₁) = 120

Expenditure on second commodity (s₂) = 180

We need to find the quantity as

120 = 6 * Q₁.              180 = 10 * Q₂

Q₁ = 120 / 6                 Q₂ = 180 / 10

Q₁ = 20                         Q₂ = 18

price elasticity of demand by percentage change method as

Eₐ = P / Q * ΔQ /ΔP

ΔQ = Q₂ - Q₁ = 18 -20 = -2

ΔP = p₂ - p₁ = 10 - 6 = 4

Eₐ = 6 / 120 * -2 / 4

Eₐ = - 0.025

Therefore, the price elasticity of demand by percentage change method is -0.025.

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Answer: TV costs $828 and DVD player costs $414.

Step-by-step explanation: let x represent the cost of tv while y represents the cost of the player. x+y = 1242. since the tv costs twice as much as the dvd player, we can say that 2y=x. now replace x with 2y in our equation and we get 2y+y= 1242,  3y = 1242. now divide both sides by 3 and y = 414. Now that we now the value of y, just sub it into our original equation, x+y = 1242. x + 414 = 1242. x = 828.

Answer: x=18 ==> C

Step-by-step explanation:

19+15+11=14+15+?

19+11=14+?

30=14+?

?=16

?+11+x=19+15+11

x+?=19+15

x+16=34

x=18 ==> C

Answer:

Step-by-step explanation:

go pathways gives all the answers

The first company shipping charge is 5 $

the second company shipping charge is 2 $

let the size of the box be 'x'

Given the first company charged as below

20$ for medium-sized and an extra 1$ for one pound

=  20x + 1

The second company charged as below

$5 for the same size of the box, plus an additional $4 per pound

= 5x + 4

The price of the two shipping options will be same.

20x + 1 = 5x + 4

20x - 5x = 4- 1

15x = 3

x= 3 / 15

x = 1/5

x = 0.2

∴ 20x+1 = 20 * 0.2 +1 = 5

   5x + 4 = 5 * 0.2 + 1 = 2

Therefore, The first company's shipping fee is $5, whereas the second company's fee is $2.

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14912 years ago, the tree that was used to kill the support pole died which is one-sixth of the carbon-14 of a modern, living tree.

There are four basic properties of logarithms:

logₐ(xy) = logₐx + logₐy.

logₐ(x/y) = logₐx - logₐy.

logₐ(xⁿ) = n logₐx.

logₐx = logₓa / logₓb.

One-sixth of a modern, living tree's carbon-14 is present in an ancient structure.

Assume that the half-life of carbon-14 is 5770 years

As per the given condition, the expression would be

[tex]A(t) = A_o}\dfrac{1}{2} ^{(\frac{t}{5770} )}\\A(t) = (0.167)A_o}[/tex]

[tex](0.167)A_o} = A_o}\dfrac{1}{2} ^{(\frac{t}{5770} )}[/tex]

[tex](0.167) = \dfrac{1}{2} ^{(\frac{t}{5770} )}[/tex]

Take the logarithms of both sides to get:

[tex]log(0.167) =log \dfrac{1}{2} ^{(\frac{t}{5770} )}[/tex]

Using logₐ(xⁿ) = n logₐx.

⇒ log 0.167= (t/5770) log (0.5)

⇒ log 0.167/ log (0.5)= (t/5770)

⇒ 3 = (t/5770)

⇒ t = 3 × 5770

⇒ t = 14912

Hence, 14912 years ago, the tree that was used to kill the support pole died.

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The total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27.

Interest is the sum of money paid for using someone else's funds. You must pay interest when you borrow money from lenders. You receive interest when you lend money to borrowers. It could be stated in terms of money or the rate of payment. You will learn more about interest in this article, including the different sorts of interest, what they are, and how to determine how much interest will be charged on a loan or a loan amount.

Given that,

At the beginning of each of her four years in college

Miranda took out a new Stafford loan.

Each loan had a principal of $5,500,

An interest rate of 7.5% compounded monthly

A duration of ten years.

Miranda paid off each loan by making constant monthly payments, starting with when she graduated.

All of the loans were subsidized.

The total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27

Therefore, the total lifetime cost for Miranda to pay off her 4 loans is: $31,337.27

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Answer: D 31,337.27

Step-by-step explanation:

Answer:

x=26, y=90

Step-by-step explanation:

This is what I did,

I knew that y = 90 because it is a Right Angle. I subtracted 90 - 64 and that got me = y 26