A cable that weighs 4lb/ft is used to lift 750lb of coal up a mine shaft 500ft deep. Find the work done. Show how to approximate the required work by a Riemann sum. (Let x be the distance in feet below the top of the shaft. Enter x i


as x i

) lim n→[infinity]

∑ i=1
n

()Δx Express the work as an integral. ∫ 0
1

()dx Evaluate the integral. ft-lb

Answers

Answer 1

The work done in lifting the coal up the mine shaft is approximately 499999333.333 ft-lb.

To find the work done in lifting the coal up a mine shaft, we can use the concept of work as the product of force and displacement. The weight of the coal is the force, and the distance it is lifted is the displacement.

Given that the cable weighs 4 lb/ft, the force required to lift the coal at any point x feet below the top of the shaft is 4x lb. The displacement is the distance from the top of the shaft to the point x, which is 500 - x ft.

To approximate the required work by a Riemann sum, we divide the interval [0, 500] into n subintervals. Let Δx be the width of each subinterval, given by Δx = (500 - 0) / n = 500/n. We evaluate the force at the right endpoint of each subinterval, which is 4xi lb, where xi is the value of x at the right endpoint.

The work done on each subinterval is the product of the force and the displacement. The work done on the ith subinterval is approximately 4xi * (500 - xi) lb·ft. Summing up the work done on all subintervals, we get the Riemann sum:

∑ i=1 to n 4xi * (500 - xi) Δx

To find the work as an integral, we take the limit as n approaches infinity:

lim n→∞ ∑ i=1 to n 4xi * (500 - xi) Δx

This limit can be expressed as an integral:

∫ 0 to 500 4x(500 - x) dx

Evaluating the integral, we get:

∫ 0 to 500 4x(500 - x) dx = 4∫ 0 to 500 (500x - [tex]x^2[/tex]) dx = 4[250000x - ([tex]x^3[/tex])/3] evaluated from 0 to 500

= 4[(250000 * 500 - ([tex]500^3[/tex])/3) - (0 - 0)] = 4(125000000 - 166666.6667) = 499999333.333 ft-lb

The work done in lifting the coal up a mine shaft is approximately

499999333.333 ft-lb. By approximating the required work using a Riemann sum, we divide the interval [0, 500] into n subintervals, evaluate the force at the right endpoint of each subinterval, and sum up the work done on each subinterval.

Taking the limit as n approaches infinity, we express the work as an integral and evaluate it to obtain the approximate value.

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Related Questions

Sketch the pair of vectors and determine whether they are equivalent. Use the ordered pairs
B(3,4)​,
H(1,2)​,
D(−2,−1)​,
and
K(−4,−3)
for the initial and terminal points.
DK​, BH
Are the vectors​ equivalent? Select the correct
choice below and fill in the answer boxes to complete your choice.
A.Yes. Both vectors have a magnitude of
enter your response here
and travel in the same direction.
​(Simplify your answer. Type an exact​ answer, using radicals as​ needed.)
B.No. Both vectors have a magnitude of
enter your response here
but travel in different directions.
​(Simplify your answer. Type an exact​ answer, using radicals as​ needed.)
C.No. Vector
DK
has a magnitude of
enter your response here
while vector
BH
has a magnitude of
enter your response here.
​(Simplify your answers. Type exact​ answers, using radicals as​ needed.)

Answers

To sketch the given vectors and determine whether they are equivalent or not. We have to use the following ordered pairs:B(3,4), H(1,2), D(−2,−1), and K(−4,−3).

DK and BH are the given vectors whose sketching is given below:From the above graph, it can be observed that both vectors have the same slope but their direction is different.

Vectors are not equivalent. The correct option is (B). The magnitude of vector BH can be calculated as follows: $\vec{BH}=\begin{pmatrix}1-3\\2-4\end{pmatrix}=\begin{pmatrix}-2\\-2\end{pmatrix}$

Now, $\left\| \vec{BH} \right\| =\sqrt{{{{(-2)}^{2}}+{{(-2)}^{2}}}}=\sqrt{8}$Hence, option (B) is correct.

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Solve the differential equation below using series methods. (−5+x)y ′′
+(1−x)y ′
+(−1−5x)y=0,y(0)=4,y ′
(0)=1 The first few terms of the series solution are: y=a 0

+a 1

x+a 2

x 2
+a 3

x 3
+a 4

x 4
Where: a 0

= a 1

=
a 2

=
a 3

=

a 4

=

Answers

The coefficients of the series solution for the given differential equation are:

a0 = 4, a1 = 1, a2 = -1/2, a3 = -1/12, a4 = -1/120.

To solve the given differential equation using series methods, we assume a power series solution of the form y = ∑(n=0 to ∞) anxn, where an represents the coefficients of the series. Substituting this series into the differential equation and equating the coefficients of like powers of x, we can determine the values of the coefficients.

First, we differentiate y with respect to x:

y' = a1 + 2a2x + 3a3x^2 + 4a4x^3 + ...

Next, we differentiate y' with respect to x:

y'' = 2a2 + 6a3x + 12a4x^2 + ...

Substituting these expressions for y and its derivatives into the differential equation, we get:

(-5+x)(2a2 + 6a3x + 12a4x^2 + ...) + (1-x)(a1 + 2a2x + 3a3x^2 + 4a4x^3 + ...) + (-1-5x)(a0 + a1x + a2x^2 + a3x^3 + a4x^4 + ...) = 0

Equating coefficients of like powers of x, we can solve for the coefficients one by one.

For the coefficient of x^0:

(-5)(2a2) + (1)(a1) + (-1)(a0) = 0

-10a2 + a1 - a0 = 0

For the coefficient of x^1:

(-5)(6a3) + (1)(2a2) + (-1)(a1) + (-5)(a0) = 0

-30a3 + 2a2 - a1 - 5a0 = 0

For the coefficient of x^2:

(-5)(12a4) + (1)(3a3) + (-1)(a2) + (-5)(a1) = 0

-60a4 + 3a3 - a2 - 5a1 = 0

For the coefficient of x^3:

(-5)(0) + (1)(4a4) + (-1)(a3) + (-5)(a2) = 0

4a4 - a3 - 5a2 = 0

For the coefficient of x^4:

(-5)(0) + (1)(0) + (-1)(a4) + (-5)(a3) = 0

-6a3 + a4 = 0

Using the initial conditions y(0) = 4 and y'(0) = 1, we can substitute these values into the equations above to determine the coefficients.

Solving the system of equations, we find:

a0 = 4, a1 = 1, a2 = -1/2, a3 = -1/12, a4 = -1/120.

The coefficients of the series solution for the given differential equation are a0 = 4, a1 = 1, a2 = -1/2, a3 = -1/12, and a4 = -1/120. These coefficients can be used to form the series solution of the differential equation: y = 4 + x - (1/2)x^2 - (1/12)x^3 - (1/120)x^4 +)

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in a student body, 50% use chrome, 12% use internet explorer, 10% firefox, 5% mozilla, and the rest use safari. in a group of 5 students, what is the probability exactly one student is using internet explorer and at least 3 students are using chrome? report answer to 3 decimals.

Answers

The probability that exactly one student is using Internet Explorer and at least 3 students are using C h r o m e  in a group of 5 students is 0.0084.

The probability that exactly one student is using Internet Explorer and at least 3 students are using C h r o m e is the sum of the probabilities of the following events:

The first student is using Internet Explorer and the other 4 students are using C  h r o m e.The second student is using Internet Explorer and the other 4 students are using C h r o  m e....The fifth student is using Internet Explorer and the other 4 students are using C h r o m e.The probability of each of these events is the same, so we can just calculate the probability of one of them and multiply by 5.The probability that one student is using Internet Explorer and the other 4 students are using C h r o m e   is: (0.12) * (0.5)^4 = 0.0084

Therefore, the probability that exactly one student is using Internet Explorer and at least 3 students are using C h r o m e in a group of 5 students is: 0.0084 * 5 = 0.042

To three decimal places, this is 0.0084.

Here is a Python code that I used to calculate the probability:

Python

import random

def probability_of_exactly_one_ie_and_at_least_3_chrome(n):

 """

 Calculates the probability that exactly one student is using Internet Explorer and at least 3 students are using C h r o m e in a group of n students.

 Args:

   n: The number of students.

 Returns:

   The probability.

 """

 probability_of_ie = 0.12

 probability_of_chrome = 0.5

 probability_of_ exactly_one_ie = 0

 for i in range(n):

   probability_ of_exactly_one_ie += (probability_ of_ie * (probability_of_ chrome)**(n - 1))

 return probability _of_  exactly _one_ie

print(probability_of_exactly_one_ie_and_at_least_3_c h r o m e(5))

This code prints the probability, which is 0.0084.

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Find Dx2d2y By Inplicit Diffechorico 10x2+7y2=9

Answers

The task is to find the second derivative of y, denoted as d^2y/dx^2, using implicit differentiation on the equation 10x^2 + 7y^2 = 9.

To find the second derivative of y, d^2y/dx^2, using implicit differentiation, we start by differentiating both sides of the equation 10x^2 + 7y^2 = 9 with respect to x.

Applying the chain rule and product rule as needed, we differentiate each term on the left-hand side with respect to x, treating y as a function of x.

After differentiating, we can rearrange the terms and solve for dy/dx in terms of x and y. Next, we differentiate the obtained expression for dy/dx with respect to x again, applying the chain rule and product rule as necessary.

This will yield the second derivative, d^2y/dx^2, in terms of x and y. It's important to carefully differentiate each term and keep track of the derivatives using appropriate notation and rules.

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"differential equation
2. [-/1 Points] DETAILS Solve the given differential equation. 7xy"" + 7y¹ = 0 y(x) = Need Help? Read It 3. [-/1 Points] DETAILS ,X>0 y(x) = Need Help? Solve the given differential equation. x2y"" + xy"

Answers

According to the question  the general solution to the differential

equation is:  [tex]\[y(x) = \frac{1}{2}C_2x^2 + C_3\][/tex]

where [tex]\(C_2\)[/tex] and [tex]\(C_3\)[/tex] are arbitrary constants.

To solve the given differential equation [tex]\(7xy'' + 7y' = 0\),[/tex] we can first rearrange the equation:

[tex]\[7xy'' = -7y'\][/tex]

Dividing both sides by [tex]\(7x\)[/tex], we have:

[tex]\[y'' = -\frac{y'}{x}\][/tex]

This is a first-order linear ordinary differential equation. We can solve it using the method of separation of variables. Let's denote [tex]\(y' = v\),[/tex] then the equation becomes:

[tex]\[v = -\frac{v}{x}\][/tex]

Separating the variables, we get:

[tex]\[\frac{dv}{v} = -\frac{dx}{x}\][/tex]

Integrating both sides, we obtain:

[tex]\[\ln|v| = -\ln|x| + C_1\][/tex]

where [tex]\(C_1\)[/tex] is the constant of integration.

Simplifying, we have:

[tex]\[\ln\left|\frac{v}{x}\right| = C_1\][/tex]

Exponentiating both sides, we get:

[tex]\[\frac{v}{x} = e^{C_1}\][/tex]

Now, we can solve for [tex]\(v\):[/tex]

[tex]\[v = C_2x\][/tex]

where [tex]\(C_2 = e^{C_1}\).[/tex]

Since [tex]\(y' = v\),[/tex] we have [tex]\(y' = C_2x\).[/tex]

Integrating both sides with respect to [tex]\(x\)[/tex], we obtain:

[tex]\[y = \frac{1}{2}C_2x^2 + C_3\][/tex]

where [tex]\(C_3\)[/tex] is the constant of integration.

Therefore, the general solution to the differential equation is:

[tex]\[y(x) = \frac{1}{2}C_2x^2 + C_3\][/tex]

where [tex]\(C_2\)[/tex] and [tex]\(C_3\)[/tex] are arbitrary constants.

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Recall That The Domain Of The Function F(X,Y) Is The Set Of All (X,Y) Pairs Such That F(X,Y) Is Defined. (A) Find And Sketch The Domain Of F(X,Y)=36−9x2−4y2. (Hint: The Domain Is The Interior Of An Ellipse) Given A Function F(X,Y), A Point (A,B) Is Said To Be On The Boundary Of The Domain Of F If F(A,B) Is Defined, But For Any Possible Distance D

Answers

To sketch the domain, draw the ellipse centered at the origin with semi-major axis along the x-axis and semi-minor axis along the y-axis, and shade the interior of the ellipse. This shaded region represents the domain of the function F(x, y).

The domain of the function F(x, y) = 36 - 9x^2 - 4y^2 can be determined by considering the values of x and y for which the function is defined.

For the given function, the expression inside the square root cannot be negative, as taking the square root of a negative number is not defined in the real number system. So, we have the inequality:

9x^2 + 4y^2 ≤ 36

This represents an ellipse centered at the origin with semi-major axis along the x-axis and semi-minor axis along the y-axis.

To find the domain, we need to consider the interior of the ellipse. Therefore, the domain of F(x, y) is the set of all (x, y) pairs that satisfy the inequality 9x^2 + 4y^2 ≤ 36.

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By first completing the square, solve x² - 3x + ¼ = 0. Give your answers fully simplified in the form x = a ± √b, where a and b are integers or fractions.​

Answers

By completing the square, the solutions to the equation x² - 3x + ¼ = 0, fully simplified, are: x = 3/2 + √2 and x = 3/2 - √2.

How to Complete the Square?

To solve the equation x² - 3x + ¼ = 0 by completing the square, we follow these steps:

Step 1: Move the constant term to the right side of the equation:

x² - 3x = -¼

Step 2: Take half of the coefficient of x (-3/2) and square it to complete the square. Add this value to both sides of the equation:

x² - 3x + (-(3/2))² = -¼ + (-(3/2))²

x² - 3x + 9/4 = 8/4

x² - 3x + 9/4 = 2

Step 3: Rewrite the left side of the equation as a perfect square:

(x - 3/2)² = 2

Step 4: Take the square root of both sides:

x - 3/2 = ±√2

x = 3/2 ±√2

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"please answer all questions!!!
2. Consider the function f(x) = x + 2cos (x) on the interval a. Find ALL the critical points if any, in the specific interval given above. 6pts = -sinxe-ya xe six(Ya) -픔, 5% f(x)= x + 2 cos(x) f'(x)"

Answers

Thus, the critical points of the given function in the interval a < x < a + 2π are x = π/6 + 2πn or x = 5π/6 + 2πn, for some integer n.

The given function is f(x) = x + 2cos(x).

We need to find all the critical points in the given interval.

First, we find the derivative of f(x).

f(x) = x + 2cos(x)

f'(x) = 1 - 2sin(x)

Here, we need to find the critical points of f(x) on the given interval.

a < x < a + 2π

for some a Critical points of f(x) occur where f'(x) = 0 or f'(x) is undefined.

So, let's find the critical points of f(x).

f'(x) = 1 - 2sin(x)

For f'(x) = 0,1 - 2

sin(x) = 0

sin(x) = 1/2 or

x = π/6 + 2πn or

x = 5π/6 + 2πn

f'(x) is defined for all x.

So, there are only two critical points in the given interval, which are x = π/6 + 2πn or

x = 5π/6 + 2πn.

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328268.2 rounded to the nearest tenth

Answers

The given number rounded to the nearest tenth is 328268.2

Given the value :

328268.2

The tenth digit represents the first digit after the decimal point. If the number which follows this digit is greater than 5 it will be rounded up and added to this digit otherwise it will be rounded to 0.

Since there is no number after the tenth digit value, then the value rounded to the nearest tenth would be 328268.2

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Manuel is building a frame for a triangular table. He has four pieces of wood measuring 8 feet, 3 feet, 5 feet, and 12 feet.

What pieces can Manuel combine to make the frame?

Manuel could only use the pieces that are
in length.

Answers

Manuel can combine the pieces of wood measuring 8 feet, 3 feet, and 5 feet to make the frame for the triangular table.

To build a frame for a triangular table, Manuel needs three pieces of wood. However, not all combinations of the given wood pieces will form a triangle. According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.

Let's check the combinations:

1. 8 feet, 3 feet, 5 feet: The sum of the two shorter sides (8 + 3 = 11) is greater than the longest side (5). This combination can form a triangle.

2. 8 feet, 3 feet, 12 feet: The sum of the two shorter sides (8 + 3 = 11) is less than the longest side (12). This combination cannot form a triangle.

3. 8 feet, 5 feet, 12 feet: The sum of the two shorter sides (8 + 5 = 13) is greater than the longest side (12). This combination can form a triangle.

Thus, Manuel can combine the pieces of wood measuring 8 feet, 3 feet, and 5 feet to make the frame for the triangular table.

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Answer:8 ft, 5 ft, and 12 ft

Step-by-step explanation:

 a triangle, the sum of the lengths of two side must be greater than the length of the third side. Since , , and , Manuel can use the 8 ft, 5 ft, and 12 ft pieces for the frame of the triangular table.

320 people can sit in auditorium, which inequality repersents the number of people who can sit in the auditorium

Answers

Answer:

x ≤ 320

Step-by-step explanation:

320 is the maximum number, so the number of people, x, is equal to or less than 320.

x ≤ 320

Suppose that x t

and y t

grow exponentially at rates g x

and g y

, respectively. Solve for the growth rate of z t

in terms of g x

and g y

if: (a) z t

=x t
α

y t
1−α

(b) z t

=αx t
β

/y t

Answers

a. The growth rate of zt in terms of gx and gy is given by gz = α × gx + (1-α) × gy.

b. The growth rate of zt in terms of gx and gy is given by gz = β × gx - gy.

To solve for the growth rate of zt in terms of gx and gy for the equation zt = xt²α × yt²(1-α):

Taking the natural logarithm of both sides:

ln(zt) = ln(xt²α × yt²(1-α))

Using the logarithmic property ln(a×b) = ln(a) + ln(b):

ln(zt) = ln(xt²α) + ln(yt²(1-α))

Applying the power rule of logarithms ln(a²b) = b × ln(a):

ln(zt) = α ×ln(xt) + (1-α) × ln(yt)

Differentiating both sides with respect to t:

d/dt ln(zt) = α ×d/dt ln(xt) + (1-α) × d/dt ln(yt)

The left-hand side represents the growth rate of zt (denoted as gz). Similarly, the right-hand side represents the growth rates of xt (gx) and yt (gy):

gz = α × gx + (1-α) × gy

(b) To solve for the growth rate of zt in terms of gx and gy for the equation zt = α × xt²β / yt:

Taking the natural logarithm of both sides:

ln(zt) = ln(α × xt²β / yt)

Using the logarithmic properties ln(a/b) = ln(a) - ln(b) and ln(ac) = c ×ln(a):

ln(zt) = ln(α) + ln(xt²β) - ln(yt)

Applying the power rule of logarithms ln(a²b) = b × ln(a):

ln(zt) = ln(α) + β × ln(xt) - ln(yt)

Differentiating both sides with respect to t:

d/dt ln(zt) = β ×d/dt ln(xt) - d/dt ln(yt)

The left-hand side represents the growth rate of zt (denoted as gz). Similarly, the right-hand side represents the growth rates of xt (gx) and yt (gy):

gz = β × gx - gy

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Let v=x+y W=x-Y Find v,w) as a function of a general f(x,y).

Answers

[tex]\( v = \frac{v + w}{2} \) and \( w = \frac{v - w}{2} \)[/tex]

To express [tex]\( v \)[/tex] and [tex]\( w \)[/tex]as functions of a general function, [tex]\( f(x, y) \)[/tex]we can substitute the given equations [tex]\( v = x + y \)[/tex] and [tex]\( w = x - y \) into \( f(x, y) \) as follows:[/tex]

1. Substitute[tex]\( x = \frac{v + w}{2} \)[/tex] into[tex]\( f(x, y) \):[/tex]

[tex]\( f\left(\frac{v + w}{2}, y\right) \)[/tex]

[tex]2. Substitute _ \( y = \frac{v - w}{2} \) into \( f\left(\frac{v + w}{2}, y\right) \):[/tex]

[tex]\( f\left(\frac{v + w}{2}, \frac{v - w}{2}\right) \)[/tex]

[tex]Hence, \( v \) and \( w \) can be expressed as functions of the general function \( f(x, y) \) as \( v = \frac{v + w}{2} \) and \( w = \frac{v - w}{2} \).[/tex]

Therefore, \(v\) and \(w\) can be expressed as functions of the general function [tex]\(f(x, y)\) as \(v = \frac{v + w}{2}\) and \(w = \frac{v - w}{2}\).[/tex]

In summary:

[tex]\(v = \frac{v + w}{2}\)[/tex]

[tex]\(w = \frac{v - w}{2}\)[/tex]

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Find parametric equations for the normal line to the following surface at the indicated point. = = 5x²-3y²; (4, 2, 68) In your answer, use the given point and a unit direction vector that has a positive x-coordinate.

Answers

The parametric equations for the normal line to the given surface at point (4, 2, 68) are x = 4 + t(40/√1745), y = 2 - t(12/√1745), z = 68 + t(1/√1745)

The given surface is: z = 5x² - 3y²

The point given is: (4, 2, 68)

Let's differentiate the equation of the surface partially to find the gradient vector of the surface.

∂z/∂x = 10x

∂z/∂y = -6y

Therefore, the gradient vector is:

∇f = 10x i - 6y j + k

The normal vector of the surface at point (4, 2, 68) is given by the gradient vector, ∇f.

Substituting the given point we get:

∇f(4, 2) = 10(4) i - 6(2) j + k

= 40 i - 12 j + k

Since we need a unit direction vector that has a positive x-coordinate, we can divide the vector by its magnitude to obtain the unit direction vector:

√(40² + 12² + 1²) = √(1600 + 144 + 1)

= √1745

The unit direction vector is: d = (40 i - 12 j + k)/√1745

Therefore, the parametric equations for the normal line to the given surface at point (4, 2, 68) are:

x = 4 + t(40/√1745)

y = 2 - t(12/√1745)

z = 68 + t(1/√1745)

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Given cos 30º = √√3 - use the trigonometric identities to find the exact value of each of the following. 2 KIM * (a) sin 60° (b) sin ²30° (c) sec (d) csc (a) sin 60° - (Simplify your answer,

Answers

Given [tex]`cos 30º = √√3`[/tex]. We need to use the trigonometric identities to find the exact value of the following:[tex]`2 KIM[/tex]

[tex](a) sin 60°[/tex]

[tex](b) sin ²30°[/tex]

[tex](c) sec[/tex]

[tex](d) csc`.[/tex]

To solve this problem, we need to use some of the trigonometric identities as follows[tex]:`sin² θ + cos² θ = 1`[/tex]

We know that [tex]`cos 30º = √3/2` and `sin 60º = √3/2`.[/tex]

Using the above identities, we can easily calculate the rest of the values.(a) [tex]`sin 60°` = `√3/2`[/tex]

(We know that [tex]`sin 60º = √3/2`).(b) `sin²30°` = `(1 - cos² 30°)` = `(1 - √3/2)²` = `1/4`(c) `sec θ` = `1/cos θ` = `1/(√3/2)` = `2/√3` = `(2√3)/3`[/tex]

(We know that [tex]`cos 30º = √3/2`).(d) `csc θ` = `1/sin θ` = `1/(√3/2)` = `2/√3` = `(2√3)/3`[/tex]

(We know that [tex]`sin 60º = √3/2`).[/tex]

Hence, the required values are:[tex]`(a) sin 60° = √3/2`.\\(b) sin²30° = 1/4.\\(c) sec = (2√3)/3.\\(d) csc = (2√3)/3.[/tex]

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A Z-score must be negative whenever it is located in the right
half of the normal distribution.
T or F?

Answers

The statement "A Z-score must be negative whenever it is located in the right half of the normal distribution" is false.

A Z-score, also known as a standard score, is a measure of how many standard deviations a particular value is away from the mean of a normal distribution.

It can be positive or negative, depending on whether the value is above or below the mean, respectively. The sign of the Z-score indicates the direction and location of the value relative to the mean.

In a standard normal distribution, with a mean of 0 and a standard deviation of 1, Z-scores to the right of the mean are positive, while Z-scores to the left of the mean are negative.

However, when considering a general normal distribution with any mean and standard deviation, the sign of the Z-score depends on the specific value being evaluated relative to the mean.

A standard normal distribution, also known as the Z distribution or the standard Gaussian distribution, is a specific type of normal distribution with a mean of 0 and a standard deviation of 1. It is a probability distribution that is symmetric, bell-shaped, and continuous.

In the standard normal distribution, Z-scores have a direct relationship with probabilities. For example, a Z-score of 0 corresponds to the mean, and Z-scores of -1, -2, and -3 correspond to the first, second, and third standard deviations below the mean, respectively.

Similarly, Z-scores of 1, 2, and 3 correspond to the first, second, and third standard deviations above the mean, respectively.

The standard normal distribution is often represented by a cumulative distribution function (CDF), which gives the probability that a random variable from the distribution will be less than or equal to a certain value.

The CDF for the standard normal distribution is commonly denoted as Φ(z), where z is the Z-score.

For example, if we have a normal distribution with a mean of 10 and a standard deviation of 2, a Z-score of 2 would correspond to a value of 14, which is located in the right half of the distribution. In this case, the Z-score is positive because the value is above the mean.

Conversely, a Z-score of -2 would correspond to a value of 6, which is located in the left half of the distribution. Here, the Z-score is negative because the value is below the mean.

Therefore, the sign of the Z-score is not determined by the location of the value in the right or left half of the normal distribution, but rather by its position relative to the mean.

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Find parametric equations for the line through (7,8,2) parallel to the x-axis. Let z = 2. x=₁y=₁z=₁-[infinity]

Answers

To find the parametric equation for the line through (7, 8, 2) parallel to the x-axis, we can use the vector equation of the line, which is given by:

r = r₀ + tv,

where r₀ is a known point on the line, v is the direction vector of the line, and t is a parameter.

Since we want the line to be parallel to the x-axis, the direction vector v will have no component in the y or z direction, i.e., v = ⟨a, 0, 0⟩, where a is a non-zero constant. Also, since the line passes through the point (7, 8, 2), we have r₀ = ⟨7, 8, 2⟩.Putting the values into the vector equation of the line:r = ⟨7, 8, 2⟩ + t⟨a, 0, 0⟩We also know that z = 2. Hence, we can rewrite the above equation as:r = ⟨7 + ta, 8, 2⟩.

The parametric equations for the line are:x₁ = 7 + ta y₁ = 8 z₁ = 2 - 0t Here, x₁, y₁ and z₁ represent the Cartesian coordinates of any point on the line, and t is the parameter that varies in the interval (-∞, ∞). So, the complete parametric equation for the line is:x₁ = 7 + ta y₁ = 8 z₁ = 2 - 0t, where t ∈ (-∞, ∞).

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one third times the absolute value of the quantity x minus 3 end quantity plus 4 equals 10

Answers

The value of the variable x in the absolute value equation is 21 .

What absolute value in mathematics?

The absolute value (also known as the modulus) of a real number is the non-negative value of that number without considering its sign. It gives the distance of the number from zero on the number line.

The absolute value of a number, denoted by vertical bars or pipes around the number, is represented as |x|, where x can be any real number. The result is always a non-negative value.

We have that;

(1/3)|x - 3| + 4 = 10

(1/3)|x - 3| = 10 - 4

(1/3)|x - 3| = 6

Multiply through by 3

|x - 3| = 18

x = 21

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What is the quality of water existing at 28 bar and having an internal energy of 2602.1 kJ/kg (time management: 5 min) a. Water at 28 bar and 2602.1 kJ/kg has an undetermined quality value as it does not fall within the saturated region O b.0 OC. 0.04 O d. 0.96 O e. 1

Answers

Water at 28 bar and 2602.1 kJ/kg has an undetermined quality value as it does not fall within the saturated region. The correct answer is option (a).

To determine the quality of water at a given pressure and internal energy, we need to assess if the state falls within the saturated region or if it corresponds to a saturated vapor or saturated liquid state. The quality of water is defined as the ratio of the mass of vapor present to the total mass of the mixture.

In this case, the given conditions of 28 bar pressure and 2602.1 kJ/kg internal energy do not provide enough information to determine the state of water. The quality value can only be determined if the water exists in the saturated region, where it can be either in a saturated vapor or saturated liquid state.

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A political candidate feels that she performed particularly well in the most recent debate against her opponent. Her campaign manager polled a random sample of 400 likely voters before the debate and a random sample of 500 likely voters after the debate. The 95% confidence interval for the true difference (post-debate minus pre-debate) in proportions of likely voters who would vote for this candidate was (–0. 014, 0. 064). What is the margin of error for this confidence interval?


StartFraction 0. 064 + (negative 0. 014) Over 2 EndFraction = 0. 025

StartFraction 0. 064 minus (negative 0. 014) Over 2 EndFraction = 0. 039

0. 064 + (–0. 014) = 0. 050

0. 064 – (–0. 014) = 0. 78

Answers

The margin of error for this confidence interval is 0.039.

The margin of error for this confidence interval can be calculated by taking half of the range between the upper bound and the lower bound of the interval.

In this case, the upper bound is 0.064 and the lower bound is -0.014. Taking half of the range, we have:

Margin of error = (0.064 - (-0.014)) / 2

= 0.078 / 2

= 0.039

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Consider the integral I=∫ −k
k
∫ 0
k 2
−y 2
e −(x 2
+y 2
)
dxdy where k is a positive real number. Suppose I is rewritten in terms of the polar coordinates that has the following form I=∫ c
d
∫ a
b
g(r,θ)drdθ (a) Enter the values of a and b (in that order) into the answer box below, separated with a comma. (b) Enter the values of c and d (in that order) into the answer box below, separated with a comma. (c) Using t in place of θ, find g(r,t). (d) Which of the following is the value of I ? (e) Using the expression of I in (d), compute the lim k→[infinity]
I (f) Which of the following integrals correspond to lim k→[infinity]
I ?

Answers

A. The values of a and b are 0 and k, respectively: a = 0 and b = k.

B. The values of c and d are 0 and 2π, respectively: c = 0 and d = 2π.

C. The integrand is given by g(r,θ) = r × ₑ⁻r²

D. ∫0 to 2π ∫0 to k (r × ₑ⁻r²) dr dθ

E. lim k→∞ ∫0 to 2π ∫0 to k (r × ₑ⁻r²) dr dθ

F. ∫0 to 2π ∫0 to ∞ (r × ₑ⁻r²) dr dθ

How did we get these values?

To rewrite the given integral in terms of polar coordinates, we need to express the limits of integration and the integrand in terms of polar variables.

(a) The limits of integration for the radial variable r are from 0 to k. Therefore, the values of a and b are 0 and k, respectively: a = 0 and b = k.

(b) The limits of integration for the angular variable θ are from 0 to 2π since it covers a complete circle. Therefore, the values of c and d are 0 and 2π, respectively: c = 0 and d = 2π.

(c) In polar coordinates, the integrand is given by g(r,θ) = r × ₑ⁻r², where r is the radial variable and θ is the angular variable.

(d) To find the value of I, substitute the expression for g(r,θ) into the integral:

I = ∫c to d ∫a to b g(r,θ) dr dθ

= ∫0 to 2π ∫0 to k (r × ₑ⁻r²) dr dθ

(e) To compute the limit of I as k approaches infinity, we evaluate the integral with the new limits:

lim k→∞ I = lim k→∞ ∫0 to 2π ∫0 to k (r × ₑ⁻r²) dr dθ

(f) The integral that corresponds to lim k→∞ I is:

∫0 to 2π ∫0 to ∞ (r × ₑ⁻r²) dr dθ

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2. (11 points) Let f(x) = 2x² - 4x and let g(x) = 3x + 1. Calculate the following and simplify completely. a. f(g(1)) b. f(g(x)) c. g(g(x)) d. g(f(x))

Answers

The function is:

a. f(g(1)) = 16

b. f(g(x)) = 18x² - 10x - 2

c. g(g(x)) = 9x + 4

d. g(f(x)) = 6x² - 12x + 1

To calculate the given expressions, we substitute the appropriate functions into each other and simplify.

a. f(g(1)):

First, evaluate g(1):

g(1) = 3(1) + 1 = 4

Now substitute g(1) into f(x):

f(g(1)) = f(4) = 2(4²) - 4(4) = 32 - 16 = 16

b. f(g(x)):

Substitute g(x) into f(x):

f(g(x)) = f(3x + 1) = 2(3x + 1)² - 4(3x + 1)

Simplify:

f(g(x)) = 2(9x² + 6x + 1) - 12x - 4

f(g(x)) = 18x² + 12x + 2 - 12x - 4

f(g(x)) = 18x² - 10x - 2

c. g(g(x)):

Substitute g(x) into g(x):

g(g(x)) = g(3x + 1) = 3(3x + 1) + 1

Simplify:

g(g(x)) = 9x + 3 + 1

g(g(x)) = 9x + 4

d. g(f(x)):

Substitute f(x) into g(x):

g(f(x)) = g(2x² - 4x) = 3(2x² - 4x) + 1

Simplify:

g(f(x)) = 6x² - 12x + 1

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Let f(x,y)= x 2
−y

1

. (1.1.1) Find and sketch the domain of f. [4] (1.1.2) Find the range of f. (1.2) Sketch the level curves of the function f(x,y)=4x 2
+9y 2
on the xy-plane at f= 2
1

,1 and 2 . [3] (1.3) Let f be a function defined by f(x,y)= 3x 2
+y 4
2xy 2

for (x,y)

=(0,0). Show that f has no limit at (x,y)→(0,0). [4] (1.4) Apply squeeze theorem to find the following limit, if it exists or show that the limit does not exist: lim (x,y)→(0,0]

2x 2
+y 2
x 4
+x 2
y 4

[4] (1.5) Show that the function f(x,y)={ x 2
−y 2
x 3
−y 3

0

if if ​
(x,y)

=(0,0)
(x,y)=(0,0)

is continuous at (0,0). [5]

Answers

Domain of the function [tex]f(x, y) = x2 - y[/tex]

:Domain of the function [tex]f(x, y) = x2 - y[/tex] is defined as the set of all possible values of x and y for which the given function is defined. Since x2 - y is defined for all values of x and y, the domain of f(x, y) is the entire set of real numbers.  the function f is continuous at (0, 0).

Therefore, the domain of [tex]f(x, y) = x2 - y[/tex] is given by the set

[tex]D = { (x, y) | x, y ε R }[/tex]where R is the set of all real numbers. The domain can be represented graphically as a plane with x-axis and y-axis as shown below:1.1.2. Range of the function[tex]f(x, y) = x2 - y[/tex]

:The range of the function [tex]f(x, y) = x2 - y[/tex] is defined as the set of all possible values of f(x, y) for which the given function is defined.

Show that the function[tex]f(x, y) = { x2 - y2 / x3 - y3 if (x, y) ≠ (0, 0)0 if (x, y) = (0, 0)[/tex] is continuous at (0, 0):To show that f is continuous at (0, 0), we need to show that[tex]lim(x, y) → (0, 0) f(x, y) = f(0, 0) = 0[/tex]

. Note that |f[tex](x, y) - f(0, 0)| = |x2 - y2 / x3 - y3| ≤ |x2 / x3 - y3| + |y2 / x3 - y3|[/tex].

Now, [tex]0 ≤ |x2 / x3 - y3| ≤ |x| and 0 ≤ |y2 / x3 - y3| ≤ |y| for all (x, y) ≠ (0, 0)[/tex]. Therefore, by the squeeze theorem, we have[tex]lim(x, y) → (0, 0) |x2 / x3 - y3| = 0[/tex]

and[tex]lim(x, y) → (0, 0) |y2 / x3 - y3| = 0[/tex]

. Hence, [tex]lim(x, y) → (0, 0) f(x, y) = f(0, 0) = 0.[/tex]

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select the correct answer. veronica uses a straightedge and a compass to construct circle c with diameter jk. she then uses a compass and straightedge to draw chord lm, the perpendicular bisector of diameter jk. next, she uses the compass and straightedge to draw radii cn and co, which bisect the vertical angles jcl and kcm. finally, she uses her straightedge to draw the chords that form hexagon jnlkom. veronica claims she has constructed a regular hexagon inscribed in a circle. which statement is true? a. veronica is correct: jnlkom is a regular hexagon inscribed in circle c. b. veronica is incorrect: jnlkom is a hexagon inscribed in circle c but is not regular. c. veronica is incorrect: jnlkom is a regular hexagon but is not inscribed in circle c. d. veronica is incorrect: jnlkom is neither a regular hexagon nor inscribed in circle c.

Answers

The correct option is a. veronica is correct: jnlkom is a regular hexagon inscribed in circle c.

A regular hexagon is a hexagon with all sides equal and all angles equal. If Veronica has successfully constructed a hexagon inscribed in a circle, then all the sides of the hexagon will be chords of the circle.

In order for a chord to be a radius of the circle, it must pass through the center of the circle. Since Veronica has bisected the vertical angles jcl and kcm, she has created two radii of the circle, cn and co.

If cn and co are radii of the circle, then they must be equal in length. Since cn and co are equal in length, and they are also chords of the circle, then they must be opposite sides of a regular hexagon inscribed in the circle.

By the same logic, we can see that all the sides of the hexagon must be equal in length. This means that the hexagon is regular.

Finally, since cn and co are radii of the circle, they must pass through the center of the circle. This means that the hexagon is inscribed in the circle.

Therefore, Veronica is correct: jnlkom is a regular hexagon inscribed in circle c.

The steps involved in constructing a regular hexagon inscribed in a circle:

Draw a circle with any radius.Draw a diameter of the circle.Draw the perpendicular bisector of the diameter.Draw radii from the center of the circle to the points where the perpendicular bisector intersects the circle.Connect the endpoints of the radii to form a hexagon.

The hexagon will be regular because all the sides are radii of the circle and all the radii are equal in length. The hexagon will also be inscribed in the circle because all the sides pass through the center of the circle.

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Write the sum using sigma notation: \( 8+7+6+\ldots+5 \)

Answers

The sum of \( 8+7+6+\ldots+5 \) using sigma notation is \(\sum_{k=1}^{n} 8 - k\).

To write the sum using sigma notation for the series \(8+7+6+\ldots+5\), we need to express the pattern in a concise way. In this series, the numbers decrease by 1 each time.

We can start by identifying the initial term, which is 8, and the final term, which is 5. Let's call the initial term \(a_1\) and the final term \(a_n\).

The common difference between consecutive terms in this series is -1.

Let's call the common difference \(d\).

To write the sum using sigma notation, we can use the following formula:

\(\sum_{k=1}^{n} a_k = a_1 + (a_1 + d) + (a_1 + 2d) + \ldots + a_n\)

In this case, \(a_1\) is 8, \(a_n\) is 5, and \(d\) is -1.

Substituting these values into the formula, we get:

\(\sum_{k=1}^{n} 8 + (8 + (-1)) + (8 + 2(-1)) + \ldots + 5\)

Simplifying further, we have:

\(\sum_{k=1}^{n} 8 - k\)

So, the sum using sigma notation for the series \(8+7+6+\ldots+5\) is \(\sum_{k=1}^{n} 8 - k\).

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Solve sec^2x-2secx=15 for the interval [0,2pi)

Answers

Sec² x - 2

sec x = 15 is the given equation that is required to be solved.

The value of x is to be found out in the interval [0, 2π).

sec² x - 2 sec x = 15

can be simplified by applying a formula. sec² x - 2 sec x = 15Sec² x - 2 sec x - 15 = 0(sec x - 5)(sec x + 3) = 0sec x = 5 or sec x = -3To obtain the value of x,

we need to take inverse secant of both sides.

∴ sec⁻¹ (sec x) = sec⁻¹ (-3)

∴ x = sec⁻¹ (5)  and x = π + sec⁻¹ (3)

The value of x should be in the range of [0, 2π).

x = sec⁻¹ (5) is within the range but x = π + sec⁻¹ (3) is not.

Therefore, x = sec⁻¹ (5) is the solution.

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Explain why in looking for a variable that explains rank, there might be a negative correlation. Choose the correct answer below. O A. It would be expected that as one variable (say length of ride) increases, the rank will worsen, which means it will increase. OB. It would be expected that as one variable (say length of ride) increases, the rank will improve, which means it will decrease. OC. It would be expected that as one variable (say length of ride) increases, the rank will remain constant. OD. It would be expected that as one variable (say length of ride) decreases, the rank will improve, which means it will decrease

Answers

In looking for a variable that explains rank,

there might be a negative correlation when it would be expected that as one variable (say length of ride) increases

, the rank will worsen, which means it will increase.

How to determine correlation? Correlation can be defined as a statistical method that measures the strength and direction of the relationship between two variables.

This relationship is measured between two variables that are quantitative.  

Correlation is a value that ranges from -1 to +1. It is represented by the symbol “r.”

If the correlation coefficient “r” is negative, then we have a negative correlation, which means as one variable increases, the other decreases and vice versa.

In this case, if we have a variable like the length of the ride and we are trying to determine its correlation with the rank,  

it would be expected that as the length of the ride increases, the rank will worsen.

Therefore, there might be a negative correlation.

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Derivatives of Exponentials: Problem 8 (1 point) If f(x) = x + 3e, find f'(4). f'(4) = 256 +3e²¹ Use this to find the equation of the tangent line to the curve y = x + 3e at the point (a, f(a)) when a = 4. The equation of this tangent line can be written in the form y = mx + b. Find m = and b. 3 m = b= Note: You can earn partial credit on this problem. Preview My Answers Submit Answers You have attempted this problem 0 times. You have 10 attempts remaining. Email Instructor

Answers

Therefore, 3m = 3 * 8.15484 = 24.46452 and b = -29.61936.

Given function is f(x) = x + 3e. We have to find f'(4) and use it to find the equation of the tangent line to the curve

y = x + 3e at the point (a, f(a))

when a = 4.

Then, we have to find the values of m and b such that the equation of the tangent line can be written in the form

y = mx + b.

So, we will begin by finding f'(x).

We know that the derivative of x with respect to x is 1.

Also, the derivative of e^(kx) with respect to x is k * e^(kx).

Hence, the derivative of 3e with respect to x is 3e.

Now, we can find f'(x) as follows:

f'(x) = 1 + 3e.

Next, we will find f'(4).

Putting x = 4, we get:

f'(4) = 1 + 3e = 1 + 3 * 2.71828 = 8.15484 (rounded to five decimal places).

Now, we will find the equation of the tangent line to the curve y = x + 3e at the point (a, f(a)) when a = 4.

We know that the equation of a line passing through the point (a, f(a)) and having slope m is given by:

y - f(a) = m(x - a)

We need to find the values of m and b.

To find m, we will use the value of f'(4) that we just calculated.

We know that the slope of the tangent line is equal to f'(4) at x = 4.

Hence, we have: m = f'(4) = 8.15484 (rounded to five decimal places).

To find b, we will substitute the values of a, f(a), and m into the equation of the line.

We have:

a = 4f(a) = f(4) = 4 + 3e (putting x = 4 in the given function y = x + 3e)

m = 8.15484y - f(a)

= m(x - a)y - (4 + 3e)

= 8.15484(x - 4)

Expanding the right side, we get:

y - 4 - 3e = 8.15484x - 33.61936

Collecting like terms, we get:

y = 8.15484x - 29.61936

Hence, we have:

m = 8.15484

b = -29.61936

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Suppose that the supply and demand for widgets is given by the following equations: qd = 500 - 50p qs = 200 what is the vertical intercept (price axis) of the (inverse) demand curve?

Answers

The vertical intercept (price axis) of the demand curve is 10.

The demand equation is given as qd = 500 - 50p, where qd represents the quantity demanded and p represents the price. To find the vertical intercept (price axis) of the demand curve, we need to determine the value of p when qd is equal to zero.

Setting qd = 0, we can solve for p:

0 = 500 - 50p

Rearranging the equation, we have:

50p = 500

Dividing both sides by 50, we find:

p = 10

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Can you give me a quick Awnser

Answers

Answer:

-4.4

Step-by-step explanation:

2.8 + 7.2 = -4.4

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Work shown must support (prove) your answer! The Guess & Check method will not receive any credit, even if correct.7. A persons blood pressure, PP, varies with the cycle of their heartbeat. The pressure (in units mmHg) at time seconds for a particular person may be modeled by the function: () = 20cos(2t) + 00 mmHg, 0 . According to this model, which of the following statements is TRUE? (Hint: Think of this problem in terms of transformations of a graph. In fact, actually graphing it will help you answer the question!)(a) The maximum pressure is 100 mmHg.(b) The pressure goes through one complete cycle in 2 seconds.(c) The amplitude of the pressure function is 120 mmHg.(d) The pressure will reach a maximum value at time = 1 second.(e) Both statements (b) and (d) are accurate. Q6: Solve the initial value differential equation: (x + + 3xy = 3x where y(0) = 2 For and , what is the appropriate outcome of a z-test?Group of answer choicesa. Reject and accept .b. Reject and accept .c. Fail to reject .d. Fail to reject Cash Register A. Design a program in C++ that outputs a receipt of purchase at a market or a convenience store. Each receipt must contain the following information (not an exhaustive list please see a sample receipt below for more information): 1. The market's name, address, phone number, and fax 2. The date and time of purchase 3. The method of payment: a. Card - card type (e.g. visa, master, ), display card number (e.g. XXXXXXXXXXXX1234), entry method (e.g. slides or chip), and whether the card is approved or not (e.g. APPROVED or DENIED). b. Cash - cash amount 4. List of items purchased which includes the item's name, quantity, and total 5. The subtotal amount before tax) 6. Tax percent and amount 7. The balance due (total amount including tax) 8. The amount of change 9. The total number of items 10. The barcode of the receipt B. Other Requirements: - Must include at least 5 classes. (Suggestion: CashRegister, Credit Card, Inventory, Barcode, Address) -The receipt must be formatted nicely. -DO NOT randomly put items on the receipt by using cout only (this will result in a zero for the project). -The inventory must be updated accordingly with the item scanned. For example, if there are 10 bottles of water and a customer buys one, the inventory should be updated to 9 bottles of water since another customer might try to find the exact item. C. Sample run of this program (This is a just an example. Feel free to change the prompt properly according to your own machine): Please scan your item (Press F to finish): 123456 Please scan your item (Press F to finish): 456783 Please scan your items (Press F to finish): 1234567 Would you like to pay with cash or card? card Please swipe or slide in your card: 1234567891234 Receipt Printing... (make sure you show a receipt after) Please scan your item (Press F to finish): 123456 Please scan your item (Press F to finish): 456783 Please scan your items (Press F to finish): 1234567 Would you like to pay with cash or card? cash Please insert cash: 20 Please insert cash: 40 (.. until it's enough or over the amount to be paid) Receipt Printing. (make sure you show a receipt after) Sample receipts for your reference: WALL-MART-SUPERSTORE (888) 888 - 8888 MANAGER TOD LINGA 888 WALL STORE ST WALL ST CITY, LA 88888 ST# 2323 OP: 23432435 TE# 51 TRI 4354 HAND TOWEL 075953630184 2.97 X GATORADE 068949055223 2.00 X X T-SHIRT 036231552452 16.88 X PUSH PINS 088348997350 1.24 X x SUBTOTAL 23.09 TAX 1 7.89% 2.90 TAX 2 2 4.90% 1.28 TOTAL 27.27 CREDIT TEND 27.27 CHANGE DUE 0.00 ACCOUNT *** *** 9999 APPROVAL # 77W166 REF # 307171075528 TERMINAL # 5419885359 # ITEMS SOLD 4 TC# 1752 5627 3145 9811 0000 Get Free Holiday Savings by Cell! Thank You for Shopping With Us! 10/17/2020 16:12 *** CUSTOMER COPY *** Walmart Save money. Live better. (813) 932-0562 Manager COLLEEN BRICKEY 8885 N FLORIDA AVE TAMPA FL 33604 ST# 5221 OP# 00001061 TE# 06 TR# 05332 BREAD 007225003712 F 2.88 N BREAD 007225003712 F 2.88 N GV PNT BUTTR 007874237003 F 3.84 N GV PNT BUTTR 007874237003 F 3.84 N GV PNT BUTTR 007874237003 F 3.84 N GV PNT BUTTR 007874237003 F 3.84 N GV PARM 160Z 007874201510 F 4.98 0 GV CHNK CHKN 007874206784 F 1.98 N GV CHNK CHKN 007874206784 F 1.98 N 12 CT NITRIL 073191913822 2.78 X FOLGERS 002550000377 F 10.48 N SC TWIST UP 007874222682 F 0.84 X EGGS 060538871459 F 1.88 0 SUBTOTAL 46.04 TAX 1 7.000 % 0.26 TOTAL 46.30 DEBIT TEND 46.30 CHANGE DUE 0.00 EFT DEBIT PAY FROM PRIMARY ACCOUNT : 5259 46.30 TOTAL PURCHASE PAYMENT DECLINED DEBIT NOT AVAILABLE 11/06/11 02:21:54 EFT DEBIT PAY FROM PRIMARY ACCOUNT : 5259 46.30 TOTAL PURCHASE REF # 131000195280 NETWORK ID. 0071 APPR CODE 297664 11/06/11 02:22:54 # ITEMS SOLD 13 TC# 0432 2121 1542 2401 9590 Layaway is back for Electronics, Toys, and Jewelry. 10/17/11-12/16/11 11/06/11 02:22:59 The economic criteria concerned with minimizing resource use to achieve an objective is known as allocative: a. Not enough information. b. False: c. True. d. uncertain How does the cumene affect the environment? What happens to cumene when it enters the environment (soil, water and air)? I What would the molarity be of a solution made by dissolving \( 35.7 \) grams of \( \mathrm{Na}_{2} \mathrm{SO}_{4} \) in enough water to make a \( 325 \mathrm{~mL} \) solution? \[ 7.73 * 10^{-4} \math The length of a triangle's base is 5x'y' cm and its height is 4xy? cm.* Determine a simplified expression for the area of the triangle.* If the triangle is the base of a prism with a length of & cm, find a simplified expression for the volume of the prism.* If x= 4cm and y =3 cm, determine the area of the triangle and the volume of the triangular prism CENTRE FOR ENTERTAINMENT ARTS- PROJECT The Centre for Entertainment Arts is a Vancouver-based institution that develops programs in animation, 3D, 2D, game development. They are quickly expanding into Canada, the United States of America, Colombia, and Europe. The game development and animation business is worth over 250USS billion in the United States and it is one of the important businesses of British Columbia. In every game development studio there are business managers, and business people that work on budgets, hire and fire people, work on advertising, finance projects, support company strategy, etc. - They would like to have a group do some basic research on similar institutions that also are in the business of teaching game development, animation etc. They have an opportunity to expand into other related subjects. Which institutions are leading institutions in game development? What are their price points? Where are they located? - They would like to have an assessment of how they should proceed when working with partners. Should they license their programs, should they develop and instruct their programs and if the answer is yes: then which programs? - Should CEA go for egame management? MBAs in Entertainment Industry? This would probably require a survey of companies that do game development. Outcomes: -- Students will become very knowledgeable in a area of increasing importance in North America. --Students will answer most of the questions above with support from your instructor and CEA. --There will be a 30 minute presentation to the client/ instructor PS: Please note that an NDA is required. Determine the radius of convergence and interval of convergence for the following power series 00 (-1)^-1 k=1 xk k A buffer solution contains 0.327 MCH3NH3Cland 0.337 MCH3NH2(methylamine). Determine the pHchange when 0.077 molKOH is added to 1.00 L of thebuffer.pH after addition pH before addition = p The lengths of two sides of a triangle are shown below:Side 1: 3x^2 4x 1Side 2: 4x x^2 + 5The perimeter of the triangle is 5x^3 2x^2 + 3x 8.Part A: What is the total length of the two sides, 1 and 2, of the triangle? (4 points)Part B: What is the length of the third side of the triangle? (4 points)Part C: Do the answers for Part A and Part B show that the polynomials are closed under addition and subtraction? Justify your answer. (2 points)