Answer:
a) 
b) 
Step-by-step explanation:
Recall that given a function f(x,y,z) then
. To find f, we will assume it exists and then we will find its form by integration.
First assume that F =
. This implies that
if we integrate with respect to x we get that
for some function g(y,z). If we take the derivative of this equation with respect to y, we get
This must be equal to the second component of F. Then

This implies that
, which means that g depends on z only. So 
Taking the derivative with respect to z and making it equal to the third component of F, we get
which implies that
which means that g(z) = K, where K is a constant. So

b) To evaluate
we can evaluate it by using f. We can calculate the value of f at the initial and final point of C and the subtract them as follows.

Recall that
so
Also
so
So 
Given:
A(3,0)
B(1,-2)
C(3,-5)
D(7,-1)
1) reflect across x=-4
essentially calculate the difference between the x=-4 line and Px and "add" it in the other direction to x=-4
A(-4-(3-(-4)),0)=A(-11,0)
B(-4-(1-(-4)),-2)=B(-9,-2)
C(-4-(3-(-4),-5))=C(11,-5)
D(-4-(7-(-4)),-1)=D(-15,-1)
2) translate (x,y)->(x-6,y+8)
A(-3,8)
B(-5,6)
C(-3,3)
D(1,7)
3) clockwise 90° rotation around (0,0), flip the x&y coordinates and then decide the signs they should have based on the quadrant they should be in
A(0,-3)
B(-2,-1)
C(-5,-3)
D(-1,-7)
D) Dilation at (0,0) with scale 2/3, essentially multiply all coordinates with the scale, the simple case of dilation, because the center point is at the origin (0,0)
A((2/3)*3,(2/3)*0)=A(2,0)
B((2/3)*1,(2/3)*-2)=B(2/3,-4/3)
C((2/3)*3,(2/3)*-5)=C(2,-10/3)
D((2/3)*7,(2/3)*-1)=D(14/3,-2/3)
Using the z-distribution, a sample of 142,282 should be taken, which is not practical as it is too large of a sample.
<h3>What is a z-distribution confidence interval?</h3>
The confidence interval is:

The margin of error is:

In which:
is the sample mean.
is the standard deviation for the population.
Assuming an uniform distribution, the standard deviation is given by:

In this problem, we have a 95% confidence level, hence
, z is the value of Z that has a p-value of
, so the critical value is z = 1.96.
The sample size is found solving for n when the margin of error is of M = 0.006, hence:





n = 142,282.
A sample of 142,282 should be taken, which is not practical as it is too large of a sample.
More can be learned about the z-distribution at brainly.com/question/25890103
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Answer:
78.54in²
Step-by-step explanation:
i used the formula