Consider F and C below. F(x, y, z) = y2 sin(z) i + 2xy sin(z) j + xy2 cos(z) k C: r(t) = t2 i + sin(t) j + t k, 0 ≤ t ≤ π (a) Find a function f such that F = ∇f. f(x, y, z) = (b) Use part (a) to evaluate C ∇f · dr along the given curve C.

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

a. she will receive a refund of $278.

step-by-step explanation:

2 & 7

step-by-step explanation:

he climbs 300 ft/hr at a constant rate. the peak is at 1,500 ft.

when going up, he reaches 600 ft in 2 hours. remember, that he is going in a constant rate. divide 600 with 300:

600/300 = 2

assuming that he takes the same route back (meaning that it's a 1500 ft route back, you can solve.

note also that you are solving when he gets back to 600 ft after he reaches the peak.

first, find the amount of hours it takes to get up the peak. divide the total amount (1500), with the ft/her (300) to get the amount of hours (x):

1500/300 = x

x = 5

it takes jamie 5 hours to get up. now, he is returning down, and reaches 600 (his current destination). remember that he climbs at a constant 300 ft/hr. divide 600 with 300:

600/300 = x

x = 2

now add the two numbers together.

x = 5 + 2 = 7

jamie takes 7 hours to go up the mountain and reach his current position when descending.

(total amount/constant rate) = (time).

remember that it is a constant rate, so in total, when he gets back down the mountain, it will take the same amount of time as when he goes up. the total amount for the round trip is: 5 + 5 = 10.

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