Here at CherryCo Pie Factory, we bake delicious pies on an assembly line.

The trickiest part to this, as it turns out, is figuring out how to heat the pies

efficiently on a conveyor belt. What we have is a setup where pies enter the oven

on a circular conveyor, spent one minute in the oven, and then leave, cooked (in

theory) to perfection.

We want to run the conveyor belt quickly in order to make our pie-cooking

efficient and minimize the number of times each pie has to be baked. The issue,

of course, is that if we run the conveyor too quickly, there won't be enough time

for the steam to vent out, and the pies will explode. We've determined that this

happens if the pies ever heat faster than 2000. We're trying to figure out at

what rate w to turn the conveyor, in revolutions per minute (rpm).

(a) (1 point) The temperature in the oven T, in degrees C, is a function of

position (x, y) in meters relative to the center of the conveyor circle:

T(x,y) = 20 +500y²

Compute the gradient VT(z,y)./n(b) (2 points) The position (x,y) of a pie traveling clockwise along the conveyor

can be given as a function of the time t in minutes:

z(t) == cos(2mwt)

y(t) = sin(2mwt)

where 0 ≤t≤ Note that the bounds depend on the rotation rate w,

but we don't need to worry about that.

What is the velocity (z'(t), y(t))? That is to say, compute the deriva-

tives of and y with respect to t.

(c) (2 points) Using the chain rule, what is the derivative of temperature

with respect to time? (You should assume w is a constant.)

(d) (3 points) What is the maximum value of this expression, and what

is the smallest positive t for which it occurs? HINT: Use the fact

3/nthat sin(20) = 2 sin(8) cos(8), along with what you know about the graph

of the sine function.

(e) (1 point) As mentioned, we need to keep at or below 2000. What is

the highest value of that guarantees this? Leave your answer in exact

form.

(f) (2 points) After implementing the above, it's a quiet Tuesday at the factory,

until someone spills pie grease all over the conveyor! The conveyor is now

very slippery. You consult a mechanical engineer, who tells you that pies will

begin flying off the conveyor if they are ever moved with a total acceleration

exceeding 9m/s². Compute the acceleration vector (r"(t), y" (t))./n(g) (1 point) What is the magnitude a of the acceleration vector? As a

reminder, magnitude is equal to the square root of the sum of the squares

of the components.

(h) (1 point) So given the pie grease, what is the highest value of that

will prevent slippage? Again, leave your answer in exact form.