Calculus Study Guide Help:

Question

I need help answering the following questions from my study guide:

1. When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.03 cm/min. At what rate is the plates area increasing when the radius is 57 cm?
2. The range R of a projectile fired from the origin over horizontal ground is the distance from the origin to the point of impact. If the projectile is fired with an initial velocity v0 at an angle a with the horizontal, then the range is given by the formula below, where g is the downward acceleration due to gravity. Find the angle a for which the range R is the largest possible. Formula: R= (v0^2)/(g) sin2a
3. Suppose that c(x)=(7x^3)-(84x^2)+22,000x is the cost of manufacturing x items. Find a production level that will minimize the average cost of making x items. At this level, the average cost of making x items is $___?

Answer 1

I will give you some hints...but you need to work out the answers yourself in order for this to a learning exercise.

a) is a related rate problem. The area of the plate is πr². To find the related rates, differentiate the function

A=πr² with respect to t.

b) is a maximum problem: differentiate the function for R with respect to θ and set it = to 0 to find the maximum.

c) is also a maximum problem: differentiate c(x) with respect to x and set it = 0 (same kind of problem as b)