In this question, work step-by-step through an optimization problem.

A craftsman wants to make a cylindrical jewelry box that has volume,V, equal to 65 cubic inches.He will make the base and side of the box out of a metal that costs 30 cents per square inch. The lid of the box will be made from a metal with a more ornate finish which costs 100 cents per square inch.

a) Writing the radius of the cylindrical box asr, and the height of the box ash, calculate the cost,C, in cents, of the metal used to produce the box in terms ofhandr (assume that the craftsman only need consider the metal that actually ends up in the box: no metal is wasted.)

b) Since the volume of the box is 65 cubic inches, complete the following constraint equation:

1. 65=
2. Using the constraint equation, rewritehin terms ofr

c) Rewrite expression for the cost of the box in terms of the single variabler

d) DifferentiateCwith respect tor, to find the derivativedC / dr

e) Find the value ofrfor which we have a potential relative extreme point ofC (give answer to 2 decimal places)

f) What is the height of the box?