In this question, you will work step-by-step through an optimization problem. A
ID: 2846385 • Letter: I
Question
In this question, you will work step-by-step through an optimization problem.
A craftsman wants to make a cylindrical jewelry box that has volume, V, equal to 70 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 200 cents per square inch.
(Assume that the craftsman only need consider the metal that actually ends up in the box: no metal is wasted.)
This is the objective function for this problem.
Using the constraint equation, rewrite h in terms of r
(Give your answer, and those that follow, correct to two decimal places.)
r =
For this question, you need not show that this is actually a relative minimum point. (But you should know how you would do this.)
h =
In this question, you will work step-by-step through an optimization problem. A craftsman wants to make a cylindrical jewelry box that has volume, V, equal to 70 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 200 cents per square inch. Writing the radius of the cylindrical box as r, and the height of the box as h, calculate the cost, C, in cents, of the metal used to produce the box in terms of h and r. (Assume that the craftsman only need consider the metal that actually ends up in the box: no metal is wasted.)Explanation / Answer
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A craftswoman wants to make a cylindrical jewelry box that has volume, V, equal to 50 cubic inches. She will make the base and side of the box out of a metal that costs 40 cents per square inch. The lid of the box will be made from a metal with a more ornate finish which costs 300 cents per square inch.
The cost is going to be based off of the surface area:
SA = 2(pi)(r^2) + 2(pi)(r)(h)
The first term is for the lid and the base
The second term is for the side of the cylindrical can
Cost = C = 40(pi*r^2 + 2(pi)(r)(h)) + 300(pi*r^2)
Volume = V = 50 = pi*r^2*h
h = 50/(pi*r^2)
C = 40(pi*r^2 + 2pi(r)(50/pi*r^2)) + 300(pi*r^2)
C = 40pi*r^2 + 4000/r + 300pi*r^2
C = 340pi*r^2 + 4000/r
dC/dr = 680pi*r - 4000/r^2 = 0
680pi*r = 4000/r^2
680pi*r^3 = 4000
pi*r^3 = 400/68 = 200/34 = 100/17
r = (100/17pi)^(1/3) PLEASE PLUG THIS VALUE INTO YOUR CALC
h = 50/(pi*r^2) plug r into this to find the value for h.
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