SOLUTION: An open-top, square bottom crate is made of two materials--the sides are made of a light material that costs $3 per square foot and the bottom is made with a heavier material that

Question 980545: An open-top, square bottom crate is made of two materials--the sides are made of a light material that costs $3 per square foot and the bottom is made with a heavier material that costs $5 per square foot. If the volume of the crate has to be 5 cubic feet, what should be the dimensions of the crate to minimize the cost of the materials to build the crate. Assume that the labor and assembly materials cost the same no matter the dimensions of the crate and excess materials can be reused.
The available multiple choice answers for my homework problem are:
1) Base is 2ft x 2ft and Height is 1.25ft
2) Base is 1.82ft x 1.82ft and Height is 1.51ft
3) Base is 2.24ft x 2.24ft and Height is 1ft
4) Base is 5ft x 1ft and Height is 1ft
Thanks.
Answer by josgarithmetic(39618) (Show Source): You can put this solution on YOUR website!
x, base dimensions
y, how tall

Bottom Material, $5 per sq ft
Sides Material, $3 per sq ft
Volume must be 5 cubic feet

A, for area of surfaces (outer or inner, but not both);
.

Let c be cost of the box.

Volume of 5 feet^3,

Use this in the cost equation:

CALCULUS PROCESS TO FIND x FOR MINIMUM COST

Find dc/dx, set equal to 0, solve and examine any or all solutions which occur; one of them should correspond to the x for minimum cost. Once found x, use it to evaluate y. Note, I did not finish this but only described it. You could probably do with no trouble.
-
If you do not yet know derivative from Calculus, then maybe just use a graphing calculator or other graph software.
RELATED QUESTIONS
An open-top box with a square base is to be made from two materials, one for the bottom... (answered by ankor@dixie-net.com)

A can in the shape of a right circular cylinder is required to have a volume of 700 cubic (answered by nerdybill)

A box with a square bottom and no top is to be made to contain 100 cubic inches. Bottom... (answered by ikleyn)

A rectangular box with square base and a fixed volume (V) is to be made from two types of (answered by Fombitz)

Cost of a Can. A can in the shape of a right circular cylinder is required to have a... (answered by josgarithmetic)

A metal box with a square bottom and top is to contain 768 cubic centimeters. The bottom... (answered by ankor@dixie-net.com)

A rectangular storage container with an open top is to have a volume of 10 m3. The length (answered by josgarithmetic)

A rectangular storage container with an open top is to have a volume of 12 cubic meters.... (answered by josmiceli)

A rectangular storage container with an open top is to have a volume of 18 cubic meters. (answered by ankor@dixie-net.com)