# Optimization of a box with volume using $x,$ $y$ and $z$

I have an optimization problem:

The base of an decorative rectangular feature with a volume of $V = 315$ ft$^3$ is to be constructed in a restaurant. The bottom is made of marble, the sides are made of glass, and the top is open. If marble costs five times as much (per unit area) as glass, find the dimensions of the feature that minimize the cost of the materials. (Assume that the length is greater than or equal to the width. Give your answers correct to at least three decimal places.)

I have createde a constraint equation that I am unsure of: $$lwh - (lw) = 315$$

The rest of the steps are a mystery to me.

I am asked to find the dimensions that would minimize cost for length, width and height.

• The top being open reduces the area needed, not the volume. – mathreadler Sep 29 '17 at 16:10
• How many sides are there and what dimensions do they have? What dimensions does the bottom have? How much area of each material will be needed? – mathreadler Sep 29 '17 at 16:11

## 1 Answer

Hint: With the bottom having dimensions $x,y$ and with the cost normalized to the cost of glass, we have $$\text{Cost} = 5xy+2yz+2xz$$ with the constraint that $$V=xyz=\text{const.}$$

Can you take it from here?

• It is usually more important to learn to set up the problem than how to solve it. – mathreadler Sep 29 '17 at 16:14
• @mathreadler I agree. I also believe that one can learn from examples though. – Bobson Dugnutt Sep 29 '17 at 16:16
• Yes, I would like to know why exactly a $cos(t)$ fits in this equation, the second part is understandable, the $5xy$ in the first equation is also a mystery to me. – Computer Sep 29 '17 at 16:21
• english word cost, not a cosine. – mathreadler Sep 29 '17 at 16:22
• oh, woops, well that certainly makes more sense. We have the scalar multiplied quantity for each dimension all being equivalent to one cost. We can determine the cost of each individual element from that equation. – Computer Sep 29 '17 at 16:25