# How to determine dimensions of a square based prism

Determine the dimensions of a square based prism box with each volume that requires the least material to make.

$$a) 512 \ \ cm^3$$ $$b) 1000\ \ cm3$$ $$c) 750\ \ cm^3$$

I attempted the question, sort of understand it. if someone can show me how to do one of them, I can most likely do the rest on my own .

For a calculus based solution,

HINT:

Let the base square have side $x$, and the height be $y$.

The volume $V = x^2y$, giving $x = \sqrt{\frac{V}{y}}$

The surface area $A = 2x^2 + 4xy = 2\frac{V}{y} + 4\sqrt V \sqrt y$

Now you need to minimise $A(y) = 2\frac{V}{y} + 4\sqrt V \sqrt y$. Can you proceed from here? Find the general form for the minimal surface area in terms of the volume, then compute it for each case.

If the dimensions of the box are $a$,$b$ and $c$, and the volume is $V$, then you want to minimize $ab+bc+ca$, given that $abc = V$. You can prove that this is achieved when $a=b=c$.

Thus, $V = abc = a^3$, so $a=b=c=\sqrt[3]{V}$.

I am purposefully leaving the details out.

HINT: Lagrange Multipiers

• You don't need a multivariable calculus approach here, since it's given that it's a square prism. – Deepak Jan 2 '16 at 5:16
• Sure, but I personally prefer to master the powerful (and easy to apply) tool and (learn to recognize when to) use it, whenever possible rather then not have have the powerful tool and instead try to solve each separate simpler case separately :) – Petar Ivanov Jan 2 '16 at 5:23