# Find dimensions of square-based prism with greatest volume

• Jan 9th 2010, 11:30 AM
kymcd
Find dimensions of square-based prism with greatest volume
Find the dimensions of the square-based prism with the greatest volume given each surface area. a) 700 cm2

How do I do this?
• Jan 9th 2010, 11:43 AM
The height is variable and the square base area are variable.

Zero volume corresponds to zero height and also to zero base area.

In between these extremes is a maximum volume.

Write the prism volume equation in terms of one variable only using the surface area information.
Differentiate the volume equation,
finding the maximum value of the volume equation.
• Jan 9th 2010, 11:48 AM
pickslides
Hi there kymcd, welcome to MHF!

First you need to consider the formula to surface area for a square based prism.

It is, $\displaystyle \text{SA} = 4xh+2x^2$

So making this equal 700 and solving for h we will be able to find an expressin for volume to terms of x.

$\displaystyle 700= 4xh+2x^2$

$\displaystyle 700-2x^2= 4xh$

$\displaystyle \frac{700}{4x}-\frac{2x^2}{4x}= h$

$\displaystyle h = \frac{175}{x}-\frac{x}{2}$

Now volume is

$\displaystyle \text{V} = x^2h$

$\displaystyle \text{V} = x^2\left(\frac{175}{x}-\frac{x}{2}\right)$

$\displaystyle \text{V} = 175x-\frac{x^3}{2}$

Now to finish it, finding the maximum make $\displaystyle \frac{dV}{dx}=0$ and solve for $\displaystyle x$

Can you take it from here?