Find dimensions of square-based prism with greatest volume

**kymcd** · January 9th 2010, 11:30 AM

Find the dimensions of the square-based prism with the greatest volume given each surface area. a) 700 cm2

How do I do this?

**Archie Meade** · January 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.

**pickslides** · January 9th 2010, 11:48 AM

Hi there kymcd, welcome to MHF!

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

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

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

$700= 4xh+2x^2$

$700-2x^2= 4xh$

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

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

Now volume is

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

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

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

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

Can you take it from here?

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