Review question

# When is the volume of this cuboid a maximum? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9260

## Suggestion

A cuboid has a total surface area of $\quantity{150} {cm^2}$ and is such that its base is a square of side $\quantity{x}{cm}$.

Show that the height, $\quantity{h}{cm}$, of the cuboid is given by $h=\frac{75-x^2}{2x}$.

How can we find the surface area of a cuboid from its sides?

Express the volume of the cuboid in terms of $x$. Hence determine, as $x$ varies, its maximum volume and show that this volume is a maximum.

Can we write the volume of the cuboid in terms of $x$ and $h$? Can we now substitute for $h$?

How can we find the maximum value?

This applet shows how the shape of the cuboid changes as we vary $x$. The surface area is fixed and the length of the red line is proportional to the volume.