Question

Economite plastics plans to produce a one-compartment vertical file by bending the long side of an 8-in. by 14-in. sheet of plastic along two lines to form a u shape. how tall should the file be in order to maximize the volume that the file can hold?

Answer 1

Volume of the file = Volume of the rectangle = length *width * height

$=l*w*h$

$V=(14-2x)*8*x\\ =112x-16x^2$

So the Volume of the file V=112x-16x^2

Now to maximize the file: Compute dV/dx and set to 0.

$\frac{dV}{dx}=112-32x$

Set dV/dx=0 implies $112-32x=0 \Rightarrow 32x=112\Rightarrow x=3.5$

Thus x=3.5 in tall should the file be in order to maximize the volume that the file can hold.