Volume of the file = Volume of the rectangle = length *width * height
[tex] =l*w*h [/tex]
[tex] V=(14-2x)*8*x\\ =112x-16x^2
[/tex]
So the Volume of the file V=112x-16x^2
Now to maximize the file: Compute dV/dx and set to 0.
[tex] \frac{dV}{dx}=112-32x\\ [/tex]
Set dV/dx=0 implies [tex] 112-32x=0 \Rightarrow 32x=112\Rightarrow x=3.5 [/tex]
Thus x=3.5 in tall should the file be in order to maximize the volume that the file can hold.
