Answer to Question #135995 in Calculus for Sean

Question

A tent in the shape of a pyramid with a square base is to be constructed from a piece of material having
a side of length 5 meters. In the base of the pyramid, let x be the distance from the center to a side (see
figure below). Find a mathematical model expressing the volume of the tent as a function of x.
(The volume of a pyramid is V = Bh, where V, B and h are the volume, base area and height
of the pyramid respectively).

Answer 1

Let $O=AC\cap BD$, $M\in AB, AM=MB, N\in CD, CN=ND.$

Then $OM=ON=x, MN=AD=2x, AM=MB=x, \medspace x\in (0, \frac{5}{2}).$

$MS+SN=5, so\medspace MS=SN=\frac{5}{2}.$

$\triangle SMO\medspace (\angle SOM=90^\circ): SO=\sqrt{MS^2-OM^2}=\sqrt{\left( \frac{5}{2}\right )^2-x^2}=\sqrt{ \frac{25}{4}-x^2}.$

$V_{SABCD}(x)=\frac{1}{3}AD^2\cdot SO=\frac{1}{3}\cdot(2x)^2\cdot\sqrt{ \frac{25}{4}-x^2}=\frac{1}{3}\cdot4x^2 \frac{\sqrt{25-4x^2}}{2}=\frac{2}{3}x^2\sqrt{25-4x^2}.$

Answer: $V(x)=\frac{2}{3}x^2\sqrt{25-4x^2}, \medspace x\in (0, \frac{5}{2}).$