Answer to Question #134495 in Calculus for xxx

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.

Answer 1

Let x be the distance from the center of the square to its side and the height of the triangular face is l. Therefore, $2x+2l= 5, \;\; l= 2.5-x.$

The height of the pyramid is $\sqrt{l^2-x^2}$ . Therefore, the volume of the pyramid is $V = \dfrac13\cdot (2x)^2 \cdot \sqrt{l^2-x^2} = \dfrac{4}{3} \cdot x^2 \cdot \sqrt{l^2-x^2} = \dfrac{4}{3} \cdot x^2 \cdot \sqrt{(2.5-x)^2-x^2} = \dfrac{4}{3} \cdot x^2 \cdot \sqrt{2.5^2 - 5x}.$