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

Since, we are given that distance of side from the center is x,

So side length of the square will be 2x

Hence it's area (Base area) $B = (2x)(2x) = 4x^2$

Let material be in shape of square then it's area will be $25 m^2$

Volume of the pyramid will be, $V = Bh$

Total area of the pyramid and material will be same.

For the inclined face, the perpendicular distance between vertex and the mid point of the side will be

$l = \sqrt{x^2+h^2}$

Then total surface area of the pyramid will be

$S = 4(\frac{1}{2}2x\sqrt{x^2+h^2}) + (2x)^2 = 4x\sqrt{x^2+h^2} + (2x)^2$

This area will be equal to surface area of the material

i.e. $S = 4x\sqrt{x^2+h^2} + (2x)^2 = 25$

solving it we get, $h = \frac{\sqrt{625 - 200x^2} }{4x}$

Then volume of the pyramid will be

$V = Bh = 4x^2 \frac{\sqrt{625 - 200x^2} }{4x} = x\sqrt{625 - 200x^2}$