Question

A right prism has a base that is an equilateral triangle. The height of the prism is equal to the height of the base. If the volume of the prism is 81, what are the lengths of the sides of the base?

Answer 1

`\large \boxed{6}`

Explanation:

The formula for the volume of a right triangular prism is

V = ½ach, where

a = the height of the base

c = the length of a side of the base, and

h = the height of the prism

In your prism, a = h, so

V = ½ch²

The base is an equilateral, so

`\begin{array}{rcl}(h)/(c) & = & (√(3))/(2)\\\\h & = & (c√(3))/(2)\\\end{array}`

Then

`\begin{array}{rcl}V & = & (1)/(2)ch^(2)\\\\81 & = & (1)/(2)* c * \left((c√(3))/(2) \right )^(2)\\\\81 & = & (1)/(2)* c^(3)*(3)/(4)\\\\81 & = & (3)/(8)c^(3)\\\\c^(3) & = &(81 * 8)/(3)\\\\ & = & 27 * 8\\c & = & 3 * 2\\c & = & \mathbf{6}\\ \end{array}\\\text{The lengths of the sides of the base are $\large \boxed{\mathbf{6}}$}`