The base of a solid right pyramid is a square with an edge length of [tex]$n$[/tex] units. The height of the pyramid is [tex]$n-1$[/tex] units. Which expression represents the volume of the pyramid?
A. [tex]$\frac{1}{3} n(n-1)$[/tex] units[tex]$^3$[/tex]
B. [tex]$\frac{1}{3} n(n-1)^2$[/tex] units[tex]$^3$[/tex]
C. [tex]$\frac{1}{3} n^2(n-1)$[/tex] units[tex]$^3$[/tex]
D. [tex]$\frac{1}{3} n^3(n-1)$[/tex] units[tex]$^3$[/tex]
Answer (1)
The base of the pyramid is a square with side length n , so the area of the base is n 2 .
The height of the pyramid is n − 1 .
The volume of a pyramid is given by V = 3 1 B h , where B is the area of the base and h is the height.
Substitute the expressions for the base area and height into the volume formula to find the volume: 3 1 n 2 ( n − 1 ) .
Explanation
Problem Analysis The problem provides the dimensions of a right pyramid with a square base and asks for the expression representing its volume. We know the base is a square with side length n , and the height of the pyramid is n − 1 .
Volume Formula The volume V of a pyramid is given by the formula: V = 3 1 B h where B is the area of the base and h is the height of the pyramid.
Base Area Calculation Since the base is a square with side length n , its area B is: B = n 2
Height Value The height of the pyramid is given as h = n − 1 .
Volume Calculation Substitute the expressions for the base area B and the height h into the volume formula: V = 3 1 ( n 2 ) ( n − 1 ) V = 3 1 n 2 ( n − 1 )
Final Answer Therefore, the volume of the pyramid is 3 1 n 2 ( n − 1 ) units 3 .
Examples
Imagine you're designing a paperweight in the shape of a pyramid with a square base. If the side of the square base is 3 cm and the height of the pyramid is 2 cm, you can calculate the volume of material needed using the formula we just derived. In this case, n = 3 and n − 1 = 2 , so the volume would be 3 1 "." 3 2 "."2 = 6 cubic centimeters. This helps you estimate the cost and weight of the paperweight before you even start building it!
