A right pyramid with a square base has a base length of $x$ inches, and the height is two inches longer than the length of the base. Which expression represents the volume in terms of $x$?
A. $\frac{x^2(x+2)}{3}$ cubic inches
B. $\frac{x(x+2)}{3}$ cubic inches
C. $\frac{x^3}{3}+2$ cubic inches
D. $\frac{x^3+2}{3}$ cubic inches
Answer (2)
The volume of a pyramid is calculated using the formula V = 3 1 B h , where B is the base area and h is the height.
The base is a square with side x , so its area is B = x 2 .
The height is given as h = x + 2 .
Substituting these into the volume formula gives V = 3 x 2 ( x + 2 ) .
3 x 2 ( x + 2 )
Explanation
Problem Analysis Let's analyze the given information. We have a right pyramid with a square base. The side length of the base is x inches, and the height of the pyramid is x + 2 inches. We need to find an expression for the volume of the pyramid in terms of x .
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 Since the base is a square with side length x , the area of the base is: B = x 2
Height The height of the pyramid is given as x + 2 , so: h = x + 2
Volume Calculation Now, substitute the expressions for B and h into the volume formula: V = 3 1 ( x 2 ) ( x + 2 ) V = 3 x 2 ( x + 2 )
Final Answer Therefore, the volume of the pyramid in terms of x is 3 x 2 ( x + 2 ) cubic inches.
Examples
Imagine you are designing a paperweight in the shape of a right pyramid with a square base. If you want the base to be 3 inches long and the height to be 5 inches (2 inches longer than the base), you can use the formula to calculate the volume of resin needed to create the paperweight. In this case, x = 3 , so the volume would be 3 3 2 ( 3 + 2 ) = 3 9 × 5 = 15 cubic inches. This helps you determine the amount of material you need for your project.
The expression for the volume of a right pyramid with a square base, where the base length is x and the height is x + 2 , is given by the formula V = 3 x 2 ( x + 2 ) . The correct answer is option A.
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