The height of a right rectangular prism is 3 units greater than the length of the base. The edge length of the square base is x units.
Which expression represents the volume of the prism, in cubic units?
A. [tex]$x^3+9$[/tex]
B. [tex]$x^3+3 x^2$[/tex]
C. [tex]$x^3+3 x+3$[/tex]
D. [tex]$x^3+6 x^2+9 x$[/tex]
Answer (1)
The problem provides a right rectangular prism with a square base of side x and height x + 3 .
The volume of the prism is calculated by multiplying the area of the base ( x 2 ) by the height ( x + 3 ).
Expanding the expression gives the volume V = x 2 ( x + 3 ) = x 3 + 3 x 2 .
The expression representing the volume of the prism is x 3 + 3 x 2 .
Explanation
Problem Analysis Let's analyze the problem. We are given a right rectangular prism with a square base of side length x . The height of the prism is x + 3 . We need to find the expression that represents the volume of this prism.
Volume Calculation The volume of a right rectangular prism is given by the formula: V = Area of base × height Since the base is a square with side length x , the area of the base is: Area of base = x 2 The height of the prism is given as x + 3 . Therefore, the volume of the prism is: V = x 2 × ( x + 3 ) Now, we expand this expression: V = x 2 × x + x 2 × 3 V = x 3 + 3 x 2
Final Answer The expression representing the volume of the prism is x 3 + 3 x 2 .
Examples
Imagine you are designing a storage box with a square base. If you want the height of the box to be 3 inches more than the side length of the base, and you let the side length of the base be 'x' inches, then the volume of the box would be represented by the expression x 3 + 3 x 2 cubic inches. This formula helps you calculate how much you can store in the box based on the dimensions you choose.
