Select the correct answer. The height of a right rectangular pyramid is equal to [tex]$x$[/tex] units. The length and width of the base are [tex]$(x+5)$[/tex] units and [tex]$\left(x-\frac{1}{2}\right)$[/tex] units. What is an algebraic expression for the volume of the pyramid?
A. [tex]$\frac{1}{3} x^3-\frac{11}{6} x^2-\frac{5}{6} x$[/tex]
B. [tex]$\frac{1}{3} x^3+\frac{3}{2} x^2-\frac{5}{6} x$[/tex]
C. [tex]$x^3-\frac{9}{2} x^2-\frac{5}{2} x$[/tex]
D. [tex]$\frac{1}{3} x^3+\frac{3}{2} x^2-\frac{5}{6} x$[/tex]
Answer (2)
Calculate the base area by multiplying the length and width: ba se A re a = ( x + 5 ) ( x − 2 1 ) = x 2 + 2 9 x − 2 5 .
Apply the formula for the volume of a pyramid: V = 3 1 × ba se A re a × h e i g h t .
Substitute the base area and height into the formula: V = 3 1 ( x 2 + 2 9 x − 2 5 ) x .
Simplify the expression to find the volume: V = 3 1 x 3 + 2 3 x 2 − 6 5 x . The final answer is 3 1 x 3 + 2 3 x 2 − 6 5 x .
Explanation
Identify Given Information First, let's identify the given information:
Height of the pyramid: x
Length of the base: x + 5
Width of the base: x − 2 1
State the Volume Formula The volume V of a right rectangular pyramid is given by the formula: V = 3 1 × ba se A re a × h e i g h t
Calculate the Base Area The base area of the rectangular base is: ba se A re a = l e n g t h × w i d t h = ( x + 5 ) ( x − 2 1 ) Let's expand this expression: ba se A re a = x 2 − 2 1 x + 5 x − 2 5 = x 2 + 2 9 x − 2 5
Calculate the Volume Now, substitute the base area and the height into the volume formula: V = 3 1 × ( x 2 + 2 9 x − 2 5 ) × x V = 3 1 x 3 + 3 1 × 2 9 x 2 − 3 1 × 2 5 x V = 3 1 x 3 + 2 3 x 2 − 6 5 x
State the Final Answer Therefore, the algebraic expression for the volume of the pyramid is: 3 1 x 3 + 2 3 x 2 − 6 5 x
Examples
Understanding the volume of a pyramid is useful in architecture and construction. For example, if you're designing a pyramid-shaped structure, you need to calculate the volume to determine the amount of material needed. Suppose you want to build a small decorative pyramid where the height is 2 meters, the length of the base is 7 meters, and the width is 1.5 meters. Using the formula, you can find the volume to estimate the amount of stone or other material required. This ensures you purchase the correct amount, avoiding waste and saving costs.
The volume of the pyramid can be calculated using the formula V = 3 1 × ba se A re a × h e i g h t . After finding the base area by multiplying the length and width, we simplify to obtain the expression for volume. The correct answer is option B: 3 1 x 3 + 2 3 x 2 − 6 5 x .
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