A solid oblique pyramid has a square base with edges measuring $x cm$. The height of the pyramid is $(x+2)$ cm. Which expression represents the volume of the pyramid?
$\frac{x^3+2 x^2}{3} cm^3$
$\frac{x^2+2 x^2}{2} cm^3$
$\frac{x^3}{3} cm^3$
$\frac{x^3+2 x^2}{2} cm^3$
Answer (2)
The volume of a pyramid is calculated using the formula V = 3 1 × ba se A re a × h e i g h t .
The base area of the square is x 2 .
Substitute the base area and height into the volume formula: V = 3 1 x 2 ( x + 2 ) .
Simplify the expression to find the volume: 3 x 3 + 2 x 2 c m 3 .
Explanation
Problem Analysis We are given a solid oblique pyramid with a square base. The length of the edges of the square base is x cm, and the height of the pyramid is ( x + 2 ) cm. We need to find the expression that represents the volume of this pyramid.
Volume Formula The volume V of a pyramid is given by the formula: V = 3 1 × ba se A re a × h e i g h t
Base Area Calculation Since the base is a square with edges of length x cm, the area of the base is: ba se A re a = x × x = x 2 c m 2
Volume Calculation The height of the pyramid is given as ( x + 2 ) cm. Substituting the base area and height into the volume formula, we get: V = 3 1 × x 2 × ( x + 2 ) V = 3 x 2 ( x + 2 ) V = 3 x 3 + 2 x 2 c m 3
Final Answer Therefore, the expression representing the volume of the pyramid is 3 x 3 + 2 x 2 c m 3 .
Examples
Imagine you're designing a paperweight in the shape of an oblique pyramid with a square base. If the side length of the base is 3 cm and the height is 5 cm, you can use the volume formula to calculate how much material you'll need. Substituting x = 3 into the volume expression 3 x 3 + 2 x 2 , we get 3 3 3 + 2 ( 3 2 ) = 3 27 + 18 = 3 45 = 15 cubic centimeters. This helps you estimate the cost and weight of the paperweight before manufacturing.
The volume of the pyramid is derived using the formula V = 3 1 × ba se A re a × h e i g h t . With a base area of x 2 and height of ( x + 2 ) , the volume simplifies to 3 x 3 + 2 x 2 c m 3 . Therefore, the correct choice is 3 x 3 + 2 x 2 c m 3 .
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