A solid right pyramid has a square base with an edge length of $x cm$ and a height of $y cm$. Which expression represents the volume of the pyramid?
A. $\frac{1}{3} x y cm^3$
B. $\frac{1}{3} x^2 y cm^3$
C. $\frac{1}{2} x y^2 cm^3$
D. $\frac{1}{2} x^2 cm^2$
Answer (2)
Recall the formula for the volume of a pyramid: V = 3 1 B h , where B is the area of the base and h is the height.
Calculate the area of the square base with side length x : B = x 2 .
Substitute the base area and height y into the volume formula: V = 3 1 ( x 2 ) ( y ) .
The volume of the pyramid is 3 1 x 2 yc m 3 .
Explanation
Problem Analysis Let's analyze the problem. We have a solid right pyramid with a square base. The side length of the square base is given as x cm, and the height of the pyramid is y cm. We need to find the expression that represents the volume of this pyramid.
Volume Formula The formula for the volume V of a pyramid is given by: 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 B is: B = x 2
Height The height of the pyramid is given as y , so h = y .
Volume Calculation Now, substitute the values of B and h into the volume formula: V = 3 1 ( x 2 ) ( y ) = 3 1 x 2 y Therefore, the volume of the pyramid is 3 1 x 2 y c m 3 .
Final Answer The expression that represents the volume of the pyramid is 3 1 x 2 y c m 3 .
Examples
Imagine you're designing a paperweight in the shape of a pyramid with a square base. If you want the base to be 5 cm on each side and the height to be 6 cm, you can use the formula V = 3 1 x 2 y to calculate the volume of material you'll need. Plugging in the values, you get V = 3 1 ( 5 2 ) ( 6 ) = 50 cubic centimeters. This helps you estimate the amount of resin or other material needed to create the paperweight, ensuring you don't waste resources.
The volume of the solid right pyramid with a square base is given by the expression 3 1 x 2 y c m 3 . This is determined by the formula for the volume of a pyramid, incorporating the area of the base and the height. Hence, the correct answer is option B: 3 1 x 2 y c m 3 .
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