A rectangular prism has a volume of 27 in.3. If a rectangular pyramid has a base and height congruent to the prism, what is the volume of the pyramid?
this is for connexus
Since both the rectangular prism and the rectangular pyramid have congruent bases and heights, their volumes are proportional.
The formula for the volume of a rectangular prism is V = lwh, where l is the length, w is the width, and h is the height.
If the volume of the prism is 27 in.3, we can let l = 3 in., w = 3 in., and h = 3 in. (since 3 x 3 x 3 = 27).
The formula for the volume of a rectangular pyramid is V = (1/3)Bh, where B is the area of the base and h is the height of the pyramid.
Since the base of the pyramid is congruent to the base of the prism, the area of the base of the pyramid is also 3 x 3 = 9 in.2.
Thus, the volume of the pyramid is:
V = (1/3)(9 in.2)(3 in.)
V = 9 in.3
Therefore, the volume of the pyramid is 9 in.3.
To find the volume of the rectangular pyramid, we need to know the dimensions of the base and the height. Since the base and height of the pyramid are congruent to the rectangular prism, we can use the given volume of the prism to find the dimensions.
Given that the volume of the rectangular prism is 27 in^3, we can express it as:
Volume of prism = length x width x height
Since the base and height are congruent, we can represent them as 'x'. Hence, the equation becomes:
27 = x * x * x
To solve for 'x', we can take the cube root of both sides of the equation:
∛27 = ∛(x * x * x)
Simplifying, we get:
3 = x
So, the dimensions of the rectangular prism are 3 x 3 x 3.
Now, to find the volume of the rectangular pyramid, we use the formula:
Volume of pyramid = (1/3) x base area x height
Since the base and height are congruent and each side of the base is 3, the base area becomes:
Base area = 3 x 3 = 9
Substituting the values into the formula, we get:
Volume of pyramid = (1/3) x 9 x 3
Simplifying, the volume of the pyramid is:
Volume of pyramid = 9 in^3