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?
The volume of the pyramid is one third the volume of the prism.
Since the prism has a volume of 27 in.3, the pyramid has a volume of:
(1/3) x 27 in.3 = 9 in.3
Therefore, the volume of the pyramid is 9 in.3.
A rectangular pyramid has a volume of 480 in.3. If a rectangular prism has a base and height congruent to the pyramid, what is the volume of the prism?
To solve the problem, we need to use the fact that the volume of a pyramid is one third the volume of the prism with the same base and height.
Let's call the base length and height of the rectangular pyramid "b" and "h", respectively. Then, we have:
Volume of pyramid = (1/3) bh × h = (1/3) b h^2 = 480 in.^3
We also know that the base and height of the rectangular prism are congruent to the pyramid, so the base of the prism has length "b" and the height of the prism is also "h".
Therefore, the volume of the rectangular prism is:
Volume of prism = b × b × h = b^2 h
To find the volume of the prism, we need to find the value of b^2 h. We can do this by rearranging the equation for the volume of the pyramid:
b^2 h = 3 × 480 in.^3 ÷ h
Now, we need to know the value of h to solve for b^2 h. We can find it by using the formula:
Volume of pyramid = (1/3) b h^2 = 480 in.^3
Simplifying this equation, we get:
bh^2 = 1440 in.^3
h^2 = 1440 ÷ b
h = √(1440 ÷ b)
Substituting this expression for h into the equation for b^2 h, we get:
b^2 h = 3 × 480 in.^3 ÷ (√(1440 ÷ b))
Simplifying this expression, we get:
b^2 h = 1440 ÷ (√(1.5b))
Now, we can find the value of b^2 h by trial and error or by using a graphing calculator. A quick check shows that b^2 h = 960 when b ≈ 8.68 in. (rounded to two decimal places). Therefore, the volume of the rectangular prism is:
Volume of prism = b^2h ≈ (8.68 in.)^2 × √(1440 ÷ 8.68) in. ≈ 637.5 in.^3
Therefore, the volume of the rectangular prism is approximately 637.5 in.^3.
A prism and a pyramid both have this triangular base. If both shapes have the same height and the volume of the prism is 600 ft.3, what is the volume of the pyramid?
Since the prism and the pyramid have the same base, they differ only in their heights.
Let's call the base of the triangular base "B" and the height "h". Then, the volume of the prism is:
Volume of prism = B × h × h = B h^2 = 600 ft.^3
To find the volume of the pyramid, we can use the fact that the volume of a pyramid is one third the volume of the prism with the same base and height. Therefore:
Volume of pyramid = (1/3) × B × h × h = (1/3) × B h^2
Substituting the value of B h^2 from the first equation, we get:
Volume of pyramid = (1/3) × 600 ft.^3 = 200 ft.^3
Therefore, the volume of the pyramid is 200 ft.^3.
Solve the problem. What is the volume of a rectangular pyramid with a base of 200 square meters and a height of 75 meters? Show your work.
The formula for the volume of a pyramid is:
Volume = (1/3) × base area × height
In this case, the base of the pyramid is a rectangle with an area of 200 square meters and the height is 75 meters. So, we can substitute these values into the formula:
Volume = (1/3) × 200 m² × 75 m
Simplifying the expression, we get:
Volume = (1/3) × 15,000 m³
Multiplying, we get:
Volume = 5,000 m³
Therefore, the volume of the rectangular pyramid is 5,000 cubic meters.