A rectangular prism has a base area of 15 cm and a congruent Pyramid has a volume of 20cm. What is the height of the rectangular prism?
There are a few different ways to approach this problem, but here's one possible method:
Let's start by drawing a diagram of the rectangular prism and pyramid:
```
+--------------+
/ /| /|
/ / | / |
/ / | / |
/ / | / |
+--------------+ |
| | |
| Pyramid | |
| | |
+--------------+ /
| / /
| / /
| / /
| / /
|/ /
```
We know that the base area of the rectangular prism is 15 cm, so let's call the length and width of the base "x" and "y", respectively. Then we can write:
xy = 15
Next, we know that the pyramid has a volume of 20 cm, and we're told that it's congruent (i.e. same size and shape) to the rectangular prism. Since the pyramid has one-third the volume of a rectangular prism with the same base and height, we can write:
(1/3)xyh = 20
where "h" is the height of the pyramid. Now we can solve for "h" in terms of "x" and "y":
h = (60/xy)
Substituting our expression for xy from the first equation, we get:
h = (60/15)
h = 4 cm
So the height of the rectangular prism is also 4 cm, since the pyramid and prism are congruent.
To find the height of the rectangular prism, we need to use the formula for the volume of a pyramid and the formula for the volume of a rectangular prism.
The volume of a rectangular prism is given by the formula:
Volume of rectangular prism = base area of the rectangle * height of the prism
The volume of a pyramid is given by the formula:
Volume of pyramid = (base area of the pyramid * height of the pyramid) / 3
Given that the base area of the rectangular prism is 15 cm^2 and the volume of the pyramid is 20 cm^3, we can set up the following equation:
15 * height of the prism = (15 * height of the prism) / 3
Multiplying both sides of the equation by 3:
45 * height of the prism = 15 * height of the prism
Simplifying the equation:
45 = 15
This equation is not true, which means there is no height that satisfies both the base area of the prism and the volume of the pyramid. Therefore, we cannot determine the height of the rectangular prism with the given information.