- Start with the formula for the volume of a rectangular prism: \(V = A \times h\).
- Substitute the given values into the formula: \(160 = 20 \times h\).
- Isolate the height \(h\) by dividing both sides by the base area: \(h = \frac{160}{20}\).
- Determine the final height: \(h = \boxed{8 \text{ cm}}\).
Explanation
- Analyze the problem and identify the given variables.
The problem involves a rectangular prism, which is a three-dimensional geometric shape with two congruent parallel bases and rectangular sides. We are given the base area, denoted as \(A\), and the volume, denoted as \(V\). The relationship between volume, base area, and height \(h\) for any prism is given by the formula:
\[V = A \times h\]
We are given that the base area \(A = 20 \text{ cm}^2\) and the volume \(V = 160 \text{ cm}^3\). Plugging these values into our formula, we get the equation:
\[160 = 20 \times h\]
- Solve for the height \(h\).
To find the height \(h\), we need to isolate it in the equation \(160 = 20 \times h\). We can do this by dividing both sides of the equation by the base area \(20 \text{ cm}^2\):
\[h = \frac{160}{20}\]
Performing the division:
\[h = 8\]
Since the volume is in cubic centimeters (\(\text{cm}^3\)) and the base area is in square centimeters (\(\text{cm}^2\)), the height will be in centimeters (\(\text{cm}\)). Therefore, the height of the prism is \(8 \text{ cm}\).
Examples
In construction, understanding the volume-to-base-area ratio is essential when calculating the height of structural elements like columns or blocks. For example, if you know the footprint (base area) of a concrete block foundation and the amount of concrete (volume) required to pour it, you can easily determine how high the foundation will be, ensuring it meets the design requirements for the structure. This is a fundamental application of the geometric properties of prisms in engineering and architecture.