Sure, let's solve this step by step.
1. Understand the Problem:
The problem involves calculating the volume of a cylinder given its radius and height. Specifically, the radius [tex]\( r = 19 \)[/tex] feet and the height [tex]\( h = 21 \)[/tex] feet. We know the formula for the volume [tex]\( V \)[/tex] of a cylinder:
[tex]\[
V = \pi r^2 h
\][/tex]
2. Substitute the Given Values:
Substitute the given radius and height into the formula:
[tex]\[
V = \pi \times 19^2 \times 21
\][/tex]
3. Perform Intermediate Calculations:
First, calculate the square of the radius:
[tex]\[
19^2 = 361
\][/tex]
Then, multiply this result by the height:
[tex]\[
361 \times 21 = 7581
\][/tex]
4. Include the Value of [tex]\( \pi \)[/tex]:
Multiply the result by [tex]\( \pi \)[/tex] (approximately 3.14159):
[tex]\[
V = 7581 \times \pi \approx 7581 \times 3.14159 \approx 23816.413906864218 \, \text{ft}^3
\][/tex]
So, the volume of the cylinder is approximately [tex]\( 23816.413906864218 \)[/tex] cubic feet.
5. Express in Terms of [tex]\( \sqrt{x} \)[/tex] ft³:
We also need to express this volume in the form [tex]\( \sqrt{x} \, \text{ft}^3 \)[/tex].
6. Find [tex]\( x \)[/tex]:
Since [tex]\( V = \sqrt{x} \)[/tex], then:
[tex]\[
x = V^2 \approx 23816.413906864218^2 \approx 567221571.3830754
\][/tex]
Therefore, the volume [tex]\( V \)[/tex] of the cylinder is [tex]\( \sqrt{567221571.3830754} \)[/tex] cubic feet.
To summarize the results:
- Radius, [tex]\( r = 19 \)[/tex] feet
- Height, [tex]\( h = 21 \)[/tex] feet
- Volume, [tex]\( V \approx 23816.413906864218 \)[/tex] cubic feet
- In the form [tex]\( \sqrt{x} \)[/tex], [tex]\( x \approx 567221571.3830754 \)[/tex]
The final answer is:
[tex]\[
V = \sqrt{567221571.3830754} \text{ ft}^3
\][/tex]
