To determine how many square feet of the back porch will be covered by the tent, we need to calculate the base area of the conical tent. Given the height and the volume of the cone, we can find the radius first and then the base area.
### Here's a step-by-step solution to the problem:
1. Identify the formula for the volume of a cone:
The formula for the volume [tex]\( V \)[/tex] of a cone is given by:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base of the cone, and [tex]\( h \)[/tex] is the height of the cone.
2. Rearrange the formula to solve for [tex]\( r^2 \)[/tex]:
To find the radius, we need to rearrange the formula to express [tex]\( r^2 \)[/tex] in terms of [tex]\( V \)[/tex], [tex]\( \pi \)[/tex], and [tex]\( h \)[/tex]:
[tex]\[
r^2 = \frac{3V}{\pi h}
\][/tex]
3. Substitute the known values into the equation:
Given:
- Height [tex]\( h = 4.5 \)[/tex] feet
- Volume [tex]\( V = 10.5 \)[/tex] cubic feet
Substitute these values into the equation:
[tex]\[
r^2 = \frac{3 \times 10.5}{\pi \times 4.5}
\][/tex]
4. Calculate the radius [tex]\( r \)[/tex]:
After calculating the above expression, we find:
[tex]\[
r^2 \approx 2.228
\][/tex]
Taking the square root to find [tex]\( r \)[/tex]:
[tex]\[
r \approx 1.493 \text{ feet}
\][/tex]
5. Calculate the base area of the cone:
The base area [tex]\( A \)[/tex] of the cone can be calculated using the formula for the area of a circle, [tex]\( A = \pi r^2 \)[/tex]:
[tex]\[
A = \pi \times 2.228
\][/tex]
After calculating this, we find:
[tex]\[
A \approx 7.0 \text{ square feet}
\][/tex]
### Conclusion:
Thus, the tent will cover approximately 7.0 square feet of the back porch. This measure, the base area, is used to determine the extent of the coverage on the back porch.
Note that the required calculation involved finding the radius of the base first, and then using it to find the base area.
