Question

Laila is setting up a conical tent in the backyard. The height of the tallest part of the tent is 7 feet. The inside of the tent has a volume of 56 cubic feet. How many square feet are covered by the base of the tent?

Answer 1

To solve for the area of the base of the conical tent, we first need to use the formula for the volume of a cone. The volume \( V \) of a cone is given by the formula:

\[
V = \frac{1}{3} \pi r^2 h
\]

where \( r \) is the radius of the base of the cone, and \( h \) is the height of the cone.

From the problem, we know:

- The volume \( V = 56 \) cubic feet.
- The height \( h = 7 \) feet.

Plugging the known values into the volume formula, we get:

\[
56 = \frac{1}{3} \pi r^2 (7)
\]

Simplifying this, we multiply both sides by 3 to eliminate the fraction:

\[
168 = 7 \pi r^2
\]

Next, we divide both sides by 7:

\[
\frac{168}{7} = \pi r^2
\]

This simplifies to:

\[
24 = \pi r^2
\]

To solve for \( r^2 \), we divide both sides by \( \pi \):

\[
r^2 = \frac{24}{\pi}
\]

Now, we can find the area \( A \) of the base of the cone, which is given by the formula for the area of a circle:

\[
A = \pi r^2
\]

Substituting the expression we found for \( r^2 \):

\[
A = \pi \left(\frac{24}{\pi}\right)
\]

This simplifies to:

\[
A = 24
\]

Thus, the area covered by the base of the tent is:

\[
\boxed{24} \text{ square feet.}
\]