Question

volume of a sphere with a surface area of 64π square meters in cubic meters

Answer 1

To find the volume of a sphere given its surface area, we can use the formulas for the surface area and volume of a sphere.

1. The formula for the surface area \( A \) of a sphere is given by:
\[
A = 4\pi r^2
\]
where \( r \) is the radius of the sphere.

2. The formula for the volume \( V \) of a sphere is given by:
\[
V = \frac{4}{3}\pi r^3
\]

Given the surface area \( A = 64\pi \) square meters, we can set up the equation:
\[
4\pi r^2 = 64\pi
\]

To solve for \( r^2 \), we divide both sides by \( 4\pi \):
\[
r^2 = \frac{64\pi}{4\pi} = \frac{64}{4} = 16
\]

Now, we take the square root to find \( r \):
\[
r = \sqrt{16} = 4 \text{ meters}
\]

Next, we can use the radius to find the volume. Plugging \( r = 4 \) into the volume formula:
\[
V = \frac{4}{3}\pi (4)^3
\]

Calculating \( 4^3 \):
\[
4^3 = 64
\]

Now substitute that back into the volume formula:
\[
V = \frac{4}{3}\pi (64) = \frac{256}{3}\pi \text{ cubic meters}
\]

Therefore, the volume of the sphere is:
\[
\frac{256}{3}\pi \text{ cubic meters} \quad \text{or approximately } 268.08 \text{ cubic meters}
\]

In conclusion, the volume of the sphere is:
\[
\frac{256}{3}\pi \text{ cubic meters}
\]