Question

The volume of a sphere is 48 m3
. What is the volume of a cylinder if its radius is the same as the sphere’s and its height is equal to the sphere’s diameter?(1 point

Answer 1

To find the volume of the cylinder, we first need to know the radius of the sphere. The formula for the volume of a sphere is:

\[
V = \frac{4}{3} \pi r^3
\]

Given that the volume of the sphere is \( 48 \, m^3 \), we can set up the equation:

\[
\frac{4}{3} \pi r^3 = 48
\]

To isolate \( r^3 \), we first multiply both sides by \( \frac{3}{4} \):

\[
\pi r^3 = 48 \times \frac{3}{4} = 36
\]

Next, divide both sides by \( \pi \):

\[
r^3 = \frac{36}{\pi}
\]

Now, to find the radius \( r \):

\[
r = \sqrt[3]{\frac{36}{\pi}}
\]

Next, we need to find the diameter of the sphere, which is \( 2r \).

The volume \( V_c \) of a cylinder is calculated using the formula:

\[
V_c = \pi r^2 h
\]

where \( h \) (the height) is equal to the diameter of the sphere \( 2r \):

\[
V_c = \pi r^2 (2r) = 2\pi r^3
\]

Now we can substitute \( r^3 \) from our earlier calculation:

\[
V_c = 2\pi \left(\frac{36}{\pi}\right)
\]

Simplifying this:

\[
V_c = 2 \times 36 = 72 \, m^3
\]

Thus, the volume of the cylinder is \( \boxed{72 \, m^3} \).