Question

A sphere has a volume of 20 cm cubed if another similar sphere has a radius 3 times larger than the first sphere what is the volume of the larger sphere in cm cubed? ( volume of a sphere is given by V= 4/3 pir^3)

Answer 1

The volume \( V \) of a sphere is given by the formula:

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

Given that the volume of the first sphere is 20 cm³, we can use this information to find the radius of the first sphere.

1. Set the volume formula equal to 20 cm³:

\[
\frac{4}{3} \pi r_1^3 = 20
\]

2. Solve for \( r_1^3 \):

\[
r_1^3 = \frac{20 \cdot 3}{4 \pi} = \frac{60}{4 \pi} = \frac{15}{\pi}
\]

3. To find the radius \( r_1 \):

\[
r_1 = \left( \frac{15}{\pi} \right)^{1/3}
\]

Now, we're told that the radius of the larger sphere \( r_2 \) is 3 times larger than that of the first sphere:

\[
r_2 = 3r_1
\]

4. Now, we can find the volume \( V_2 \) of the larger sphere using its radius:

\[
V_2 = \frac{4}{3} \pi r_2^3
\]
\[
= \frac{4}{3} \pi (3r_1)^3
\]
\[
= \frac{4}{3} \pi (27r_1^3)
\]
\[
= 27 \cdot \frac{4}{3} \pi r_1^3
\]

Since we already know that \( \frac{4}{3} \pi r_1^3 \) equals 20 cm³:

\[
V_2 = 27 \cdot 20 = 540 \text{ cm}^3
\]

Thus, the volume of the larger sphere is:

\[
\boxed{540} \text{ cm}^3
\]