Question

A sphere has a volume of 20 cm3 if another similar sphere has a radius 3 times larger than the first sphere what is the volume of the larger sphere in cm3 ( Volume of a sphere is given by V=43πr3 ) _____cm3

Answer 1

To find the volume of the larger sphere, we first need to determine the relationship between the volumes of similar spheres based on their radii.

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

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

Given that the first sphere has a volume of \( 20 \, \text{cm}^3 \), we can denote the radius of the first sphere as \( r_1 \). Thus, we have:

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

Now, if the second sphere has a radius \( r_2 \) that is 3 times larger than \( r_1 \), we have:

\[
r_2 = 3r_1
\]

Now we compute the volume of the larger sphere (the second sphere), \( V_2 \):

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

Calculating \( (3r_1)^3 \):

\[
(3r_1)^3 = 27r_1^3
\]

Now substituting this back into the volume formula for the second sphere:

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

Since we know that \( \frac{4}{3} \pi r_1^3 = 20 \, \text{cm}^3 \):

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

Thus, the volume of the larger sphere is:

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