Question

Carlos has built a snowman consisting solely of 3 spherical snowballs.

The bottom snowball has a radius of 3 ft, the middle snowball has a radius of 2 ft, and the top snowball has a radius of 1 ft.

What is the total volume of the snowman?



(1 point)
Responses

288π ft3
288π ft3

48π ft3
48π ft3

36π ft3
36π ft3

216π ft3

Answer 1

To find the total volume of the snowman made of three spherical snowballs, we need to calculate the volume of each snowball and then sum the volumes. The formula for the volume \( V \) of a sphere is given by:

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

where \( r \) is the radius of the sphere.

1. **Volume of the bottom snowball (radius = 3 ft):**
\[
V_1 = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 = 36 \pi \text{ ft}^3
\]

2. **Volume of the middle snowball (radius = 2 ft):**
\[
V_2 = \frac{4}{3} \pi (2)^3 = \frac{4}{3} \pi \times 8 = \frac{32}{3} \pi \text{ ft}^3
\]

3. **Volume of the top snowball (radius = 1 ft):**
\[
V_3 = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \times 1 = \frac{4}{3} \pi \text{ ft}^3
\]

Now, we sum the volumes of all three snowballs:

\[
V_{total} = V_1 + V_2 + V_3 = 36\pi + \frac{32}{3}\pi + \frac{4}{3}\pi
\]

First, we convert \( 36\pi \) to have a common denominator of 3:

\[
36\pi = \frac{108}{3}\pi
\]

Now, we can sum the volumes:

\[
V_{total} = \frac{108}{3}\pi + \frac{32}{3}\pi + \frac{4}{3}\pi = \frac{108 + 32 + 4}{3}\pi = \frac{144}{3}\pi = 48\pi \text{ ft}^3
\]

Thus, the total volume of the snowman is:

\[
\boxed{48\pi \text{ ft}^3}
\]