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?
The Origin of the Snowman (2020) – Kern Valley Highlights
a
\large 48\pi ft^3
b
\large 216\pi ft^3
c
\large 36\pi ft^3
d
\large 288\pi ft^3
c
36\pi ft^3c
36\pi ft^3c
36\pi fteterhrwth3y6rtyjtmvtrh0eygm00gyowtnvnygebgrc
error gate 404
c
36\pi ft^3
huh
To find the total volume of the snowman, we need to calculate the volume of each snowball separately and then add them together.
To calculate the volume of a sphere, we use the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where V is the volume and r is the radius.
For the bottom snowball, the radius is 3 ft.
\[ V_1 = \frac{4}{3} \pi (3^3) = \frac{4}{3} \pi 27 = 36 \pi ft^3 \]
For the middle snowball, the radius is 2 ft.
\[ V_2 = \frac{4}{3} \pi (2^3) = \frac{4}{3} \pi 8 = \frac{32}{3} \pi ft^3 \]
For the top snowball, the radius is 1 ft.
\[ V_3 = \frac{4}{3} \pi (1^3) = \frac{4}{3} \pi 1 = \frac{4}{3} \pi ft^3 \]
Now, we can add up the volumes of each snowball to find the total volume of the snowman:
\[ V_{total} = V_1 + V_2 + V_3 = 36 \pi + \frac{32}{3} \pi + \frac{4}{3} \pi = \frac{108}{3} \pi + \frac{32}{3} \pi + \frac{4}{3} \pi = \frac{144}{3} \pi = 48 \pi ft^3 \]
Therefore, the total volume of the snowman is \(\boxed{\textbf{(a) } 48\pi ft^3}\).