To determine approximately how many spheres the size of Pluto are needed to equal the volume of Earth, we need to understand the volumes of both spheres using the formula for the volume of a sphere, [tex]\( V = \frac{4}{3} \pi r^3 \)[/tex].
1. Volume of Earth and Pluto:
- Given:
- Radius of Earth, [tex]\( r_{\text{Earth}} = 6,371 \)[/tex] kilometers
- Radius of Pluto, [tex]\( r_{\text{Pluto}} = 1,161 \)[/tex] kilometers
2. Calculate the Volume of Earth:
- Volume of Earth, [tex]\( V_{\text{Earth}} = \frac{4}{3} \pi (6,371)^3 \)[/tex]
- The computed volume of Earth is [tex]\( 1,083,206,916,845.7535 \)[/tex] cubic kilometers.
3. Calculate the Volume of Pluto:
- Volume of Pluto, [tex]\( V_{\text{Pluto}} = \frac{4}{3} \pi (1,161)^3 \)[/tex]
- The computed volume of Pluto is [tex]\( 6,555,189,764.967643 \)[/tex] cubic kilometers.
4. Calculate the Ratio of Earth's Volume to Pluto's Volume:
- Volume ratio, [tex]\( \frac{V_{\text{Earth}}}{V_{\text{Pluto}}} = \frac{1,083,206,916,845.7535}{6,555,189,764.967643} \)[/tex]
- The volume ratio is approximately [tex]\( 165.24417380479915 \)[/tex].
5. Determine the Number of Pluto-sized Spheres Equivalent to Earth:
- Since the volume ratio is approximately [tex]\( 165.24417380479915 \)[/tex], it takes about 165 Pluto-sized spheres to equal the volume of one Earth.
Given the choices:
A. 121
B. 154
C. 165
D. 176
E. 189
The correct answer is:
C. 165
