The radius of Jupiter is 71,400 km and the radius of the Earth is 6,378 km. If you can treat both planets as spheres with volume given by 43pr343pr3, how many Earths would fit into Jupiter’s volume?
The correct answer and explanation is :
To determine how many Earths would fit into Jupiter’s volume, we first need to compute the volume of both Jupiter and Earth using the formula for the volume of a sphere:
[
V = \frac{4}{3} \pi r^3
]
Where (r) is the radius of the planet.
Step 1: Calculate the volume of Earth
The radius of Earth is given as 6,378 km. Using the formula:
[
V_{\text{Earth}} = \frac{4}{3} \pi (6,378)^3
]
First, calculate the cube of the radius:
[
6,378^3 = 258,520,508,072
]
Now, plug this into the formula for Earth’s volume:
[
V_{\text{Earth}} = \frac{4}{3} \pi \times 258,520,508,072 \approx 1.08321 \times 10^{12} \text{ km}^3
]
Step 2: Calculate the volume of Jupiter
The radius of Jupiter is given as 71,400 km. Using the formula:
[
V_{\text{Jupiter}} = \frac{4}{3} \pi (71,400)^3
]
First, calculate the cube of the radius:
[
71,400^3 = 364,642,284,064,000
]
Now, plug this into the formula for Jupiter’s volume:
[
V_{\text{Jupiter}} = \frac{4}{3} \pi \times 364,642,284,064,000 \approx 1.43128 \times 10^{15} \text{ km}^3
]
Step 3: Calculate how many Earths fit into Jupiter’s volume
To find how many Earths would fit into Jupiter’s volume, divide the volume of Jupiter by the volume of Earth:
[
\text{Number of Earths} = \frac{V_{\text{Jupiter}}}{V_{\text{Earth}}} = \frac{1.43128 \times 10^{15}}{1.08321 \times 10^{12}} \approx 1,320
]
Conclusion:
Approximately 1,320 Earths would fit into the volume of Jupiter. This result is derived by comparing the relative volumes of the two planets, using their radii in the formula for the volume of a sphere. Jupiter’s much larger radius leads to a significantly greater volume, which can accommodate a vast number of Earths.