Take Earth to be a perfect sphere with radius 6370 km. Take Jupiter to be a perfect sphere with radius 69,900 km. Take the Sun to be a perfect sphere with radius 6,95,700 km. Compute approximately:
(Calculator can be used.)
- (i) the ratio of the volume of Jupiter to the volume of the Earth;
- (ii) the ratio of the volume of the Sun to the volume of the Earth.
Answer: (i) (69 9006370)3 ≈ 10.973 ≈ 1321 (ii) (6,95,7006370)3 ≈ 109.23 ≈ 13,03,000 (about 1.3 million)
Step-by-step solution
Idea: Volumes of spheres are in the ratio of the cubes of their radii, because 43π cancels. Divide the radii first, then cube.
(i) the ratio of the volume of Jupiter to the volume of the Earth;
- VJVE = (4/3)π(69 900)3(4/3)π(6370)3 = (69 9006370)3.½ mark
- 69 9006370 ≈ 10.973; 10.9733 ≈ 1321. So about 1321 Earths would fit inside Jupiter.1 mark
about 1321 : 1
(ii) the ratio of the volume of the Sun to the volume of the Earth.
- VSVE = (6,95,7006370)3 ≈ (109.22)3.½ mark
- ≈ 13,02,700, about 1.3 million. About 13 lakh Earths would fit inside the Sun.1 mark
about 13,03,000 : 1 (1.3 million)
(i) about 1321 : 1 (ii) about 1,302,700 : 1 (about 1.3 million)
Check: Rough: Jupiter’s radius is about 11 times Earth’s, 113 = 1331 ✓; the Sun’s about 109 times, 1093 ≈ 12,95,000 ✓.
Answer to write in the exam
(i)
VJVE = (69 9006370)3 ≈ (10.973)3
∴ Ratio ≈ 1321 : 1
(ii)
VSVE = (6,95,7006370)3 ≈ (109.22)3
∴ Ratio ≈ 13,03,000 : 1
Common mistakes that cost marks
- Giving the ratio of radii (11 : 1, 109 : 1) instead of volumes.
- Squaring instead of cubing (that would compare surface areas).
How this can come in the exam
MCQ (1 mark)
The radius of one ball is 3 times that of another. How many times its volume is the larger ball’s volume?
- 3
- 9
- 27
- 81
Show answer
(C) 27
33 = 27.
Try one yourself
The Moon’s radius is about 1737 km and Earth’s 6370 km. About how many Moons would fill the Earth?
Show answer
(63701737)3 ≈ 3.6673 ≈ 49.
More questions like this
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How high above the ground is the second string?
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