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If the radius of a sphere is increased by 2 cm, then its surface area increases by 352 cm2. The radius of the sphere before the increase was :
- a)3 cm
- b)4 cm
- c)5 cm
- d)6 cm
Correct answer is option 'D'. Can you explain this answer?
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If the radius of a sphere is increased by 2 cm, then its surface area...
⇒ r = 6 cm
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If the radius of a sphere is increased by 2 cm, then its surface area...
To solve this problem, let's use the formula for the surface area of a sphere:
Surface Area = 4πr^2
Let's assume the radius of the sphere before the increase is 'r' cm.
1. Surface area before the increase:
Surface Area1 = 4πr^2
2. Surface area after the increase:
The radius is increased by 2 cm, so the new radius is (r + 2) cm.
Surface Area2 = 4π(r + 2)^2
3. Difference in surface area:
We are given that the surface area increased by 352 cm^2.
Surface Area2 - Surface Area1 = 352
Substituting the values from step 1 and step 2:
4π(r + 2)^2 - 4πr^2 = 352
Now, let's simplify the equation and solve for 'r'.
4. Expanding and simplifying the equation:
4π(r^2 + 4r + 4) - 4πr^2 = 352
4πr^2 + 16πr + 16π - 4πr^2 = 352
16πr + 16π = 352
5. Simplifying further:
16πr = 352 - 16π
16πr = 352 - 50.27 (approximately)
16πr = 301.73 (approximately)
6. Solving for 'r':
r = 301.73 / (16π)
r ≈ 6 cm
Therefore, the radius of the sphere before the increase was 6 cm.
Hence, the correct answer is option (D) 6 cm.
Surface Area = 4πr^2
Let's assume the radius of the sphere before the increase is 'r' cm.
1. Surface area before the increase:
Surface Area1 = 4πr^2
2. Surface area after the increase:
The radius is increased by 2 cm, so the new radius is (r + 2) cm.
Surface Area2 = 4π(r + 2)^2
3. Difference in surface area:
We are given that the surface area increased by 352 cm^2.
Surface Area2 - Surface Area1 = 352
Substituting the values from step 1 and step 2:
4π(r + 2)^2 - 4πr^2 = 352
Now, let's simplify the equation and solve for 'r'.
4. Expanding and simplifying the equation:
4π(r^2 + 4r + 4) - 4πr^2 = 352
4πr^2 + 16πr + 16π - 4πr^2 = 352
16πr + 16π = 352
5. Simplifying further:
16πr = 352 - 16π
16πr = 352 - 50.27 (approximately)
16πr = 301.73 (approximately)
6. Solving for 'r':
r = 301.73 / (16π)
r ≈ 6 cm
Therefore, the radius of the sphere before the increase was 6 cm.
Hence, the correct answer is option (D) 6 cm.
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If the radius of a sphere is increased by 2 cm, then its surface area increases by 352 cm2. The radius of the sphere before the increase was :a)3 cmb)4 cmc)5 cmd)6 cmCorrect answer is option 'D'. Can you explain this answer?
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