If the surface area of a sphere is reduced to one-ninth of the area, its radius reduces toa)One-fourthb)One-thirdc)One-fifthd)One-ninthCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Answer

Surface area of a sphere = 4πr²
∴ Surface area of a sphere ∝∝ (Square of radius)

Since the surface area of a sphere is reduced to one-ninth, radius will reduce to One-third.

Answer

Explanation:

To solve this problem, we need to use the formula for the surface area of a sphere and see how the radius affects it when it is reduced.

The formula for the surface area of a sphere is given by:

S = 4πr^2

Where S is the surface area and r is the radius of the sphere.

Let's assume the original surface area of the sphere is S1 and the original radius is r1. According to the question, the surface area is reduced to one-ninth of the original area, so the new surface area is S2 = S1/9.

We need to find the new radius, which we'll call r2.

Step 1: Set up the equation using the formula for surface area:

S1 = 4πr1^2
S2 = 4πr2^2

Step 2: Substitute the given values:

S2 = S1/9
4πr2^2 = S1/9

Step 3: Divide both sides of the equation by 4π to isolate r2:

r2^2 = (S1/9)/(4π)

Step 4: Simplify the equation:

r2^2 = S1/(9 * 4π)

Step 5: Take the square root of both sides to solve for r2:

r2 = √(S1/(9 * 4π))

Step 6: Simplify the equation further:

r2 = √(S1/(36π))

Step 7: We can see that the radius reduces to 1/6 of the original radius. Therefore, the correct answer is option 'B', which states that the radius reduces to one-third.

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