Question:easy

If the volume of a sphere is divided by its surface area, we obtain 27 cm. The radius of the sphere is

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For any sphere, volume over surface area simplifies to r divided by 3. So set r over 3 equal to 27 and multiply, do not divide.
Updated On: Jul 17, 2026
  • 9 cm.
  • 81 cm.
  • 27 cm.
  • 24 cm.
Show Solution

The Correct Option is B

Solution and Explanation

Step 1: Set up the equation exactly as the sentence reads.
The sentence "volume divided by surface area gives 27" translates straight into
\[ \frac{\frac{4}{3}\pi r^3}{4\pi r^2} = 27 \]
Here $r$ is the radius in centimetres. Nothing else is unknown, so this single equation settles the problem.

Step 2: Clear the fraction step by step instead of cancelling at sight.
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{4}{3}\pi r^3 \times \frac{1}{4\pi r^2} = 27 \]
Group the like parts. The numerical part is $\frac{4}{3} \times \frac{1}{4} = \frac{1}{3}$. The $\pi$ divides out completely. The power of $r$ is $r^{3-2} = r^1$. So the left side becomes
\[ \frac{1}{3}r = 27 \]

Step 3: Solve for r.
Multiply both sides by 3:
\[ r = 81 \]
So the radius measures 81 cm.

Step 4: Test it with real numbers.
Take $\pi \approx 3.14$ and $r = 81$. Then $r^2 = 6561$ and $r^3 = 531441$.
Surface area $\approx 4 \times 3.14 \times 6561 \approx 82406$ square cm.
Volume $\approx \frac{4}{3} \times 3.14 \times 531441 \approx 2224969$ cubic cm.
Dividing, $\frac{2224969}{82406} \approx 27$. The condition checks out numerically as well as algebraically.

Step 5: Why the tempting answers are wrong.
The number 27 in the question is a decoy: it makes 27 cm look like the obvious answer, but the ratio for a sphere is $\frac{r}{3}$, not $r$. If the radius were 27, the ratio would be 9, not 27.
If the radius were 9, the ratio would be 3.
If the radius were 24, the ratio would be 8.
Only a radius of 81 gives $\frac{81}{3} = 27$, so option (B) stands.

Final Answer:
The sphere's radius is 81 cm.
\[ \boxed{81} \]
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