Question

The number of solid spheres, each of diameter $6$ cm, that can be made by melting a solid metal cylinder of height $45$ cm and diameter $4$ cm is:

• A. $3$
• B. $5$
• C. $4$
• D. $6$

Hint:

When a solid is melted and reshaped, the total volume remains constant. Always equate the original and new volumes.

Answer 1

The Correct Option is A

Step 1: Write the given data.
Diameter of cylinder $= 4$ cm $\Rightarrow$ radius $r_1 = 2$ cm.
Height of cylinder $h = 45$ cm.
Diameter of each sphere $= 6$ cm $\Rightarrow$ radius $r_2 = 3$ cm.

Step 2: Apply the volume formula.
When a solid is melted and recast into other solids, the volumes remain equal. Hence, \[ \text{Volume of cylinder} = n \times \text{Volume of one sphere} \] \[ \pi r_1^2 h = n \times \frac{4}{3}\pi r_2^3 \]
Step 3: Substitute the given values.
\[ \pi (2)^2 (45) = n \times \frac{4}{3} \pi (3)^3 \] \[ 180\pi = n \times \frac{4}{3} \pi \times 27 \] \[ 180 = n \times 36 \]
Step 4: Solve for $n$.
\[ n = \frac{180}{36} = 5 \] Wait — recheck: \( 4 \times 27 = 108 \), not \(36\). Let's fix properly: \[ 180 = n \times \frac{4}{3} \times 27 \] \[ 180 = n \times 36 \Rightarrow n = 5 \] Yes, it’s correct.

Step 5: Conclusion.
Hence, the number of solid spheres formed is $5$.