Question

A copper wire having length of 243 m and diameter 4 mm was melted to form a sphere. Find the diameter of the sphere thus formed?

• A. 16 cm
• B. 24 cm
• C. 18 cm
• D. 22 cm
• E. 20 cm

Hint:

When melting and reshaping, volume remains constant.

Answer 1

The Correct Option is C

Step 1: Volume of Wire:
Wire is cylindrical. Length = 243 m = 24300 cm (since 1 m = 100 cm)
Diameter = 4 mm = 0.4 cm, so radius = 0.2 cm.
Volume of wire = \(\pi r^2 h = \pi \times (0.2)^2 \times 24300 = \pi \times 0.04 \times 24300\)
\[ = \pi \times 972 \text{ cm}^3 \]

Step 2: Volume of Sphere:
Volume of sphere = \(\frac{4}{3} \pi R^3\)
Equating volumes:
\[ \frac{4}{3} \pi R^3 = 972 \pi \]
\[ \frac{4}{3} R^3 = 972 \]
\[ R^3 = 972 \times \frac{3}{4} = 243 \times 3 = 729 \]
\[ R = \sqrt[3]{729} = 9 \text{ cm} \]
Diameter = \(2R = 18\) cm.