Question

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

• A. 17 m
• B. 18 cm
• C. 15 cm
• D. 20 cm

Answer 1

The Correct Option is B

To find the diameter of the sphere formed from a melted copper wire, we have to use the concept of volume conservation. This means that the volume of the copper wire before melting is equal to the volume of the sphere after melting.

Step 1: Volume of the copper wire

The copper wire is cylindrical. The formula for the volume of a cylinder is:

\(V = \pi r^2 h\)

where

• \(r\) is the radius of the cylinder,
• \(h\) is the height (or length) of the cylinder.

The diameter of the wire is given as 4 mm, so the radius \((r)\) is 2 mm or 0.2 cm (since 1 cm = 10 mm).

The length of the wire \((h)\) is 243 m or 24300 cm.

Substitute the values to find the volume of the wire:

\(V = \pi \times (0.2)^2 \times 24300 = \pi \times 0.04 \times 24300 = 972\pi \, \text{cm}^3\)

Step 2: Volume of the sphere

The formula for the volume of a sphere is:

\(V = \frac{4}{3}\pi r^3\)

where \(r\) is the radius of the sphere.

Since the volumes are equal, we set them equal to solve for \(r\):

\(\frac{4}{3}\pi r^3 = 972\pi\)

Simplify by dividing both sides by \(\pi\):

\(\frac{4}{3}r^3 = 972\)

Multiply both sides by \(\frac{3}{4}\) to isolate \(r^3\):

\(r^3 = 972 \times \frac{3}{4} = 729\)

Take the cube root of both sides to solve for \(r\):

\(r = \sqrt[3]{729} = 9 \, \text{cm}\)

Step 3: Calculate the diameter of the sphere

The diameter of the sphere is twice the radius:

\(D = 2 \times r = 2 \times 9 = 18 \, \text{cm}\)

Therefore, the diameter of the sphere is 18 cm. Thus, the correct option is:

18 cm