A copper sphere of diameter 12 cm is drawn into a wire of diameter 4 mm. What is the length (in cm) of the wire? (Where $\pi =\frac{22}{7}$ )

The correct answer is 7200.

Step-by-Step Explanation:

When a solid metal sphere is drawn into a wire, its volume remains conserved (constant). Therefore, the volume of the cylindrical wire is equal to the volume of the original copper sphere.

Step 1: Find the volume of the copper sphere
Given the diameter of the sphere = 12 cm.
Radius of the sphere (R) = 12 / 2 = 6 cm.

The volume of a sphere is given by the formula:
$V_{\text{sphere}}=\frac{4}{3}\pi R^3$
Substituting R = 6 cm:
$V_{\text{sphere}}=\frac{4}{3}\times \pi \times {\left(6\right)}^3$
$V_{\text{sphere}}=\frac{4}{3}\times \pi \times 216=288\pi {\text{cm}}^3$

Step 2: Express the volume of the wire (cylinder)
Given the diameter of the wire = 4 mm.
Converting millimeter to centimeter (since 1 cm = 10 mm):
Diameter in cm = 4 / 10 = 0.4 cm.
Radius of the wire (r) = 0.4 / 2 = 0.2 cm.
Let h be the length (height) of the wire in centimeters.

The volume of a cylinder is given by the formula:
$V_{\text{wire}}=\pi r^{2}h$
Substituting r = 0.2 cm:
$V_{\text{wire}}=\pi \times {\left(0.2\right)}^2\times h=0.04\pi h{\text{cm}}^3$

Step 3: Equate both volumes to solve for the length (h)
$0.04\pi h=288\pi$
Dividing both sides by π:
$0.04h=288$
Solving for h:
$h=\frac{288}{0.04}=\frac{28800}{4}=7200\text{cm}$

Therefore, the length of the wire is 7200 cm.