Three spherical balls of radius 2 cm, 4 cm, and 6 cm are melted to form a new spherical ball. In this process, there is a loss of 25% of the material. What is the radius (in cm) of the new ball?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 6
We are given:
- Radii of three spheres: $r_1 = 2\,\text{cm}, r_2 = 4\,\text{cm}, r_3 = 6\,\text{cm}$
- Loss of 25% of material when melted
We need to find the radius R of the new sphere.
Step 1: Calculate the total initial volume
Volume of a sphere:
$V = \frac{4}{3}\pi r^3$
$V_{\text{total}} = \frac{4}{3}\pi (2^3 + 4^3 + 6^3) = \frac{4}{3}\pi (8 + 64 + 216) = \frac{4}{3}\pi \cdot 288$
$V_{\text{total}} = 384 \pi \,\text{cm³}$
Step 2: Account for 25% loss
$V_{\text{new}} = 75\% \text{ of } 384\pi = 0.75 \cdot 384\pi = 288 \pi$
Step 3: Find radius of new sphere
$\frac{4}{3}\pi R^3 = 288 \pi ⇒R^3 = 288 \cdot \frac{3}{4} = 216$
$R = \sqrt[3]{216} = 6\,\text{cm}$