Question

A solid metallic sphere of radius 12 cm is melted to form three smaller spheres. If the radii of two spheres are 6 cm and 8 cm, find the radius of the third sphere.

• A. 10 cm
• B. 9 cm
• C. 11 cm
• D. 10.5 cm

Answer 1

Answer:

A. 10 cm

To find the radius of the third sphere, we use the principle of conservation of volume. The volume of the large sphere is equal to the sum of the volumes of the three smaller spheres.

1. Volume Formula for a Sphere

The volume ($V$) of a sphere of radius $r$ is given by:

$V = \frac{4}{3}\pi r^3$

2. Set Up the Equation

Let the radius of the large sphere be $R = 12\text{ cm}$, and the radii of the three smaller spheres be $r_1 = 6\text{ cm}$, $r_2 = 8\text{ cm}$, and $r_3$ (unknown).

$\frac{4}{3}\pi R^3 = \frac{4}{3}\pi r_1^3 + \frac{4}{3}\pi r_2^3 + \frac{4}{3}\pi r_3^3$

Dividing both sides by $\frac{4}{3}\pi$, we get:

$R^3 = r_1^3 + r_2^3 + r_3^3$

3. Substitute the Values and Solve

$12^3 = 6^3 + 8^3 + r_3^3$

Calculate the cubes:

• $12^3 = 1,728$

• $6^3 = 216$

• $8^3 = 512$

Substitute these back into the equation:

$1,728 = 216 + 512 + r_3^3$

$1,728 = 728 + r_3^3$

$r_3^3 = 1,728 - 728$

$r_3^3 = 1,000$

Taking the cube root on both sides:

$r_3 = \sqrt[3]{1000} = 10\text{ cm}$

The radius of the third sphere is 10 cm.