Question

A solid metal sphere is melted and smaller spheres of equal radii are formed. 10% of the volume of the sphere is lost in the process. The smaller spheres have a radius which is \(\frac{1}{9}\text{th}\) the larger sphere. If 10 litres of paint were needed to paint the larger sphere,how many litres are needed to paint all the smaller spheres?

• A. 90
• B. 81
• C. 9
• D. 810

Answer 1

The Correct Option is B

1. Volume of the Larger Sphere (V):
Let the radius of the larger sphere be \(r\). The volume of the larger sphere is given by \(V = \frac{4}{3}\pi r^3\).

2. Volume Lost in Process:
10% of the volume is lost in the process. So, the remaining volume is \(90\%\) of the original volume:
Remaining volume =\(0.9 \times \frac{4}{3}\pi r^3\).

3. Volume of Smaller Sphere (V'):
The radius of the smaller spheres is \(\frac{1}{9}\) of the radius of the larger sphere, which is \(\frac{1}{9}r\). The volume of each smaller sphere is given by:
\(V' = \frac{4}{3}\pi \left(\frac{1}{9}r\right)^3 = \frac{4}{3} \cdot \frac{1}{729}\pi r^3\).

4. Number of Smaller Spheres:
The number of smaller spheres that can be formed from the original sphere is:
Number of smaller spheres = \(\frac{0.9 \times \frac{4}{3}\pi r^3}{\frac{4}{3} \cdot \frac{1}{729}\pi r^3} = 0.9 \cdot 729 = 656.1\) (approx).

5. Total Volume of Smaller Spheres:
The total volume of all the smaller spheres is:
Total volume = \(656.1 \times \frac{4}{3} \cdot \frac{1}{729}\pi r^3 = \frac{1752}{729}\pi r^3\)cubic units.

6. Ratio of Volumes:
The ratio of the volume of the larger sphere to the total volume of all the smaller spheres is:
Ratio = \(\frac{\frac{4}{3}\pi r^3}{\frac{1752}{729}\pi r^3} = \frac{2430}{1}\).

7. Paint Needed for Smaller Spheres:
Since 10 litres of paint were needed to paint the larger sphere, the amount of paint needed to paint each smaller sphere is:
Paint needed per smaller sphere = \(\frac{10}{243}\) litres.

8. Total Paint Needed:
The total amount of paint needed to paint all the smaller spheres is:
Total paint needed = \(656.1 \times \frac{10}{243} = 81\) litres.

So, the simplified solution confirms that 81 litres of paint are needed to paint all the smaller spheres.