The problem involves calculating the amount of paint needed for several smaller spheres formed by melting a larger sphere, considering volume loss and changes in radius.

Relating Surface Areas of Spheres

Let the radius of the large sphere be R and the radius of each smaller sphere be r. We are given that r = R/10.

The surface area of a sphere is given by the formula A = 4π * radius².

• Surface area of the large sphere: A_large = 4πR²
• Surface area of one small sphere: A_small = 4πr²

The ratio of the surface area of a small sphere to the large sphere is:

\( \frac{A_{small}}{A_{large}} = \frac{4\pi r^2}{4\pi R^2} = \left(\frac{r}{R}\right)^2 \)

Substituting r = R/10:

\( \frac{A_{small}}{A_{large}} = \left(\frac{R/10}{R}\right)^2 = \left(\frac{1}{10}\right)^2 = \frac{1}{100} \)

This means the surface area of one small sphere is 1/100th of the large sphere's surface area. Since paint usage is proportional to surface area, one small sphere needs 1/100th the paint of the large sphere.

Calculating the Number of Smaller Spheres

The volume of a sphere is given by the formula V = (4/3)π * radius³.

• Volume of the large sphere: V_large = (4/3)πR³
• Volume of one small sphere: V_small = (4/3)πr³

The ratio of the volumes is:

\( \frac{V_{small}}{V_{large}} = \frac{(4/3)\pi r^3}{(4/3)\pi R^3} = \left(\frac{r}{R}\right)^3 \)

Substituting r = R/10:

\( \frac{V_{small}}{V_{large}} = \left(\frac{1}{10}\right)^3 = \frac{1}{1000} \)

So, the volume of one small sphere is 1/1000th of the large sphere's volume.

During melting, 5% of the volume is lost. This means 95% of the large sphere's volume is available to form smaller spheres.

Volume available = 0.95 * V_large.

The number of smaller spheres (N) is:

\( N = \frac{\text{Volume available}}{V_{small}} = \frac{0.95 \times V_{large}}{V_{large} / 1000} = 0.95 \times 1000 = 950 \)

Therefore, 950 smaller spheres are formed.

Total Paint Calculation

Paint needed for the large sphere = 14 liters. This corresponds to A_large.

Paint needed for one small sphere corresponds to A_small.

Paint per small sphere = 14 liters * (A_small / A_large) = 14 * (1/100) = 0.14 liters.

Total paint needed for all 950 smaller spheres:

Total Paint = Number of small spheres * Paint per small sphere

Total Paint = 950 * 0.14 liters

Total Paint = 950 * (14 / 100) liters

Total Paint = 9.5 * 14 liters

Total Paint = 133 liters.