The distance between the centers of two externally touching circles is 20 cm, and the difference in the areas

The distance between the centers of two externally touching circles is 20 cm, and the difference in the areas of their surfaces is 160P cm ^ 2. Determine the radii of the balls.

The condition is not quite correctly written. Are balls or circles touching?
The solution is the same; consider both cases.
1. Balls.
R1, R2 are the radii of the balls.
The surface areas of the balls are equal:
S1 = 4πR1², S2 = 4πR2².
R1 + R2 = 20
4πR1² – 4πR2² = 160π
R1² – R2² = 40 – the formula for the difference of squares.
(R1 + R2) * (R1 – R2) = 40
20 * (R1 – R2) = 40
R1 – R2 = 2
Received the system:
R1 + R2 = 20
R1 – R2 = 2
2 * R1 = 22
R1 = 11 (cm), R2 = 9 (cm).
Answer: the radii of the balls are 11 cm and 9 cm.
2. Circles.
R1, R2 – radii of the circle.
The areas of the circles are equal:
S1 = πR1², S2 = πR2².
R1 + R2 = 20
πR1² – πR2² = 160π
R1² – R2² = 160 – the formula for the difference of squares.
(R1 + R2) * (R1 – R2) = 160
20 * (R1 – R2) = 160
R1 – R2 = 8
R1 + R2 = 20
2 * R1 = 28
R1 = 14 (cm), R2 = 6 (cm).
Answer: the radii of the circles are 14 cm and 6 cm.