The radius of one of the two circles with a common center is 5 cm greater than the radius of the other.

The radius of one of the two circles with a common center is 5 cm greater than the radius of the other. The area of the ring enclosed between these circles is 1.25 of the area of the small circle. Find Radii

1. We will assume that the radius of the smaller circle is x cm. Then, if the radius of the larger circle is 5 cm larger, then it is (x + 5) cm.

2. The area of ​​the smaller circle:

S1 = nx²;

3. The area of ​​the larger circle S2 = n (x + 5) ²;

4. According to the condition, two circles have a common center. In this case, the area of ​​the ring located between the two circles is equal to 1.25 of the area of ​​the smaller circle. I.e:

S2 – S1 = 1.25 * S1;

n (x + 5) ² – nx² = 1.25 * nx²;

n (x² + 10x + 25) – nx² – 1.25 * nx² = 0;

n (x² + 10x + 25) – nx² – 1.25 * nx² = 0;

We divide each term by n:

x² + 10x + 25 – x² – 1.25x² = 0;

– 1.25x² + 10x + 25 = 0;

1.25x² – 10x – 25 = 0;

Let’s calculate the discriminant:

D = b² – 4ac = (- 10) ² – 4 * 1.25 * (- 25) = 100 + 125 = 225;

D ›0 means:

x1 = (- b – √D) / 2a = (10 – √225) / 2 * 1.25 = (10 – 15) / 2.5 = – 5 / 2.5 = – 2, does not fit.

x2 = (- b + √D) / 2a = (10 + √225) / 2 * 1.25 = (10 +15) / 2.5 = 25 / 2.5 = 10;

Therefore, the radius of the smaller circle is 10 cm, and the larger one is 10 + 5 = 15 cm;

Answer: the radii of the circles are 10 cm and 15 cm, respectively.