A sector with a radius of 5 cm and an angle of 150 degrees is equal to another sector

A sector with a radius of 5 cm and an angle of 150 degrees is equal to another sector with a central angle of 250 degrees. Find the radius of the second sector.

The area of the sector is determined by the formula:

S = n * r ^ 2 * a ° / 360 °.

Then the area of the first sector is

S1 = n * 5 ^ 2 * 150 ° / 360 °,

and the area of the second sector is

S2 = n * R ^ 2 * 250 ° / 360 °.

By hypothesis, the sectors are of equal size, i.e., S1 = S2, therefore

n * 5 ^ 2 * 150 ° / 360 ° = n * R ^ 2 * 250 ° / 360 °;

5 ^ 2 * 150 ° = R ^ 2 * 250 °,

whence the radius of the second sector is

R = 5 * √ (150/250) = 5 * √15 / 5 = √15 cm.

Answer: the radius of the second sector is √15 cm.