The area of a circular ring enclosed between two concentric circles is 286 cm². Find the radii of the two circles, given that their difference is 7 cm.

Given: The area of a circular ring = 286 cm²

The difference in radii of the two circles = 7 cm

Let the radius of the larger circle = R cm

Let the radius of the smaller circle = r cm

Thus, we are given: R - r = 7 ……………….(1)

The area of the ring is given by the difference between the areas of the two circles:

⇒ πR² - πr² = 286

⇒ π(R² - r²) = 286

⇒ $\dfrac{22}{7}$ (R² - r²) = 286

⇒ R² - r² = $\dfrac{7 \times 286}{22}$

⇒ (R - r)(R + r) = $\dfrac{2,002}{22}$

⇒ (R - r)(R + r) = 91

⇒ 7 x (R + r) = 91

⇒ R + r = $\dfrac{91}{7}$

⇒ R + r = 13 ……………….(2)

Adding both equations (1) and (2), we get:

$\begin{matrix} & R & - & r & = & 7 \ & R & + & r & = & 13 \ \ \hline & & & 2R & = & 20 \ & & & R & = & \dfrac{20}{2} \ & & & R & = & 10 \end{matrix}$

Substituting in equation (1), we get

⇒ 10 - r = 7

⇒ 10 - 7 = r

⇒ r = 3

Hence, the radii of the two concentric circles are 10 cm and 3 cm.