The sum of the radii of two circles is 84 cm and the difference of their areas is 5544 cm². Calculate the radii of the two circles.

Let r₁ and r₂ be the radii of the two circles.

Given,

Sum of the radii of two circles is 84 cm.

r₁ + r₂ = 84 …….(1)

Given,

Difference of their areas is 5544 cm².

Area of circle₁ - Area of circle₂ = 5544

⇒ πr₁² - πr₂² = 5544

⇒ $\dfrac{22}{7}$ (r₁² - r₂²) = 5544

⇒ r₁² - r₂² = $\dfrac{5544 × 7}{22}$

⇒ r₁² - r₂² = $\dfrac{38808}{22}$

⇒ r₁² - r₂² = 1764

⇒ (r₁ + r₂) (r₁ - r₂) = 1764

Substituting the value from equation (1) in above equation, we get :

⇒ 84(r₁ - r₂) = 1764

⇒ r₁ - r₂ = $\dfrac{1764}{84}$

⇒ r₁ - r₂ = 21 ……(2)

Adding equation (1) and (2), we get :

⇒ r₁ + r₂ + r₁ - r₂ = 84 + 21

⇒ 2r₁ = 84 + 21

⇒ 2r₁ = 105

⇒ r₁ = $\dfrac{105}{2}$ = 52.5 cm

Substituting value of r₁ in equation (1), we get :

⇒ r₁ + r₂ = 84

⇒ 52.5 + r₂ = 84

⇒ r₂ = 84 - 52.5

⇒ r₂ = 31.5 cm.

Hence radii of the two circles = 52.5 cm and 31.5 cm.