Question

the radius of the bigger circle is 9cm and the area of the shaded region is twice that of the smaller circle then how long is the radius of the smaller circle ?​

Answer 1

6.32

Explanation:

Area of bigger circle = π(9)² = 81π cm²

Area of smaller circle = π(9)²/2 = 81π/2 = 40.5πcm²

Area scale factor = (Linear Scale factor)²

81π/40.5π = 9²/x² {cancel the π}

81/40.5 = 81/x² (cross multiplication)

81x² = 40.5 × 81

√x² = √40.5 (square root both sides)

x = 6.32

Answer 2

the radius of the smaller circle = 3√3.

Explanation:

let C' be the bigger circle

and C be the smaller circle and r its radius.

Area(C') = π × (9² - r²)

Area(C) = π × r²

Area(shaded region) = area(C') - area(C)

= π × (9² - r²)

the statement “the area of the shaded region is twice that of the smaller circle ” means π × (9² - r²) = 2 × π × r²

Now, let’s solve the equation:

π × (9² - r²) = 2 × π × r²

⇔ π × (9² - r² - 2r²) = 0

⇔ π × (81 - 3r²) = 0

⇔ 81 - 3r² = 0

⇔ 81 = 3r²

⇔ r² = 27

⇔ r = √27

⇔ r = 3√3