Question

If the diameters of the concentric circles shown in the figure below are in the ratio 1 : 2 : 3 then find the ratio of the areas of three regions.

Answer 1

The diameters of concentric circles are in ratio 1 : 2 : 3.
let the diameters be d₁ = 1x, d₂ = 2x, d₃ = 3x.
So, the radii are $r_1=\frac{x}{2},r_2=x,r_3=\frac{3x}{2}.$
The areas of three regions are given be
$$\begin{gathered} A_1=\mathrm{\pi }{r_1}^2=\mathrm{\pi }{\left(\frac{x}{2}\right)}^2=\frac{\mathrm{\pi }{x}^2}{4}, \\ {\mathrm{A}}_2=\mathrm{\pi }{r_2}^2-\mathrm{\pi }{r_1}^2=\mathrm{\pi }{x}^{\mathit{2}}-\frac{\mathrm{\pi }{x}^2}{4}=\frac{3\mathrm{\pi }{x}^2}{4}, \\ {\mathrm{A}}_3=\mathrm{\pi }{r_3}^2-\mathrm{\pi }{r_2}^2=\mathrm{\pi }{\left(\frac{3x}{2}\right)}^2-\mathrm{\pi }{x}^2=\frac{5\mathrm{\pi }{x}^2}{4} \end{gathered}$$
So, their ratio is $A_1:A_2:A_3=\frac{\mathrm{\pi }{x}^{\mathit{2}}}{4}:\frac{3\mathrm{\pi }{x}^2}{4}:\frac{5\mathrm{\pi }{x}^2}{4}=1:3:5.$