Question

An archery target has three regions formed by three concentric circles as shown in figure. If the diameters of the concentric circles are in the ratios 1 : 2 : 3 , then find the ratio of the areas of three regions.

Answer 1

Let the three regions be A, B and C.

The diameters are in the ratio 1 : 2 : 3.

Let the diameters be 1x, 2x and 3x

Then the radius will be `x/2, (2x)/2  and (3x)/2`

Area of region `A = pi r_A^2 = pi(x/2)^2 = pix^2/4` 

Area of region `B = pi r _B^2-pi r _A^2 = pi(x)^2-pi(x/2)^2 = (3pi(x)^2)/4`

Area of region C = `pi r_C^2-pi r_B^2-pi r_A^2 = pi((3x)/2)^2-pi(x)^2-pi(x/2)^2 = pi((3x)/2)^2-(3pix^2)/4 = (5pix^2)/4` 

Thus, ratio of the areas of regions A, B and C  will be 

`pix^2/4 : (3pi(x)^2)/4 : (5pi x^2)/4`

⇒ `1 : 3 : 5`