Question

find of the An archery target has three regions formed by three concentric circles as shown in Fig 15.8. If the diameters of the concentric circles are in the ratio `1:2:3`. , then find the ratio of the areas of three regions Fig. 15.8

Answer 1

Let the diameters of concentric circles be k, 2k and 3k.
`:.` Radius of concentric circles are`(k)/(2), k and (3k)/(2)`
`:.` Area of innir circle, `A_(1)= pi((k)/(2))^(2) = (k^(2)pi)/(4)`
`:.` Area of middle region, `A_(2)= pi(k)^(2) - (k^(2)pi)/(4)=(3k^(2)pi)/(4)`
[`because` area of ring = `pi(R^(2) - r^(2))` , where R is radius of outer ring and r is radius of inner ring]
and area of outer region, `A_(3) = pi ((3k)/(2))^(2) - pik^(2)`
= `(9pik^(2))/(4) - pik^(2) = (5pik^(2))/(4)`
`:.` Required ratio = `A_(1) : A_(2) : A_(3)`
= `(k^(2)pi)/(4) : (3k^(2)pi)/(4): (5pik^(2))/(4) = 1 : 3 : 5`