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An archery target has three regions formed by three concentric circles in figure. If the diameters of the concentric circles are in the ratio $1 : 3 : 5$ , then find the ratio of the areas of three regions.

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1 : 8 : 16

2 : 5 : 7

None of these

1 : 1 : 7

answer is A.

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Detailed Solution

Given that, an archery target has three regions forms by three concentric circles, and the diameters of the concentric circles are in the ratio

1 : 3 : 5

We know that, the area of circle is

π r 2

.
Let the diameter of concentric circles be

k, 3 k, 5 k

.
Then the radius of the concentric circles is

k 2, 3 k 2, 5 k 2 .

Calculating the area of inner circle .

A 1 = π r 2

⇒ π k 2 2

⇒ π k 2 4

We know that, the area of the ring use formula

π R 2 − r 2

, where is the radius of the outer ring and r is the radius of inner ring.
Calculating the area of middle ring.

A 2 = π R 2 − r 2

⇒ π 3 k 2 2 − π k 2 2

⇒

9 π k 2 4 − π k 2 4

⇒ 8 π k 2 4

Calculating the area of outer ring.

A 3 = π R 2 − r 2

⇒ π 5 k 2 2 − π 3 k 2 2

⇒ 25 π k 2 4 − 9 π k 2 4

⇒ 16 π k 2 4

Comparing the ratio of the area of the all three regions, to find the required ratio,

A 1 : A 2 : A 3

\Rightarrow

π k 2 4 : 8 π k 2 4 : 16 π k 2 4

\Rightarrow

1 : 8 : 16
Therefore, ratio of the areas of three regions is 1:8:16.
Hence the correct option is 1.

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