The areas of three circles are in the ratio 4 : 9 : 25. Find the ratio of their radii.
(a) 3 : 3 : 5
(b) 2 : 3 : 5
(c) 5 : 3 : 3
(d) None of these
Answer (2)
The ratio of the radii of the circles, given their areas in the ratio 4 : 9 : 25, is 2 : 3 : 5. This is derived using the formula for the area of a circle, which is related to the radius. Therefore, the correct answer is (b) 2 : 3 : 5.
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To find the ratio of the radii of three circles given their areas ratio as 4:9:25, let's start by using the formula for the area of a circle, which is given by:
A = π r 2
where A is the area and r is the radius of the circle.
Given the areas are in the ratio 4:9:25, we can express the areas of the three circles as 4 x , 9 x , and 25 x respectively, where x is a constant.
Since the area is proportional to the square of the radius, equate the square of the radii with the area to get:
π r 1 2 = 4 x
π r 2 2 = 9 x
π r 3 2 = 25 x
Now, isolate the radii:
For the first circle: r 1 2 = π 4 x ⇒ r 1 = π 4 x = π 2 x
For the second circle: r 2 2 = π 9 x ⇒ r 2 = π 9 x = π 3 x
For the third circle: r 3 2 = π 25 x ⇒ r 3 = π 25 x = π 5 x
Therefore, the ratio of radii becomes: π 2 x : π 3 x : π 5 x
This simplifies to: 2 : 3 : 5
So, the correct option is (b) 2:3:5.
