In the figure below, N congruent semicircles lie on the diameter of a

Let each small semicircle be radius 'r'
Then total small semicircle area nπr^2/2..........1

Area of big circle shaded part
[π(nr/2)^2]-nπr^2/2...........2


Both ratio is 1:18

Solving above will give answer 19

C option

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The key is that the entire diameter/radius of the large semicircle is made up of each of the N individual diameters/radii of the smaller circles.

D = larger diameter

and

d = smaller diameter

Since we are finding the proportional area between similar shapes (every circle is similar to every other circle), we just need to use the ratio of corresponding lengths (whether that is the diameters or the radii)

Larger semicircle’s diameter is ——-> D = (N) (d)


With the total area of the smaller semicircles: first you find one individual area, then multiply that area by the number of semicircles —- (N) * (d)^2

However, with the overall larger semicircle, first you take one of the radii and multiply it by the N number of smaller circles.

Then you use that larger radius input to find the area

D = (N) (d) ———>

so area of larger semicircle would be:

(D)^2 = (Nd)^2 = (N)^2 * (d)^2


To see the logic at play, try using a much smaller example.

Let there be only N = 3 small semicircles each with a radius of 1.

Total area of semicircles = (1)^2 + (1)^2 + (1)^2 ——-> (3) * (1)

N * (area of 1 small semicircle)

So to get the area of the 3 smaller semicircles, we are taking each area individually and multiplying the result of one individual smaller semicircle’s Area by the N = 3 number of semicircles.


However for the large semicircle, we are taking the entire, combined radii of all 3 semicircles first. Then we are finding the area.

If radius of each semicircle was d = 1 ——> then overall radius would be:

(N - number of semicircles) (d = 1)

= Nd = (3) (1) = 3

Now we use this radius once to find the whole semicircle area.

(3)^2 = 9

In algebraic form this is the same as:

(Nd)^2 ———> rule of exponents

(N)^2 (d)^2 = area of entire larger semicircle

Then you set up the proportion as laid out in the problem:

(Area of smaller circles)
______________________
(Area entire circle) - (Area of smaller circles)

= 1 / 18

The rest of the Algebra is shown in nick1816 answer.





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