Two circles $k_1$ and $k_2$ intersect in such a way that outside their intersection is 10% of the area of the circle $k_1$ and 60% the area of the circle $k_2$. What is the ratio of the radii of the circles $k_1$ and $k_2$? What is the sum of the radii if the area of the figure is $94\pi$?

Two circles $k_1$ and $k_2$ intersect in such a way that outside their intersection is 10% of the area of the circle $k_1$ and 60% the area of the circle $k_2$. What is the ratio of the radii of the circles $k_1$ and $k_2$?What is the sum of the radii if the area of the figure is $94\pi$?

Question:

Two circles $k_1$ and $k_2$ intersect in such a way that outside their intersection is 10% of the area of the circle $k_1$ and 60% the area of the circle $k_2$. What is the ratio of the radii of the circles $k_1$ and $k_2$? What is the sum of the radii if the area of the figure is $94\pi$?


Answer:

Since, outside their intersection is 10% of the area of the circle $k_1$ and 60% the area of the circle $k_2$, 
$\therefore$ 90%$(\pi r_1^2)$=40%$(\pi r_2^2)$
$9\pi r_1^2=4\pi r_2^2$
$\frac{r_1^2}{r_2^2}=\frac{4}{9}$ ____________ (1)
$\frac{r_1}{r_2}=\frac{2}{3}$
$r_1:r_2=2:3$
From (1),
$r_1^2=\frac{4}{9}r_2^2$ _________________ (2)
Now, the area of figure is 94$\pi$.
$\therefore \pi r_1^2$+40% $(\pi r_2^2)$=94$\pi$ 
$\therefore r_1^2+\frac{2}{5}r_2^2=94$ 
$\therefore\frac{4}{9}r_2^2+\frac{2}{5}r_2^2=94$ 
$\therefore\frac{38}{45}r_2^2=94$
$\therefore r_2^2=111.32$ 
$\therefore r_2=10.55$ 
$\therefore r_1^2=\frac{4}{9}\times 111.32$ 
$\therefore r_1^2=49.48$ 
$\therefore r_1=7.03$ 
$\therefore r_1+r_2=7.03+10.55=17.58$ 

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