A target shown in the figure consists of three concentric circles of radii 3, 7 and 9 cm respectively. A dart is thrown and lands on the target. What is the probability that the dart will land on the shaded region? "
Given:
A target shown in the figure consists of three concentric circles of radii 3, 7 and 9 cm respectively.
A dart is thrown and lands on the target.
To do:
We have to find the probability that the dart will land on the shaded region.
Solution:
From the figure,
$\mathrm{OA}=3 \mathrm{~cm}, \mathrm{OB}=7 \mathrm{~cm}, \mathrm{OC}=9 \mathrm{~cm}$
We know that,
Area of a circle of radius $r=\pi r^2$.
Area of the circle with radius $\mathrm{OC}=\pi(9)^{2}$
$=81 \pi$ Area of the shaded region $=\pi(\mathrm{OB})^{2}-\pi(\mathrm{OA})^{2}$
$=\pi(7)^{2}-\pi(3)^{2}$
$=49 \pi-9 \pi$
$=40 \pi$
Therefore, Probability that the dart will land on the shaded region $=\frac{\text { Area of the shaded region } }{\text { Area of the outer circle }}$
$=\frac{40 \pi}{81 \pi}$
$=\frac{40}{81}$
The probability that the dart will land on the shaded region is $\frac{40}{81}$.
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