(i)-A circle is inscribed in a square.A point inside the square is randomly selected. What is the probability that the point is inside the circle as well as square?
Answers
Step-by-step explanation:
\begin{lgathered}Let's\ find\ the\ areas\ of\ the\ square\ and\ the\ circle.\ The\ ratio\ of \\ the\ area\ of\ the\ circle\ to\ that\ of\ the\ square\ is\ the\ probability. \\ \\ Let\ each\ side\ of\ the\ square\ be\ 1\ unit. \\ \\ \therefore\ Area\ would\ be\ 1\ unit^2. \\ \\ Diameter\ of\ the\ circle\ would\ be\ the\ side\ of\ the\ square. \\ \\ Diameter = 1\ unit \\ \\ Radius = \frac{1}{2} unit \\ \\ Area = \pi r^2 \\ \\ = (\frac{1}{2})^2\pi = \frac{1}{4}\pi\ unit^2 = \frac{\pi}{4}\ unit^2 \\ \\ \\\end{lgathered}Let′s find the areas of the square and the circle. The ratio ofthe area of the circle to that of the square is the probability.Let each side of the square be 1 unit.∴ Area would be 1 unit2.Diameter of the circle would be the side of the square.Diameter=1 unitRadius=21unitArea=πr2=(21)2π=41π unit2=4π unit2
\begin{lgathered}\\ \\ \\ Probability = \frac{Area\ of\ circle}{Area\ of\ square} \\ \\ = \frac{\frac{\pi}{4}}{1} = \frac{\pi}{4} \\ \\ \\ \therefore\ \frac{\pi}{4}\ is\ the\ answer. \\ \\ \\\end{lgathered}Probability=Area of squareArea of circle=14π=4π∴ 4π is the answer.
\begin{lgathered}\\ \\ \\ Thank\ you. \\ \\ \\ \#adithyasajeevan\end{lgathered}Thank you.#adithyasajeevan
Answer:
If a is the radius of the circle, the area of the inscribed square =2a
2
and p
1
=
πa
2
2a
2
=
π
2
,p
2
=1−p
1
=
π
π−2
π<4 and so π−2<2 which gives p
1
>p
2
.
p
1
2
−p
2
2
=(p
1
−p
2
)(p
1
+p
2
)=
π
4−π
<
3
1
as 3<π<4.