In the figure shown above, PQRS is a square. The shaded portion is formed by the intersection of sectors of circles with radius equal to the side of the square and centers at S and Q.
The probability that any point picked randomly within the square falls in the shaded area is ___________.
To find the probability that a randomly picked point within the square PQRS falls in the shaded area, we need to calculate the area of the shaded region and divide it by the area of the square.
Step 1: Calculate the area of square PQRS
If the side of the square PQRS is \( r \), then the area of the square \( A_{\text{square}} \) is given by:
\(A_{\text{square}} = r^2\)
Step 2: Calculate the area of one sector
The shaded area consists of two sectors of circles with centers at \( S \) and \( Q \), each with a radius equal to the side of the square, \( r \). Each sector is a quarter of a circle, so the area of one sector \( A_{\text{sector}} \) is given by:
\(A_{\text{sector}} = \frac{1}{4} \times \pi r^2 = \frac{\pi r^2}{4}\)
Step 3: Calculate the area of the two sectors
Since there are two such sectors, the total area covered by the sectors is:
\(A_{\text{sectors}} = 2 \times \frac{\pi r^2}{4} = \frac{\pi r^2}{2}\)
Step 4: Calculate the area of the shaded region
The shaded area is the region common to both sectors, forming one overlapping sector. By symmetry and geometrical consideration, this shaded region will be the intersection of both sectors.
This intersection can be found by noting that the overlapping sector is essentially the area covered by both minus the extra count of the quarter circles:
\(A_{\text{shaded}} = r^2 - (r^2 - \frac{\pi r^2}{2}) = r^2 - \frac{\pi r^2}{2}\)
Step 5: Calculate the probability
The probability that a randomly chosen point inside the square falls in the shaded area is given by the ratio of the area of the shaded region to the area of the square:
\(\text{Probability} = \frac{A_{\text{shaded}}}{A_{\text{square}}} = \frac{r^2 - \frac{\pi r^2}{2}}{r^2} = 1 - \frac{\pi}{2}\)
Thus, the probability is:
\(\frac{\pi}{2} - 1\)
Conclusion
Therefore, the probability that any point picked randomly within the square falls in the shaded area is \(\frac{\pi}{2} - 1\).
Important Questions from Area Under Curve
- A function $y(x)$ is defined in the interval $[0, 1]$ on the x-axis as
$y(x) = \begin{cases} 2 & \text{if } 0 \le x < \frac{1}{3} \\ 3 & \text{if } \frac{1}{3} \le x < \frac{3}{4} \\ 1 & \text{if } \frac{3}{4} \le x \le 1 \end{cases}$
Which one of the following is the area under the curve for the interval $[0, 1]$ on the x-axis? - The area of the region bounded by the parabola $x = -y^2$ and the line $y = x + 2$ equals
The work done by the force $F = (x + y)\hat{i} - (x^2 + y^2)\hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in $\vec{OX}$ and $\vec{OY}$ directions, respectively, along the upper half of the circle $x^2 + y^2 = 1$ from $(1,0)$ to $(-1,0)$ in the $xy$-plane is
- If $f(x) = 2 \ln(\sqrt{e^x})$, what is the area bounded by $f(x)$ for the interval $[0, 2]$on the x-axis?
- Define $[x]$ as the greatest integer less than or equal to $x$, for each $x \in (-\infty,\infty)$. If $y = [x]$, then area under $y$ for $x \in [1,4]$ is
