Answer:
[tex]Pr = \frac{8 - \pi}{8}[/tex]
Step-by-step explanation:
Given
See attachment for square
Required
Probability the [tex]dart\ lands[/tex] on the shaded region
First, calculate the area of the square.
[tex]Area = Length^2[/tex]
From the attachment, Length = 6;
So:
[tex]A_1 = 6^2[/tex]
[tex]A_1 = 36[/tex]
Next, calculate the area of the unshaded region.
From the attachment, 2 regions are unshaded. Each of this region is quadrant with equal radius.
When the two quadrants are merged together, they form a semi-circle.
So, the area of the unshaded region is the area of the semicircle.
This is calculated as:
[tex]Area = \frac{1}{2} \pi d[/tex]
Where
[tex]d = diameter[/tex]
d = Length of the square
[tex]d =6[/tex]
So, we have:
[tex]Area = \frac{1}{2} \pi (\frac{d}{2})^2[/tex]
[tex]Area = \frac{1}{2} \pi (\frac{6}{2})^2[/tex]
[tex]Area = \frac{1}{2} \pi (3)^2[/tex]
[tex]Area = \frac{1}{2} \pi *9[/tex]
[tex]Area = \pi * 4.5[/tex]
[tex]Area = 4.5\pi[/tex]
The area (A3) of the shaded region is:
[tex]A_3 = A_1 - A_2[/tex] ---- Complement rule.
[tex]A_3 = 36 - 4.5\pi[/tex]
So, the probability that a dart lands on the shaded region is:
[tex]Pr = \frac{A_3}{A_1}[/tex] i.e. Area of shaded region divided by the area of the square
[tex]Pr = \frac{36 - 4.5\pi}{36}[/tex]
Factorize:
[tex]Pr = \frac{4.5(8 - \pi)}{36}[/tex]
Simplify
[tex]Pr = \frac{8 - \pi}{8}[/tex]
