We can use geometric models to solve certain types of probability problems. In geometric probability, points on a segment or in a region of a plane represent outcomes. The geometric probability of an event is a ratio that involves geometric measures such as length or area. Consider the given diagram.
We are told that a point in the figure is chosen at random, and want to find the probability that the point lies in the shaded region. The probability that the point is in the shaded region is the ratio of the area of the shaded region to the area of the figure.
P(The point is in the shaded region)= [0.8em]
Area of the shaded region/Area of the figure
We will find the area of the shaded region and the area of the entire figure one at a time. Then, we will find their ratio.
Area of the Figure
The figure is a circle with a radius of 5 units. We can substitute 5 for r in the formula for the area of a circle.
The area of the circle is about 78.54 square units.
Area of the Figure: 78.54square units
Area of the Shaded Region
The area of the shaded region is the difference between the area of a regular pentagon and the area of a circle. Let's start by recalling the formula for the area of a regular pentagon
A_p=1/2ans
In this formula a is the apothem of the polygon, n the number of sides, and s is the side length. Notice that when we connect the apothem with the radius of the circle and half of the side of the polygon, a right triangle is formed.
A full turn measures 360^(∘) and a regular pentagon can be divided into 5 isosceles triangles. Therefore, the vertex angle of one of these triangles measures 3605= 72^(∘). Furthermore, an apothem bisects each of these angles. Therefore, the angle between the radius and the apothem has a measure of 722= 36^(∘).
Now, we can use trigonometric ratios to calculate the value of the apothem and the side length of the pentagon. First, let's use the cosine ratio to find a.
cos 36^(∘)=a/5 ⇔ a=5cos36^(∘)
The apothem a of the pentagon is equal to 5cos36^(∘). Next, we can use the sine ratio to find the length of the side s of the polygon.
The area of the pentagon is equal to about 59.44 square units. To find the area of the shaded region, which is formed by the difference between the circle and the pentagon, we need subtract A_p from A_c.
The shaded area is equal to 19.10 square units
Shaded Area: 19.10square units
Probability
As previously mentioned, the probability that the point is in the shaded region is the ratio of the area of the shaded region to the area of the figure. Since we already know both areas, we can find their ratio.
