Exercise 30 Page 935

We can use geometric models to solve certain types of probability exercises. 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. 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 rectangle. We also know that the radius of each circle is $3$ units. Therefore its diameter is $3(2)=6$ units. Notice that there are exactly $\text{three}$ circles along the length and $\text{two}$ circles along the width of the rectangle. This means that the length of the rectangle is $6(3)=18$ units and its width is $6(2)=12$ units.

Since the length $\ell$ of the rectangle is $18$ and its width $w$ is $12,$ we can substitute these values in the formula for the area of a rectangle. The area of the rectangle is $216$ square units.

Area of the Shaded Region

We know that the area of the shaded region is the difference between the area of the rectangle and the area of the circles. Let's focus on one of these circles. Since each circle has radius of $3$ units, we can substitute this value in the formula for the area of a circle. The area of one circle is $9\pi .$ To find the area of the shaded region, which is formed by the difference between the rectangle and six congruent circles, we need to multiply $A_c$ by $6$ and subtract it from $A_r.$ The shaded area is equal to about $46.35$ square units.

Probability

As previously mentioned, the probability that the point is in the shaded region is the ratio of the $\text{area}$ $\text{of}$ $\text{the}$ $\text{shaded}$ $\text{region}$ to the $\text{area}$ $\text{of}$ $\text{the}$ $\text{figure}.$ Since we already know both areas, we can find their ratio. The probability that a point from the figure chosen at random is in the shaded region is about $0.21,$ which can be also written as $21\%.$