Exercise 17 Page 672

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 Shaded Region

Notice that the radius of the circle is 3 inches, and that the arc of the sector measures 80^(∘). With this information, we can use the formula for the area of a circle's sector to find the area of the shaded region. Let's do it!

area of sector = θ/360^(∘)*π r^2

SubstituteII

θ= 80^(∘), r= 3

area of sector = 80^(∘)/360^(∘)*π* 3^2

▼

Simplify right-hand side

CalcPow

Calculate power

area of sector = 80^(∘)/360^(∘)*π* 9

CommutativePropMult

Commutative Property of Multiplication

area of sector = 80^(∘)/360^(∘)* 9π

MoveRightFacToNum

a/c* b = a* b/c

area of sector = 720^(∘)/360^(∘) π

CalcQuot

Calculate quotient

area of sector = 2π

We have found the area of the shaded region. Shaded Area: 2π square inches

Area of the Figure

The figure is a circle with a radius of 3 inches. To find its area, we will use the formula for the area of a circle. Area of the Figure: π* 3^2=9πsquare inches

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.

P=Area of the shaded region/Area of the figure

SubstituteValues

Substitute values

P=2π/9π

ReduceFrac

a/b=.a /π./.b /π.

P=2/9

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Convert to percent

UseCalc

Use a calculator

P=0.222222...

RoundDec

Round to 2 decimal place(s)

P≈ 0.22

WritePercent

Convert to percent

P≈ 22 %

The probability that a point from the figure chosen at random is in the shaded region is 29, or about 22 %.