Exercise 5 Page 671

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 outside the shaded region. The probability that the point is outside the shaded region is the ratio of the area of the region that is not shaded to the area of the figure.

P (The point is outside the shaded region) = \frac{Area of the region that is not shaded}{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 can calculate the area of the region that is not shaded. Finally, we will find their ratio.

Area of the Shaded Region

The shaded region is a circle with diameter

18

centimeters. Therefore, its radius is

\frac{1 8}{2} = 9

centimeters. Let's substitute this value in the formula for the area of a circle.

A = π r^{2}

Substitute

$r = 9$

A = π (9^{2})

▼

Evaluate right-hand side

CalcPow

Calculate power

A = π (81)

CommutativePropMult

Commutative Property of Multiplication

A = 81 π

Therefore, the shaded area is

81 π

square centimeters.

Area of the Figure

The figure is a regular hexagon with side lengths of $10.4$ centimeters and an apothem of $9$ centimeters.

To find the area of this hexagon, we first need to calculate its perimeter. Since this is a regular hexagon, the perimeter is six times its side length.

P = 6 \cdot 10.4 \Leftrightarrow P = 62.4 cm

Let's substitute these values in the formula for the area of a regular polygon.

A = \frac{P \cdot a}{2}

SubstituteII

$P = 62.4$ , $a = 9$

A = \frac{6 2 . 4 ( 9 )}{2}

▼

Evaluate right-hand side

Multiply

A = \frac{5 6 1 . 6}{2}

CalcQuot

Calculate quotient

A = 280.8

Therefore, the area of the figure is

280.8

square centimeters.

Area of the Region That Is Not Shaded

Now, we can find the area of the region that is not shaded. To do so, we will subtract the area of the shaded region from the area of the figure.

Area of the region that is not shaded = 280.8 - 81 π

▼

Evaluate right-hand side

Multiply

Area of the region that is not shaded = 280.8 - 254.469004 \dots

SubTerm

Subtract term

Area of the region that is not shaded = 26.330995 \dots

RoundDec

Round to $2$ decimal place(s)

Area of the region that is not shaded \approx 26.33

The area of the region that is not shaded is about

26.33

square centimeters.

Probability

As previously mentioned, the probability that the point is outside the shaded region is the ratio of the

area

of

the

region

that

is

not

shaded

to the

area

of

the

figure .

Since we already know both areas, we can find their ratio.

P = \frac{Area of the region that is not shaded}{Area of the figure}

SubstituteValues

Substitute values

P = \frac{2 6 . 3 3}{2 8 0 . 8}

▼

Convert to percent

CalcQuot

Calculate quotient

P = 0.093767 \dots

RoundDec

Round to $2$ decimal place(s)

P \approx 0.09

WritePercent

Convert to percent

P \approx 9 %

The probability that a point from the figure chosen at random is in the region that is not shaded is about

0.09,

or

9 % .