Exercise 21 Page 674

We throw a dart at the given board. Hitting any point inside the board is equally likely.

We will order the likelihoods of the dart landing in the given regions from least likely to most likely. To do so we will compare the geometric probabilities of the options. We will first calculate the areas of the regions in each option. Then we will find the probabilities of the dart landing on these regions. Let's look at the steps that we will go through.

1. We will find the probability of the green region.
2. We will find the probability of the not-blue region.
3. We will find the probability of the red region.
4. We will find the probability of the not-yellow region.
5. We will order the options.

We will start by calculating the probability of the green region.

Probability of Green Region

The area of the green region is the area of two green triangles. Since the side length of the smaller square is 6 inches, the length of the base of the triangles is also 6 inches. Now we will find the height of the triangles.

By using the Segment Addition Postulate we will find the height of one of the green triangles. 18 in. = h + 6 in. + hFrom here we will calculate h.

18 = h + 6 + h

▼

Solve for h

SubEqn

LHS-6=RHS-6

12 = h + h

AddTerms

Add terms

12 = 2 h

DivEqn

.LHS /2.=.RHS /2.

6 = h

RearrangeEqn

Rearrange equation

h=6

Using the length of the base (6 inches) and the height (6 inches), we can find the area of one of the green triangles. Therefore, we will multiply the expression by 2. Area of a Green Region: 2* 12(6)(6)=36in.^2 We found the area of the green region. To find the probability of the dart landing in this region, we need to evaluate the total area of the figure. The board is a square that has 18-inch side lengths, so let's find the area of the board. Area of the Figure 18^2=324in.^2 With this information we will find the probability.

P(Green)=36/324

▼

Evaluate right-hand side

ReduceFrac

a/b=.a /36./.b /36.

P(Green)=1/9

UseCalc

Use a calculator

P(Green)=0.11111...

RoundDec

Round to 2 decimal place(s)

P(Green)≈ 0.11

We will continue with the probability of the dart landing on a not-blue region.

Probability of Not Blue Region

Notice that the area of the not-blue region is the area of the yellow circle. Because one of the side lengths of the bigger square is 18 inches, the diameter of the big yellow circle is also 18 inches. Therefore, its radius is 182= 9 inches.

We will substitute this value of the radius into the formula for the area of a circle.

Area of yellow circle=π r^2

Substitute

r= 9

Area of yellow circle=π ( 9^2)

▼

Evaluate right-hand side

CalcPow

Calculate power

Area of yellow circle=π (81)

CommutativePropMult

Commutative Property of Multiplication

Area of yellow circle=81π

By using the area of the yellow circle and the area of the board we will find the probability.

P(not Blue)=81π/324

▼

Evaluate right-hand side

ReduceFrac

a/b=.a /81./.b /81.

P(not Blue)=π/4

UseCalc

Use a calculator

P(not Blue)=0.78539...

RoundDec

Round to 2 decimal place(s)

P(not Blue)≈ 0.79

It is time to find the probability of the dart landing on a red region.

Probability of Red Region

We will find the area of the red region, which is the smaller circle. We are given the side length of the smaller square as 6 inches. The red circle fits the smaller square, so the diameter of the red circle is 6 inches. With this information the radius is 62= 3 inches. We are ready to find the area of the circle.

Area of inner circle=π r^2

Substitute

r= 3

Area of inner circle=π ( 3^2)

▼

Evaluate right-hand side

CalcPow

Calculate power

Area of inner circle=π (9)

CommutativePropMult

Commutative Property of Multiplication

Area of inner circle=9π

With the information of the area of the red region and the area of the board, we will evaluate the probability.

P(Red)=9π/324

▼

Evaluate right-hand side

ReduceFrac

a/b=.a /9./.b /9.

P(Red)=π/36

UseCalc

Use a calculator

P(Red)=0.08727...

RoundDec

Round to 2 decimal place(s)

P(Red)≈ 0.09

Finally, we will find the probability for the last option.

Probability of Not Yellow Region

Note that finding the probability of landing in the yellow area is easier than finding the probability of landing in the not yellow region. Therefore we will find the probability of the complement of landing in the not-yellow region. We found the area of yellow region and the probability of the dart landing on the yellow region in a previous exercise. Area of the Yellow Region = (81 π - 72) inches^2 P(yellow)≈ 0.56 We will use the complement of the event to find the probability.

P(not Yellow)=1-P(Yellow)

Substitute

P(Yellow)= 0.56

P(not Yellow)=1- 0.56

SubTerm

Subtract term

P(not Yellow)=0.44

We can finally order the options.

Order of Options

We know the probabilities of the options.

	Events	Probability
A.	Dart landing in the green region	0.11
B.	Dart landing in not blue region	0.79
C.	Dart landing in red region	0.09
D.	Dart landing in not yellow region	0.44

From here we can order the likelihoods of the options from least likely to most likely. The probability that the dart is landing in the red region is the least likely option, whereas the probability that the dart is landing in the not-blue region is the most likely option. C,A,D,B