Exercise 13 Page 673

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 we throw a dart at the board shown, and want to find the probability that the dart lands in the yellow region. Therefore, the probability is the ratio of the area of the yellow region to the area of the figure. P(The dart lands in the yellow region)= [0.8em] Area of the yellow region/Area of the figure We will find the area of the yellow region and the area of the entire figure one at a time. Then, we will find their ratio.

Area of the Yellow Region

Notice that the side length of the square is 18 inches. Therefore, the diameter of the big yellow circle is also 18 inches and its radius is 182= 9 inches. 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=81π

Notice that inside this big circle, we also have two green triangles and a white square. These are not yellow, so we have to subtract their areas from the area of the big yellow circle. The side length of the white square is 6 inches. To find its area, we need to find the square of the side length. Area of the White Square: 6^2= 36in.^2 We still have to look for the area of green triangles.

We can find the height of the green triangle using the Segment Addition Postulate. 18 in. = h + 6 in. + h Now, we can use this equation to calculate h.

18 = h + 6 + 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

Moreover, from the diagram we can see that the base of the green triangle is 6 inches long. Using these values, we can find the area using the formula for area of a triangle. Area of a Green Triangle: 12(6)(6)=18in.^2 Finally, to find the area of the yellow region, we will subtract the area of white square and two green triangles from the area of the big yellow circle. Yellow Area: & 81π - 36 - 2 * 18 = & (81 π - 72) inches^2

Area of the Figure

The figure is a square with 18 inch sides. To find its area, we need to find the square of the side length. Area of the Figure: 18^2=324inches^2

Probability

As previously mentioned, the probability that the dart lands in the yellow region is the ratio of the area of the yellow region to the area of the figure. Since we already know both areas, we can find their ratio.

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

SubstituteValues

Substitute values

P=81π-72/324

FactorOut

Factor out 9

P=9(9π-8)/324

ReduceFrac

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

P=9π-8/36

▼

Convert to percent

UseCalc

Use a calculator

P = 0.56317594...

RoundDec

Round to 2 decimal place(s)

P ≈ 0.56

WritePercent

Convert to percent

P ≈ 56 %

The probability that a dart thrown at the board lands on the yellow area is about 0.56 or about 56 %.