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Big Ideas Math Geometry, 2014

BI

Big Ideas Math Geometry, 2014 View details

Cumulative Assessment

Exercise 10 Page 719

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.

B

Practice makes perfect

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 our dart is equally likely to hit any point inside the given squared board, and want to find the probability that the point lies in the yellow region. The probability of that event is the ratio of the area of the yellow region to the area of the board. P(The point is in the yellow region)= [0.8em] Area of the yellow region/Area of the board

We will find the area of the yellow region and the area of the entire board one at a time. Then, we will find their ratio.

Area of the Yellow Region

Notice that the width of the yellow region is 2 inches and the radius of the most inner circle is also 2 inches. The radius of the middle circle is the sum of the inner circle's radius and the width of the yellow area. Let's call the radius of the middle circle, r_1. r_1=2+2=4 Let's call the radius of the inner circle, r_2. We will calculate the area of the yellow region by subtracting the area of the inner circle from the area of the middle circle. Since we know the radiuses of both circles, let's substitute those values in the formula for the area of a circle.

A=π r_1^2-π r_2^2

r_1= 4, r_2= 2

A=π 4^2-π 2^2

▼

Evaluate right-hand side

Calculate power

A=π 16-π 4

CommutativePropMult

Commutative Property of Multiplication

A=16π-4π

Subtract term

A= 12π

Area of the Board

The given board is a square with the outer circle inscribed in it. To find its area, we need to find its side length. To do it, we will find the length of the diameter of the outer circle. Since it is inscribed in the square, its diameter will have the same length as the square's side. The radius of the outer circle is the sum of the middle circle's radius and the width of the blue area. Outer Circle Radius: 4+2=6 The side length of the board, which is the same as the length of the diameter of the blue circle, will be twice the length of this radius. Side of the Board: 2* 6=12 To find the area of the board, we need to find the square of this side length. Area of the Board: 12^2= 144 in^2

Probability

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

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

SubstituteValues

Substitute values

P=12π/144

a/b=.a /12./.b /12.

P=π/12

The probability that a dart lands in the yellow area is π12. This corresponds to answer B.

Subchapter links

Exercises p.718-719