1. Lesson 97
Concentric Circles

2. Vocabulary/Review
𝐴=𝜋 𝑟 2 & 𝐶=2𝜋𝑟 The equation of a circle with radius r and center (h, k) is 𝑥−ℎ 2 + 𝑦−𝑘 2 = 𝑟 2 Concentric Circles are coplanar circles with the same center Annulus is the region between 2 concentric circles

3. Write the equations of concentric circles
Need center & radius
(-2, 3); r = 2; r = 4
𝑥 𝑦−3 2 =4
𝑥 𝑦−3 2 =16

4. Formula for area of the annulus
𝐴=𝜋 𝑟 2
How will you find the shaded region?
𝐴=𝜋 𝑟 −𝜋 𝑟 2 2 Or
𝐴=𝜋 𝑟 − 𝑟 2 2

5. Find the exact area of the shaded region
𝐴=𝜋 𝑟 1 2 −𝜋 𝑟 2 2 𝐴=𝜋 7 2 −𝜋 3 2 𝐴=49𝜋−9𝜋 𝐴=40𝜋 𝑐𝑚 2

6. Alternate Method
𝐴=𝜋 𝑟 1 2 −𝜋 𝑟 2 2 𝐴=𝜋 7 2 −𝜋 3 2 Factor 𝐴=𝜋 7 2 − 3 2 Difference of Squares 𝐴=𝜋 7+3 7−3 𝐴=𝜋 10 4 𝐴=40𝜋 𝑐𝑚 2

7. Application: Targets
The bull’s eye of the target has a radius of 2 inches. Each annulus beyond is 1 inch wide. What is the probability of scoring :
100 points?
75 points?
50 points?
25 points?
We need to find each circle’s radius

8. Application: Targets
The bull’s eye of the target has a radius of 2 inches. Each annulus beyond is 1 inch wide. What is the probability of scoring :
100 points?
75 points?
50 points?
25 points?
Now find the area of each region

9. Application: Targets
The bull’s eye of the target has a radius of 2 inches. Each annulus beyond is 1 inch wide. What is the probability of scoring :
100 points?
75 points?
50 points?
25 points? 4𝜋
9𝜋−4𝜋=5𝜋
16𝜋−9𝜋=7𝜋
25𝜋−16𝜋=9𝜋
Now find the probability

10. Application: Targets 100 points? 4𝜋 25𝜋 = 4 25 75 points? 5𝜋 25𝜋 = 1 5
4𝜋 25𝜋 = 4 25
75 points?
5𝜋 25𝜋 = 1 5
50 points?
7𝜋 25𝜋 = 7 25
25 points?
9𝜋 25𝜋 = 9 25

11. Questions?
The area of the bull’s eyes appears to be larger then each annulus. This is known as the Bull’s Eye Illusion. Which is why the smaller areas are worth more points on a target. However, you believe your chances of getting a bull’s eye are greater than the other regions.