A Carousel Problem

Ann is making plans to build a carousel for a new amusement park. The specifications are that the carousel be built of wood, be at least 40’ in diameter and that it should seat at least 20 people at a time.

Ann plans to make the 40’ solid flooring of large oak planks that are 1’ wide. Clearly, she will need 40 planks, but she hopes to save money by only buying the minimum length she needs for each plank. The middle two planks must be 40’ long, but she will need to shave off a curved part of this plank in order to create the circular base she needs. (See the attached Sketchpad illustration.)

1. How much shorter from the center (vertical) line is point B than point A?

.025 ft shorter

2. So what must the length of the next two planks be?

19.975 ft

3. What is the measure of the angle ÐAOB in degrees?

2.8660º

4. What is the measure of ÐAOB in radians?

.05002

5. What is the length of the arc AB?

1.0004

6. How much shorter from the center line is point C than point B?

.0753 ft

7. So what must the length of the next two planks be?

19.8997 ft

8. What is the measure of ÐAOC in degrees?

5.739º

9. What is the measure of ÐAOC in radians?

.1002

10. So what is the measure of ÐBOC in degrees/radians?

2.8732º/.0501

11. And what is the length of arc BC?

1.0029

12. Use your knowledge of recursion and sequences on your calculator to complete the table on the back. Feel free to write a program if you wish.

Board Number / Longest edge / Longest length from center line / Shortest length from center line / Central angle measure / Board’s arc length
1 / 40 / 20 / 19.975’ / 2.86598° / 1.00042
2 / 39.95’ / 19.975’ / 19.900’ / 2.8732° / 1.0029
3 / 39.8’ / 19.900’ / 19.774’ / 2.8878° / 1.0080
4 / 39.548’ / 19.774’ / 19.596’ / 2.9100° / 1.0158
5 / 39.192’ / 19.596’ / 19.365’ / 2.9406° / 1.0264
6 / 38.73’ / 19.365’ / 19.079’ / 2.98009° / 1.0402
7 / 38.158’ / 19.079’ / 18.735’ / 3.0297° / 1.0576
8 / 37.47’ / 18.735’ / 18.3303 / 3.0909° / 1.0789
9 / 36.6606’ / 18.3303 / 17.861’ / 3.1655° / 1.1050
10 / 35.722’ / 17.861’ / 17.321’ / 3.2563° / 1.13667
11 / 34.642’ / 17.321’ / 16.703’ / 3.3670° / 1.1753
12 / 33.406’ / 16.703’ / 16’ / 3.5029° / 1.2227
13 / 32’ / 16’ / 15.199’ / 3.6717° / 1.2817
14 / 30.398’ / 15.199’ / 14.283’ / 3.8854° / 1.3563
15 / 28.566’ / 14.283’ / 13.229’ / 4.1634° / 1.4533
16 / 26.458’ / 13.229’ / 12’ / 4.5397° / 1.5847
17 / 24’ / 12’ / 10.536’ / 5.0816° / 1.7738
18 / 21.072’ / 10.536’ / 8.718’ / 5.9464° / 2.0757
19 / 17.436’ / 8.718’ / 6.245’ / 7.6471° / 2.6693
20 / 12.49’ / 6.245’ / 0’ / 18.1949° / 6.3512

13. Find and explain the sum of the angle measures.

The sum is 90°, so the combined central angle measures of the boards will form one quarter, or 90°, of the platform.

14. Find and explain the sum of the arc lengths.

The sum is 31.4159, or 10π, meaning that the arc lengths are equal to one quarter of 40π, which is the circumference of the platform.

15. Ann has 24 horses that she would like to place on her carousel. If she places them

2’ from the edge of the carousel, what is the distance between the horses?

4.7124 ft.

How many horses do you think Ann should place on her carousel?

Explain your answer.

Eight. The horses should not be too close together so the ride won’t be cramped. Having only eight horses spaced equally two feet from the edge of the carousel will leave about 14.137 feet between each rider, which is plenty of room.

16. If she wants the carousel to make 2 revolutions per minute, how fast will those

horses be traveling in miles per hour?

2.5704 mph

17. If she wants the horses to travel 5 miles per hour, how fast must the carousel

turn? It must make 3.8905 revolutions per minute.