I. to Understand the Mathematics Involved in Designing a Fair Race Track

Racetrack Design Project

Objectives:

I. To understand the mathematics involved in designing a fair race track.

II. To design a four lane oval track with straight-aways and semicircular ends.

In this project we will be using the following definition for the phrase head start. If the runner in the innermost lane begins at the start line, the head start for each other runner is the distance he/she is ahead of the start line.

To help you address the first objective, complete each of the tables below. Notice what is the same and what is different about the information used in each of the tables.

Table 1: Circular Track, Inner Radius 50 m, Lane Width 1 m

Lane Radius (m) Total Distance Head Start (m) Final Distance Around Track Terms of π/Nearest Tenth Terms of π/Nearest Tenth Terms of π/Nearest Tenth 1 50 100π / 314.2 0π / 0.0 100π / 314.2 2 51 3 4

Table 2: Circular Track, Inner Radius 65 m, Lane Width 1 m

Lane Radius (m) Total Distance Head Start (m) Final Distance Around Track Terms of π/Nearest Tenth Terms of π/Nearest Tenth Terms of π/Nearest Tenth 1 65 130π / 408.4 0π / 0.0 130π / 408.4 2 66 3 4

Table 3: Circular Track, Inner Radius 65 m, Lane Width 1.5 m

Lane Radius (m) Total Distance Head Start (m) Final Distance Around Track Terms of π/Nearest Tenth Terms of π/Nearest Tenth Terms of π/Nearest Tenth 1 65 130π / 408.4 0π / 0.0 130π / 408.4 2 66.5 3 4 Table 4: Oval Track, Inner Radius 30 m, Straightaway 100 m, Lane Width 1m

Lane Radius (m) Length of Total Distance Around Head Start (m) Final Distance Straight-Away (m) Track Terms of π/Nearest Tenth Terms of π/Nearest Tenth Terms of π/Nearest Tenth 1 30 100 200+60π / 388.5 0π / 0.0 200+60π / 388.5 2 31 3 4

Table 5: Oval Track, Inner Radius 30 m, Straightaway 200 m, Lane Width 1 m

Lane Radius (m) Length of Total Distance Around Head Start (m) Final Distance Straight-Away (m) Track Terms of π/Nearest Tenth Terms of π/Nearest Tenth Terms of π/Nearest Tenth 1 30 200 400+60π / 588.5 0π / 0.0 400+60π / 588.5 2 31 3 4 Now that you’ve completed the tables, answer each of the following questions in three complete sentences each.

1. Does the inner radius affect what the head starts will be for the other runners? (Refer to Tables 1 &2)

2. Does the width of the lane affect what the head start will be for the other runners? (Refer to Tables 2&3)

3. Do the lengths of the straight-aways affect what the head start will be for the other runners? (Refer to Tables 4 &5)

4. Based on the information you’ve learned how can you determine the head start of a runner in any race? Designing your own race track!

Your teacher will assign you one of the tracks from the table below for you to design. Once you know which track, create a table like the one in Tables 4 & 5 using your assigned numbers and include it with your four paragraphs from above. You will need to determine the following details of your track.

1. The length of the straight-aways.

2. What you want your lane width to be.

3. The head start for each of the runners in the outer lanes. (The runner in the inner lane will start and finish at the same place.)

4. The scale you will use to draw a nice, neat scale drawing of your track to show to the class and turn in.

5. A creative setting or theme for your track.

Design Radius of Inner Loop Length of Race (m) (m) A 35m 800 B 25 400 C 50 800 D 40 600 E 30 500 F 37.5 700 G 50 400

Some Suggestions to help:

1. Complete your tables on the handout. 2. Make a table for your design (first row only for now) – it should look like Table 4 or 5 3. Decide what you’d like your scale to be – look at a ruler for this and consider how large you would like for your final draft to be – centimeters recommended instead of inches. 4. Draw a rough draft of track – include how long your straightaways will be in centimeters (or inches) and how long your radius/diameter will be in centimeters (or inches) – see if these lengths are reasonable or adjust your scale. 5. Determine what you’d like your lane width to be and calculate what that would look like based on your scale – should you make your lanes that big/small in your picture? if not… adjust your scale on your rough draft or change your lane widths – your lane can be as wide as you’d like to make it! 6. Once you’ve made your decision and envisioned what your final drawing will look like, finish the last 3 rows of your table using the lane width you have chosen. 7. When you draw your final draft begin with a rectangle made of your 2 straightaways and 2 diameters. Then put a semicircle on each end. 8. DO NOT FORGET TO PUT YOUR SCALE ON YOUR DRAWING!! 9. For creativity points – add some color and landscaping/contestants/signs etc to your track!