Question

The school track has eight lanes. Each lane is 1.25 meters wide. The arc at each end of the track is 180. The distance of the home straight and the radii for the arcs in the 1st 4 lanes are given.

S=85m

r1=36.5m

r2=37.75m

r3=39m

r4=40.25m


Part one: Find the radii of lanes 5 through 8 of the track. Show your work.


Part two: If Max ran around lane one, how far did he run? Show your work and explain your solution.


Part three: Max wants to run a total of three laps around the track, choose two additional lanes (2-8) for him to run and find the distance around those two lanes. Show your work and round to the hundredths.


Part 4: Based on your lane choices in part three, what was the total distance Max ran in the three laps around the track?

Answer 1

Part one: r5 = 41.5 m , r6 = 42.75 m , r7 = 44 m , r8 = 45.25

Part two: Max ran 399.34 m in lane one

Part three: The distance in lanes 3 and 7 are 415.04 m and 446.46 m

Part four: Max ran in the three laps 1260.84 m around the track

Explanation:

* lets explain how to solve the problem

- The school track has eight lanes

- Each lane is 1.25 meters wide

- The arc at each end of the track is 180° , that means the arc at each

end is a semi-circle

- The distance of the home straight for all lanes is 85 m

- The radius of the first lane is 36.5 m

∵ The width of each lane is 1.25

∴ The radius of the second lane = 36.5 + 1.25 = 37.75

- That means the radius of each lane increased by 1.25 then the

previous lane

∴ The radius of each lane = the radius of the previous lane + 1.25

# Part one:

∵ The radius of the 4 lane is 40.25

∴ The radius of the 5th lane = 40.25 + 1.25 = 41.5 m

∴ The radius of the 6th lane = 41.5 + 1.25 = 42.75 m

∴ The radius of the 7th lane = 42.75 + 1.25 = 44 m

∴ The radius of the 8th lane = 44 + 1.25 = 45.25 m

* r5 = 41.5 m , r6 = 42.75 m , r7 = 44 m , r8 = 45.25

- The length of each lane is the lengths of the 2 end arcs and 2

home straight distance

∵ The arc is a semi-circle

∵ The length of the semi-circle is πr

∴ The length of the 2 arcs is 2πr

∵ The length of the home straight distance is 85 m

∴ The length of each lane = 2πr + 2 × 85

∴ The length of each lane = 2πr + 170

# Part two:

- Max ran around lane one

∵ The radius of lane one = 36.5 m

∵ The distance of each lane = 2πr + 170

∴ The distance of lane one = 2π(36.5) + 170 = 399.34 m

* Max ran 399.34 m in lane one

# Part three:

- We will chose lanes 3 and 7

∵ The distance of each lane = 2πr + 170

∵ The radius of lane 3 = 39

∵ The radius of lane 7 is 44

∴ The distance of lane 3 = 2π(39) + 170 = 415.04 m

∴ The distance of lane 7 = 2π(44) + 170 = 446.46 m

* The distance in lanes 3 and 7 are 415.04 m and 446.46 m

# Part four:

- To find the total distance that Max ran in the 3 laps ad the answers

in part two and part three

∵ Max ran 399.34 m in lane one

∵ Max ran 415.04 m in lane three

∵ Max ran 446.46 m in lane seven

∴ The total distance of the 3 lanes = 399.34 + 415.04 + 446.46

∴ The total distance of the 3 lanes = 1260.84

* Max ran in the three laps 1260.84 m around the track