Question

track a 400 - meter olympic - size track can be modeled with a rectangle and two semicircles. c. each lane is 1.22 meters wide. if the athlete runs in the center of the inside lane, how far has she traveled after a single lap to the nearest meter? d. how far inside the track should the athlete be positioned to run exactly 400 meters? round your answer to the nearest centimeter.

Explanation:

Step1: Analyze the track components

The track is composed of two straight - parts and two semi - circles. The length of the straight - part is 84.39 m and the diameter of the inner semi - circles is 73 m.

Step2: Calculate the circumference of the inner semi - circles

The two semi - circles together form a full circle. The formula for the circumference of a circle is $C = \pi d$, where $d = 73$ m. So $C=\pi\times73\approx3.14\times73 = 229.22$ m.

Step3: Calculate the total length of the two straight - parts

The length of each straight - part is 84.39 m, so the total length of the two straight - parts is $2\times84.39=168.78$ m.

Step4: Calculate the length of the inner - most track

The length of the inner - most track $L = 229.22+168.78 = 398$ m.

Step5: Let the distance inside the track be $x$

The difference between the 400 - m track and the inner - most track length is due to the adjustment in the circular part. Let the new diameter of the circular part for a 400 - m track be $d'$. The length of the straight - parts remains the same. So $400=2\times84.39+\pi d'$. First, simplify the right - hand side: $2\times84.39 = 168.78$. Then $400 - 168.78=\pi d'$, so $231.22=\pi d'$. Solving for $d'$, we get $d'=\frac{231.22}{\pi}\approx\frac{231.22}{3.14}=73.64$ m. The original diameter is 73 m. The difference in diameter is $73.64 - 73=0.64$ m. The distance inside the track (radius difference) is $\frac{0.64}{2}=0.32$ m. Since 1 m = 100 cm, $0.32$ m = 32 cm.

Answer:

c. 398 m
d. 32 cm