Question

track a 400-meter olympic-size track can be modeled with a rectangle and two semicircles.
a. if an athlete runs around the track once, how far has the athlete traveled to the nearest meter?
m

Explanation:

Step1: Calculate the circumference of the circular part

The two semicircles make a full circle. The diameter of the circle is 73 m, so the radius \( r = \frac{73}{2} \) m. The circumference of a circle is \( C = \pi d \) (where \( d \) is the diameter), so \( C = \pi \times 73 \approx 3.1416 \times 73 \approx 229.3468 \) m.

Step2: Calculate the length of the two straight parts

Each straight part is 84.39 m, so two straight parts have a length of \( 2 \times 84.39 = 168.78 \) m.

Step3: Calculate the total distance around the track

Add the circumference of the circle and the length of the two straight parts: \( 229.3468 + 168.78 \approx 398.1268 \approx 398 \) m? Wait, no, wait. Wait, the problem says it's a 400 - meter track, maybe my calculation is wrong? Wait, let's recalculate the circle circumference. Wait, \( \pi\times73\): 733.1415926535 = 733 + 730.1415926535 = 219+10.3362637 = 229.3362637. Then two straight parts: 84.392 = 168.78. Then total is 229.3362637 + 168.78 = 398.1162637, which is approximately 398 m? But the track is supposed to be 400 - meter. Wait, maybe the diameter is different? Wait, maybe I misread the diagram. Wait, the diagram has 73 m (diameter of the semicircles) and 84.39 m (length of the straight part). Wait, maybe the correct calculation: Wait, let's check again. Wait, maybe the formula is correct. So the total distance is the perimeter of the track, which is the circumference of the circle (from the two semicircles) plus twice the length of the straight section. So \( C = \pi d+ 2l \), where \( d = 73 \) m, \( l = 84.39 \) m. So \( C=\pi\times73 + 2\times84.39\). Let's compute \( \pi\times73\): 733.1416 = 229.3368. Then 284.39 = 168.78. Sum: 229.3368+168.78 = 398.1168, which is approximately 398 m. But the problem says "a 400 - meter Olympic - size track", maybe there's a mistake in the given dimensions? Or maybe I misread the numbers. Wait, maybe the length of the straight part is 85.39? Let me check the diagram again. The user's diagram: "73 m" and "84.39 m". So with the given numbers, the calculation is as above. So the athlete runs around the track once, so the distance is the perimeter of the track, which we calculated as approximately 398 m (to the nearest meter). Wait, but maybe my calculation of the circle is wrong. Wait, the two semicircles: the diameter is 73 m, so the circumference of the full circle is \( \pi d=3.1416\times73 = 229.3368 \). Then two straight sections: 84.39*2 = 168.78. Total: 229.3368 + 168.78 = 398.1168, which rounds to 398 m.

Answer:

\( \boxed{398} \)