Questions: The Ironman Triathlon is a race that consists of three parts: a 2.4 mile swim followed by a 112 mile bike race and then a 26.2 mile marathon. A participant swims steadily at 2 mph, cycles steadily at 20 mph, and then runs steadily at 9 mph. Assuming that no time is lost during the transition from one stage to the next, find a formula for the distance, d, covered in miles, as a function of the elapsed time t in hours, from the beginning of the race. Graph the function.

Solution Steps

The participant swims at 2 mph for 2.4 miles, cycles at 20 mph for 112 miles, and runs at 9 mph for 26.2 miles. We need to calculate the time taken for each segment.

• Swimming: \[ t_{\text{swim}} = \frac{2.4 \text{ miles}}{2 \text{ mph}} = 1.2 \text{ hours} \]

• Cycling: \[ t_{\text{cycle}} = \frac{112 \text{ miles}}{20 \text{ mph}} = 5.6 \text{ hours} \]

• Running: \[ t_{\text{run}} = \frac{26.2 \text{ miles}}{9 \text{ mph}} \approx 2.9111 \text{ hours} \]

The total time for the race is the sum of the times for each segment: \[ t_{\text{total}} = t_{\text{swim}} + t_{\text{cycle}} + t_{\text{run}} = 1.2 + 5.6 + 2.9111 \approx 9.7111 \text{ hours} \]

The distance function \(d(t)\) is piecewise:

• For \(0 \leq t < 1.2\), the participant is swimming: \[ d(t) = 2t \]

• For \(1.2 \leq t < 6.8\), the participant is cycling: \[ d(t) = 2.4 + 20(t - 1.2) \]

• For \(6.8 \leq t \leq 9.7111\), the participant is running: \[ d(t) = 114.4 + 9(t - 6.8) \]

Final Answer