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This problem can be solved using a variation of Snell's Law, where we minimize the total time the triathlete spends swimming and running. 

### Given:
- Distance from shore \(a = 140.0 \, \text{m}\)
- Distance from shore to bike \(b = 70.0 \, \text{m}\)
- Total horizontal distance \(x + y = 177.9 \, \text{m}\)
- Swimming speed \(v_{\text{swim}} = 1.416 \, \text{m/s}\)
- Running speed \(v_{\text{run}} = 8.85 \, \text{m/s}\)

### Objective:
We need to find the value of \(x\) (the distance the athlete swims parallel to the shore) to minimize the total time.

### Step-by-step approach:

1. **Time for Swimming:**
   - The swimming part consists of the horizontal distance \(x\) and the vertical distance \(a\), so the swimming distance is \(\sqrt{x^2 + a^2}\).
   - Time spent swimming \(T_{\text{swim}} = \frac{\sqrt{x^2 + a^2}}{v_{\text{swim}}}\).

2. **Time for Running:**
   - The running part is from where the athlete exits the water to the bike, a horizontal distance \(y = 177.9 - x\).
   - Time spent running \(T_{\text{run}} = \frac{y}{v_{\text{run}}} = \frac{177.9 - x}{v_{\text{run}}}\).

3. **Total time:**
   - The total time \(T(x)\) is the sum of swimming time and running time:
     \[
     T(x) = \frac{\sqrt{x^2 + a^2}}{v_{\text{swim}}} + \frac{177.9 - x}{v_{\text{run}}}
     \]

4. **Minimize the total time:**
   - To minimize the total time, we can differentiate \(T(x)\) with respect to \(x\) and set it equal to 0. Then, solve for \(x\).

I'll now calculate the optimal value of \(x\) and the minimum time.The optimal value of \(x\), the distance the triathlete swims along the shore, is approximately **22.69 meters**.

The minimum time required for the triathlete to reach the bicycle is approximately **117.70 seconds**.

Would you like more details on how the solution was derived, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. How does Snell's Law relate to this optimization problem?
2. What would happen if the swimming speed and running speed were closer to each other?
3. How does changing the distance \(a\) (swimming leg) affect the total time?
4. What would be the result if the swimming speed were faster than the running speed?
5. How can you use calculus to find minimum or maximum values in other real-life problems?

**Tip:** The derivative of a function helps find points of local minima or maxima, which is key in optimization problems.

A triathlete on the swimming leg of a triathlon is 140.0 m from the shore. The triathlete's bike is 70.0 m from the shore on the land. The component of her distance from the bicycle along the shoreline is 177.9 m. If the triathlete's running speed is 8.85 m/s and swimming speed is 1.416 m/s, calculate the value of x so that the triathlete reaches her bike in the least amount of time.

In this problem, we calculate the optimal path for a triathlete who must swim and run to her bike. Using Snell's Law for minimizing time, calculus, and optimization techniques, the solution shows how to balance the athlete's swimming and running speeds for the least total time.

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