Alice is competing in a cycling race on an extremely windy day. The race is an 'out and back again' course, so the wind is against Alice in one direction and assisting her in the other. For the first half of the race the wind is blowing against Alice, slowing her down by 4 km per hour. Given that on a normal day Alice could maintain a pace of 36 km per hour and that this race took her 4 hours and 57 minutes, calculate the total distance of the course.

Let's break down the problem and solve it step by step.

1. **Variables and Known Information**:
   - Normal speed of Alice without wind, \( v \): 36 km/h
   - Wind effect (against): -4 km/h
   - Wind effect (with): +4 km/h
   - Total time, \( t \): 4 hours 57 minutes = 4.95 hours (since 57 minutes is 0.95 of an hour)

2. **Effective Speeds**:
   - Against the wind: \( v_1 = 36 - 4 = 32 \) km/h
   - With the wind: \( v_2 = 36 + 4 = 40 \) km/h

3. **Distance Calculation**:
   Let \( d \) be the total distance of the course (one way). The total time for the course, \( t \), consists of the time against the wind, \( t_1 \), and the time with the wind, \( t_2 \).

   The time to travel one way against the wind:
   \[ t_1 = \frac{d}{v_1} = \frac{d}{32} \]

   The time to travel one way with the wind:
   \[ t_2 = \frac{d}{v_2} = \frac{d}{40} \]

   The total time for the race:
   \[ t = t_1 + t_2 = \frac{d}{32} + \frac{d}{40} \]

4. **Equation for Total Time**:
   \[ 4.95 = \frac{d}{32} + \frac{d}{40} \]

5. **Solving for \( d \)**:
   We need to find a common denominator to combine the fractions:

   \[ \frac{d}{32} + \frac{d}{40} = \frac{40d}{1280} + \frac{32d}{1280} = \frac{72d}{1280} \]

   \[ 4.95 = \frac{72d}{1280} \]

   Solving for \( d \):

   \[ 4.95 = \frac{72d}{1280} \]

   \[ 4.95 \times 1280 = 72d \]

   \[ 6336 = 72d \]

   \[ d = \frac{6336}{72} \]

   \[ d = 88 \]

So, the total distance of the course (one way) is 88 km, making the entire 'out and back' course 176 km.

**Would you like a detailed explanation of any part of this solution?**

Here are 8 follow-up questions to expand your understanding:

1. How would the total time change if the wind speed increased by 2 km/h?
2. What would the total distance be if Alice’s normal speed was 40 km/h instead?
3. How much time did Alice spend cycling with the wind?
4. How much time did Alice spend cycling against the wind?
5. What is the average speed for the entire race?
6. If the wind was only blowing against Alice the entire time, how would that affect her race time?
7. How would a change in Alice’s normal pace affect the calculation?
8. How can we use this information to predict Alice's performance in future races under similar conditions?

**Tip**: In problems involving varying speeds, carefully distinguish between the time spent at each speed and the distances covered.

Learn how to calculate the total distance of a cycling race affected by wind. Given Alice's normal speed and wind conditions, find the total distance of the 'out and back' course. Solve step-by-step with detailed formulas and calculations.

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