To solve this problem, we first need to calculate the father's rate of speed and then use that to determine the son's speed and how far the son will travel in 9 hours.

Step 1: Find the Father's Speed

The father walks 36 miles in 12 hours. So, the father's rate of speed is:

$\text{Father's speed} = \frac{\text{Distance}}{\text{Time}} = \frac{36 \, \text{miles}}{12 \, \text{hours}} = 3 \, \text{miles per hour}.$

Step 2: Find the Son's Speed

The son's speed is twice the father's speed. Therefore, the son's speed is:

$\text{Son's speed} = 2 \times 3 \, \text{miles per hour} = 6 \, \text{miles per hour}.$

Step 3: Find How Far the Son Will Travel in 9 Hours

Now, we calculate how far the son will travel in 9 hours at 6 miles per hour:

$\text{Distance traveled by the son} = \text{Speed} \times \text{Time} = 6 \, \text{miles per hour} \times 9 \, \text{hours} = 54 \, \text{miles}.$

Answer:

The bicycle rider will travel 54 miles in 9 hours.

So, the correct answer is 54 miles.

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Would you like to go over more details on this problem or have any other questions? Here are some related questions to explore:

1. How does the concept of "rate of speed" affect the time and distance traveled?
2. What would happen to the total distance if the son's speed were increased by 50%?
3. How could you calculate the time it would take the son to travel the 36 miles at his current rate of speed?
4. How would the problem change if both the father and the son were traveling at the same speed?
5. What factors could affect how far a cyclist can travel compared to walking?

Tip: Always remember that the relationship between speed, distance, and time is linear, so you can use the formula $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ to find any missing value if two are known.