To solve this problem, we can approach it step by step.

Step 1: Calculate the Distance You Travel in One Revolution

1. Outer Horse (Your Horse): The distance from the center to your horse is the radius $r = 21 \, \text{ft}$.

   • The circumference $C$ of a circle (which represents one revolution around the carousel) is given by: $C = 2 \pi r$
   • For your horse: $C_{\text{outer}} = 2 \pi \times 21 = 42 \pi \approx 131.95 \, \text{ft}$

2. Inner Horse (Cousin's Horse): The radius for your cousin’s horse is $r = 14 \, \text{ft}$.

   • For your cousin's horse: $C_{\text{inner}} = 2 \pi \times 14 = 28 \pi \approx 87.96 \, \text{ft}$

Step 2: Calculate Your Linear Speed

Since the ride takes 150 seconds to complete 12 revolutions, we can first find the time per revolution and then use that to find the linear speed.

1. Time Per Revolution:

   • The total time for 12 revolutions is 150 seconds, so the time per revolution is: $\text{Time per revolution} = \frac{150}{12} = 12.5 \, \text{seconds}$

2. Linear Speed Calculation:

   • Linear speed $v$ is given by the distance traveled per revolution divided by the time per revolution.
   • For your horse: $v_{\text{outer}} = \frac{C_{\text{outer}}}{\text{Time per revolution}} = \frac{131.95}{12.5} \approx 10.56 \, \text{ft/s}$

Summary of Results

• Distance you travel in one revolution: approximately $131.95 \, \text{ft}$
• Your linear speed: approximately $10.56 \, \text{ft/s}$

Would you like further details or explanations on any part of this solution?

Here are some additional questions that explore related concepts:

1. How would the speed change if the radius of the carousel were increased?
2. What would your cousin’s linear speed be on the inner horse?
3. How does the linear speed relate to angular velocity on a carousel?
4. How would the time for each revolution change if the carousel slowed down?
5. What effect does distance from the center have on total distance traveled?

Tip: Remember, linear speed depends on both the radius and the angular speed (revolutions per time), so moving closer or farther from the center affects how fast you travel!