https://file.aiquickdraw.com/mathbot/file/1727189217213uirtksar.jpg

Here are the results based on the analysis of the motorcycle's distance over time:

1. **Graph and Mathematical Equation**:  
   A cubic interpolation was applied to model the data, which shows the trend of the motorcycle's acceleration. The graph reflects this function, fitting the known data points smoothly.

2. **Where will the motorcycle be at 4 seconds?**  
   At \( t = 4 \) seconds, the motorcycle will be approximately **56.65 meters** from the starting line. This value was interpolated from the data.

3. **When is the motorcycle 126 meters from the start?**  
   The motorcycle reaches 126 meters at around **5.99 seconds** (close to 6 seconds). This value was also interpolated from the data.

4. **What is the distance at 12 seconds?**  
   At \( t = 12 \) seconds, the motorcycle will be approximately **543.63 meters** from the starting line. This value was extrapolated beyond the available data.

5. **Interpolation and Extrapolation**:  
   - Interpolation was used for the 4-second and 126-meter cases since these fall within the given data range.
   - Extrapolation was used for the 12-second case, as it goes beyond the last data point (10 seconds).

Would you like further details on the calculations or explanations?

### Additional Related Questions:
1. How would you derive the equation from a polynomial fit instead of cubic interpolation?
2. Can you determine the velocity at different times from the graph?
3. What would happen to the motorcycle's distance if the acceleration was constant?
4. How does the cubic interpolation compare to a linear one in this context?
5. How can the function be used to predict times or distances beyond 12 seconds?

**Tip**: For better accuracy, always check the range of data when using interpolation and extrapolation methods.

Una moto de carreras arranca desde la línea de salida. Se mide el tiempo que tarda en pasar por determinados puntos de la recta de aceleración y los valores obtenidos son: t (s) [2, 3, 5, 7, 8, 10], e (m) [14, 32, 88, 170, 220, 350]. a) Realiza la representación gráfica y determina la función matemática que la describe. b) ¿Dónde se encontrará a los cuatro segundos? c) ¿Cuándo está a 126 metros de la salida? d) ¿A qué distancia del punto de salida está a los doce segundos? e) ¿En cuáles de los apartados anteriores has interpolado y cuándo has extrapolado?

Analyze the motion of a motorcycle as it accelerates over time, with questions involving polynomial interpolation, determining distances at specific times, and extrapolating future positions. This is a high school-level problem exploring the concepts of acceleration and function fitting.

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