https://file.aiquickdraw.com/mathbot/file/1727188346582arhgiphv.jpg

### Solution Breakdown:

1. **(a) Graphical Representation and Mathematical Function:**
   The best fit for the data is a quadratic function:
   \[
   s(t) = 3.583t^2 - 1.233t + 2.986
   \]
   This function describes the relationship between time (\(t\)) and distance (\(s\)).

2. **(b) Distance at 4 seconds:**
   Using the quadratic function, the distance at \(t = 4\) seconds is approximately:
   \[
   s(4) \approx 55.38 \, \text{meters}
   \]

3. **(c) Distance at 12 seconds:**
   Extrapolating the quadratic function to \(t = 12\) seconds gives:
   \[
   s(12) \approx 504.08 \, \text{meters}
   \]

4. **(d) Distance from the starting point at 12 seconds:**
   This is the same as the distance calculated in part (c): 504.08 meters.

5. **(e) Interpolation vs. Extrapolation:**
   - **Interpolation**: When the predicted values lie within the range of given data. For instance, finding the distance at 4 seconds (which is between 2 and 10 seconds) involves interpolation.
   - **Extrapolation**: Predicting values outside the given range. The calculation for 12 seconds is an extrapolation because it lies outside the measured data (the data goes up to 10 seconds).

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### Related Questions:
1. What is the physical interpretation of each term in the quadratic function?
2. How does extrapolation differ from interpolation in terms of reliability?
3. Could we fit a different model (e.g., cubic) and compare results?
4. How would this analysis change if we had more data points?
5. What assumptions are we making about the motion of the motorcycle?

### Tip:
Always be cautious when using extrapolation, as the further from the known data, the less reliable the prediction becomes.

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] y s(m) [14, 32, 88, 170, 220, 350].

Learn how to analyze the motion of a racing motorcycle using quadratic equations. This problem covers key concepts like interpolation, extrapolation, and the quadratic formula, using data points for time and distance. Suitable for students in Grades 9-12.

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