A car travels a distance given by f(t)=3t² (where f(t) is the distance in miles after t hours). Find the average speed of the car from hour 1 to hour 4.
*

The average speed of the car over an interval is found using the formula:

\[
\text{Average Speed} = \frac{\text{Change in Distance}}{\text{Change in Time}}
\]

Given the function \( f(t) = 3t^2 \), we calculate the distance at \( t = 1 \) and \( t = 4 \):

1. Distance at \( t = 1 \):
   \[
   f(1) = 3(1)^2 = 3 \text{ miles}
   \]

2. Distance at \( t = 4 \):
   \[
   f(4) = 3(4)^2 = 3(16) = 48 \text{ miles}
   \]

Now, the change in distance (\( \Delta f \)) is:
\[
\Delta f = f(4) - f(1) = 48 - 3 = 45 \text{ miles}
\]

The change in time (\( \Delta t \)) is:
\[
\Delta t = 4 - 1 = 3 \text{ hours}
\]

Thus, the average speed is:
\[
\text{Average Speed} = \frac{45 \text{ miles}}{3 \text{ hours}} = 15 \text{ miles per hour}
\]

### Final Answer:
The average speed of the car from hour 1 to hour 4 is **15 miles per hour**.

Would you like further details or have any questions?

### Related Questions:
1. How do you find the instantaneous speed of the car at a specific time \( t \)?
2. What is the significance of the derivative of the function \( f(t) \) in this context?
3. How would the average speed change if the function was \( f(t) = 3t^3 \) instead?
4. What is the total distance traveled by the car from hour 0 to hour 4?
5. Can you explain the difference between average speed and instantaneous speed?

### Tip:
The average speed over an interval can always be calculated by taking the total distance traveled divided by the total time taken, regardless of the function form.

A car travels a distance given by f(t)=3t² (where f(t) is the distance in miles after t hours). Find the average speed of the car from hour 1 to hour 4.

Learn how to find the average speed of a car over an interval using the quadratic distance function f(t)=3t². This example involves calculating the change in distance and time, suitable for high school students studying quadratic functions and motion problems.

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