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.