To estimate the distance driven between 2:00 pm and 8:00 pm, we need to approximate the area under the velocity curve from $x = 2$ to $x = 8$, where $f(x)$ gives the velocity at various times after noon. The table provides values at different intervals, allowing us to use a midpoint Riemann sum or trapezoidal rule for approximation.

Data Interpretation

• 2:00 pm corresponds to $x = 2$.
• 8:00 pm corresponds to $x = 8$.

The function values between these times are:

• $f(2) = 47$
• $f(4) = 56$
• $f(6) = 61$
• $f(8) = 65$

Using the Trapezoidal Rule

The Trapezoidal Rule estimates the area by averaging adjacent values over each interval and multiplying by the interval width. Here, each interval width is 2 hours.

$\text{Distance} \approx \frac{2}{2} \left( f(2) + 2 \cdot f(4) + 2 \cdot f(6) + f(8) \right)$ Substituting values: $= 2 \left( 47 + 56 + 61 + 65 \right)$ $= 2 \times 229 = 458 \text{ miles}$

So, the estimated distance driven between 2:00 pm and 8:00 pm is 458 miles.

Would you like further details on this estimation or have any questions?

Here are some related questions:

1. What other methods could be used to estimate distances when given velocity data?
2. How would using a midpoint Riemann sum compare to the trapezoidal rule in accuracy for this problem?
3. What assumptions are we making about the function between data points?
4. How would the result change if we had more data points within this interval?
5. How can the integral of a velocity function be used to determine total distance traveled?

Tip: Estimations improve with more data points or smaller intervals, especially when velocity changes are non-linear.