Speedometer readings for a motorcycle at 12-second intervals are given in the table.
t (s)	0	12	24	36	48	60
v (ft/s)	30	27	24	22	24	27
(a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals.

To estimate the distance traveled by the motorcycle during the given time period using the velocities at the beginning of the time intervals, we can apply the left Riemann sum method. This involves summing the products of each time interval duration and the velocity at the beginning of each interval.

Given:
- Time intervals (\( \Delta t \)) = 12 seconds
- Velocities at the beginning of each interval: 30, 27, 24, 22, 24, and 27 ft/s

Here's the step-by-step calculation:

1. Identify the velocities at the beginning of each interval:
   - \( v(0) = 30 \) ft/s
   - \( v(12) = 27 \) ft/s
   - \( v(24) = 24 \) ft/s
   - \( v(36) = 22 \) ft/s
   - \( v(48) = 24 \) ft/s

2. Multiply each velocity by the duration of the interval (12 seconds):

\[
\begin{aligned}
\text{Distance}_1 &= v(0) \times \Delta t = 30 \times 12 = 360 \text{ ft} \\
\text{Distance}_2 &= v(12) \times \Delta t = 27 \times 12 = 324 \text{ ft} \\
\text{Distance}_3 &= v(24) \times \Delta t = 24 \times 12 = 288 \text{ ft} \\
\text{Distance}_4 &= v(36) \times \Delta t = 22 \times 12 = 264 \text{ ft} \\
\text{Distance}_5 &= v(48) \times \Delta t = 24 \times 12 = 288 \text{ ft} \\
\end{aligned}
\]

3. Sum these distances to get the total estimated distance traveled:

\[
\text{Total Distance} = \text{Distance}_1 + \text{Distance}_2 + \text{Distance}_3 + \text{Distance}_4 + \text{Distance}_5 
\]

\[
\text{Total Distance} = 360 + 324 + 288 + 264 + 288 = 1524 \text{ ft}
\]

So, the estimated distance traveled by the motorcycle during this time period is \( 1524 \) feet.

Would you like more details or have any questions about the calculations?

Here are some related questions for further exploration:
1. How would the distance estimate change if we used the velocities at the end of each interval?
2. What if we used the midpoint velocities of each interval for the estimate?
3. How could we use a trapezoidal sum for this estimation?
4. What is the average velocity of the motorcycle over the 60 seconds?
5. How would you convert the distance from feet to meters?
6. How can we use calculus to find the exact distance traveled?
7. What are the advantages of using different methods (left sum, right sum, midpoint, trapezoidal)?
8. How does the speed variation affect the accuracy of the distance estimate?

**Tip:** When estimating areas under curves, comparing different numerical methods (like Riemann sums and trapezoidal rules) can provide insights into the best method for a given data set.

Learn how to estimate the distance traveled by a motorcycle using the left Riemann sum method. This problem involves applying the method to velocities measured at 12-second intervals. Suitable for advanced high school students.

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