Speedometer readings for a vehicle (in motion) at 7 -second ntervals are given in the table. \begin{tabular}{|c|c|} \hline$t(\mathrm{sec})$ & $v(\mathrm{ft} / \mathrm{s})$ \\ \hline 0 & 27 \\ \hline 7 & 27 \\ \hline 16 & 27 \\ \hline 21 & 16 \\ \hline 28 & 21 \\ \hline 35 & 26 \\ \hline \end{tabular} Estimate the distance traveled by the vehicle during this 35-second period using the velocities at the beginning of the time intervals. Distance traveled $\approx$ feet Give another estimate using the velocities at the end of the time periods. Distance traveled feet

Step 1 ：Given the speedometer readings for a vehicle (in motion) at 7-second intervals as shown in the table below:

Step 2 ：\[\begin{tabular}{|c|c|} \hline t(\mathrm{sec}) & v(\mathrm{ft} / \mathrm{s}) \\ \hline 0 & 27 \\ \hline 7 & 27 \\ \hline 16 & 27 \\ \hline 21 & 16 \\ \hline 28 & 21 \\ \hline 35 & 26 \\ \hline \end{tabular}\]

Step 3 ：We are asked to estimate the distance traveled by the vehicle during this 35-second period using the velocities at the beginning of the time intervals.

Step 4 ：The distance traveled by a vehicle can be estimated by multiplying the velocity by the time interval. In this case, we can use the velocities at the beginning and end of each time interval to estimate the distance traveled during that interval. We can then sum up these estimates to get the total distance traveled during the 35-second period.

Step 5 ：Let's denote the time intervals as \(t = [ 0, 7, 16, 21, 28, 35]\) and the corresponding velocities as \(v = [27, 27, 27, 16, 21, 26]\). The time intervals are \(dt = [7, 9, 5, 7, 7]\).

Step 6 ：By multiplying the velocity at the beginning of each time interval by the length of the interval, we get the estimated distance traveled during that interval. Summing up these estimates, we get the total estimated distance traveled during the 35-second period as \(d_{\text{begin}} = 826\) feet.

Step 7 ：Similarly, by multiplying the velocity at the end of each time interval by the length of the interval, we get another estimate for the distance traveled during that interval. Summing up these estimates, we get another total estimated distance traveled during the 35-second period as \(d_{\text{end}} = 841\) feet.

Step 8 ：Final Answer: The estimated distance traveled by the vehicle during this 35-second period using the velocities at the beginning of the time intervals is approximately \(\boxed{826}\) feet. Another estimate using the velocities at the end of the time periods is approximately \(\boxed{841}\) feet.