Question

in order to estimate the length of the runway, a passenger on an airplane jotted down some velocity data during takeoff from the on - board entertainment screen. from the resulting table given below, calculate (\frac{o_{8}+u_{8}}{2}) to find his estimate. round any intermediate calculations, if needed, to no less than six decimal places, and round your final answer to three decimal places.

time (s)	6	12	18	24	30	36	42	48
v (mph)	30	79	117	150	172	204	223	230

Explanation:

Step1: Identify $O_8$ and $U_8$

$O_8$ is the left Riemann sum, $U_8$ is the right Riemann sum. Time interval $\Delta t = 6$ s $= \frac{6}{3600}$ h $= \frac{1}{600}$ h.

Step2: Calculate left sum $O_8$

Sum left endpoints, multiply by $\Delta t$:
$O_8 = \frac{1}{600} \times (30 + 79 + 117 + 150 + 172 + 204 + 223 + 230)$
$O_8 = \frac{1}{600} \times 1105 = 1.841667$

Step3: Calculate right sum $U_8$

Sum right endpoints, multiply by $\Delta t$:
$U_8 = \frac{1}{600} \times (79 + 117 + 150 + 172 + 204 + 223 + 230 + 230)$
$U_8 = \frac{1}{600} \times 1405 = 2.341667$

Step4: Compute the average

Find $\frac{O_8 + U_8}{2}$:
$\frac{1.841667 + 2.341667}{2} = \frac{4.183334}{2}$

Answer:

1.092 miles