you and a companion are driving a twisty stretch of road in a car with a speedometer but no odometer. to…

you and a companion are driving a twisty stretch of road in a car with a speedometer but no odometer. to find out how long this road is, you record the cars velocity at 10 - second intervals. a. estimate the length of the road using left - endpoint values. ft time (s) velocity (ft/s) 0 0 10 30 20 18 30 41 40 14 50 40 60 24 time (s) velocity (ft/s) 70 46 80 11 90 38 100 10 110 21 120 35

you and a companion are driving a twisty stretch of road in a car with a speedometer but no odometer. to find out how long this road is, you record the cars velocity at 10 - second intervals. a. estimate the length of the road using left - endpoint values. ft time (s) velocity (ft/s) 0 0 10 30 20 18 30 41 40 14 50 40 60 24 time (s) velocity (ft/s) 70 46 80 11 90 38 100 10 110 21 120 35

Answer

Explanation:

Step1: Recall left - endpoint Riemann sum formula

The left - endpoint Riemann sum for approximating the distance $d$ given velocity $v(t)$ over intervals $[t_{i},t_{i + 1}]$ with $\Delta t=t_{i+1}-t_{i}$ is $d\approx\sum_{i = 0}^{n - 1}v(t_{i})\Delta t$. Here, $\Delta t = 10$ s.

Step2: Identify the left - endpoint velocities

The left - endpoint velocities for the 12 intervals (from $t = 0$ to $t=120$ s with $\Delta t = 10$ s) are $v(0)=0$, $v(10)=30$, $v(20)=18$, $v(30)=41$, $v(40)=14$, $v(50)=40$, $v(60)=24$, $v(70)=46$, $v(80)=11$, $v(90)=38$, $v(100)=10$, $v(110)=21$.

Step3: Calculate the sum

[ \begin{align*} d&\approx10\times(0 + 30+18 + 41+14+40+24+46+11+38+10+21)\ &=10\times(0+30 + 18+41+14+40+24+46+11+38+10+21)\ &=10\times293\ &=2930 \end{align*} ]

Answer:

$2930$