Chapter 6.5, Problem 14E

a.

To determine

To find the average velocity of the car during the first 12 seconds.

Answer to Problem 14E

The average velocity of the car during the first 12 seconds is $45.33\text{ km/h}$

Explanation of Solution

Given:velocity of the car during the first 12 seconds

Calculation:

Use the midpoint rule with $n=3$ ,we can write

  ${\displaystyle \underset{0}{\overset{12}{\int }}v\left(t\right)}dt=\frac{12-0}{3}\left[v\left(2\right)+v\left(6\right)+v\left(10\right)\right]$

Use the graph to substitute the values of $v\left(2\right),v\left(6\right),v\left(10\right)$

  $\begin{array}{l}=4\left[20+50+66\right]\\ =544\end{array}$

Average of velocity:

  $\begin{array}{l}=\frac{1}{12}{\displaystyle \underset{0}{\overset{12}{\int }}v\left(t\right)}dt\\ =\frac{1}{12}\left(544\right)\\ =45.33\text{ km/h}\end{array}$

Thus the average velocity of the car during the first $12$ seconds is

b.

To determine

Find the time of the instantaneous velocity equal to the average velocity.

Answer to Problem 14E

The time of the instantaneous velocity equal to the average velocity is $t\approx 5.2$

Explanation of Solution

From part (a), know that average velocity is $45.33\text{ km/h}$

Use the graph to see that the average velocity is $45.33\text{ km/h when }t\approx 5.2$