Chapter 6, Problem 51RE

(a)

To determine

To calculate:The total fuel consumption during the hour.

Answer to Problem 51RE

The total fuel consumption is 2.42 gal

Explanation of Solution

Given information:

During a trip, a passenger recorded the fuel consumption every 5 minutes for a fun hour of travel.

  

Formula used:

  $h=\frac{b-a}{n}$

Calculation:

In the trapezoidal rule, if [ a, b ] is partitioned into n subintervals of equal length $h=\frac{b-a}{n}$ , the graph of f on [ a, b ] can be approximated by a straight line segment at each subinterval. The region between the curve and the x-axis is then approximated by the trapezoids, the area of each trapezoid being the length of its horizontal “altitude" times the average of two vertical “bases".

That is,

  $\begin{array}{l}{\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=h\frac{y_0+y_1}{2}+h\frac{y_1+y_2}{2}+\mathrm{.....}+h\frac{y_{n-1}+y_n}{2}\\ =h\left(\frac{y_0}{2}+y_1+y_2+\mathrm{....}+y_{n-1}+\frac{y_n}{2}\right)\\ {\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=\frac{h}{2}\left(y_0+2y_1+2y_2+\mathrm{.....}+2y_{n-1}+y_n\right)\end{array}$

Where,

  $y_0=f\left(a\right),y_1=f\left(x_1\right)\mathrm{..........}{y}_{n-1}=f\left(x_{n-1}\right),y_n=f\left(b\right)$

Now, apply the trapezoidal rule with

  $\begin{array}{l}n=12\\ \left[a,b\right]=\left[0,60\right]\end{array}$

So,

  $\begin{array}{l}h=\frac{b-a}{n}\\ =\frac{60-0}{12}\\ h=5\end{array}$

Now, since

  $\begin{array}{l}5\min =5\times \frac{1}{60}\text{hr}\\ 5\min =\frac{1}{12}\text{hr}\end{array}$

Now,

  $\begin{array}{l}\text{total fuel consumption}=\frac{\frac{1}{12}}{2}\left(\begin{array}{l}2.5+2\left(2.4\right)+2\left(2.3\right)+2\left(2.4\right)+2\left(2.4\right)\\ +2\left(2.5\right)+2\left(2.6\right)+2\left(2.5\right)+2\left(2.4\right)+2\left(2.3\right)\\ +2\left(2.4\right)+2\left(2.4\right)+2.3\end{array}\right)\\ =\frac{58}{24}\\ \text{total fuel consumption}=2.42\text{gal}\end{array}$

Conclusion:

The total fuel consumption is 2.42 gal

(b)

To determine

To calculate:The fuel efficiency based on the portion of the trip.

Answer to Problem 51RE

The fuel efficiency is approximately equal to 24.83 miles per gallon

Explanation of Solution

Given information:

During a trip, a passenger recorded the fuel consumption every 5 minutes for a fun hour of travel.

  

Formula used:

  $\text{fuel efficiency}=\frac{\text{distance covered in one hour}}{\text{total fuel consumed in one hour}}$

Calculation:

Fuel efficiency is given by

  $\begin{array}{l}\text{fuel efficiency}=\frac{\text{distance covered in one hour}}{\text{total fuel consumed in one hour}}\\ =\frac{60\text{miles}}{2.42\text{gallon}}\\ \text{fuel efficiency}=24.83\text{miles per gallon}\end{array}$

The fuel efficiency is approximately equal to 24.83 miles per gallon

Conclusion:

The fuel efficiency is approximately equal to 24.83 miles per gallon