Chapter 4.1, Problem 68E

a.

To determine

Find the speed of the bicycle in feet per second and miles per hour.

Answer to Problem 68E

  $\boxed{\frac{35}{11}miles/hour}$ , $\boxed{\frac{14\pi }{3}feet/\sec }$

Explanation of Solution

Given information:

The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are $4$ inches, $2$ inches, and $14$  inches, respectively. A cyclist pedals at a rate of $1$ revolution per second.

  

Find the speed of the bicycle in feet per second and miles per hour.

Calculation:

The radii of the pedal sprocket,

  $Radius=4inches$

Circumference = $2\pi r$

  $2\pi \times 4=8\pi$

The radii of wheel sprocket,

  $Radius=2inches$

Circumference = $2\pi r$

  $2\pi \times 2=4\pi$

When pedal sprocket complete one revolution the wheel sprocket complete two revolutions.

The pedal sprocket turns, $2\pi \times 1=2\pi radian/\sec ond$ ,

The wheel sprocket turns,

  $2\pi \times 2=4\pi radian/\sec ond$

So that the central angle traced by bicycle wheel is $4\pi radian/\sec ond$

  $\theta =4\pi ,t=1\sec$

Linear speed determine the speed of the bicycle with radius $r=14inches$

  $\begin{array}{l}linear,speed=\frac{s}{t}\\ =\frac{r\theta }{t}\\ =\frac{14\times 4\pi }{1\sec }\\ =56inches/\sec \end{array}$

We know that,

  $\begin{array}{l}1inch/\sec =\frac{1}{12}feet/\sec \\ linear,speed=\frac{56\pi }{12}feet/\sec \\ speed=\frac{14\pi }{3}feet/\sec \end{array}$

  $\begin{array}{l}1feet/\sec =\frac{3600}{5280}miles/hour\\ speed=\frac{14\pi \times 3600}{3\times 5280}miles/hour\\ speed=\frac{35}{11}miles/hour\end{array}$

Hence speed is $\boxed{\frac{35}{11}miles/hour}$ , $\boxed{\frac{14\pi }{3}feet/\sec }$ .

b.

To determine

Write a function for the distance $d$ (in miles) a cyclist travels in terms of the number $n$ of revolutions of the pedal sprocket.

Answer to Problem 68E

  $\boxed{\frac{\left(56\pi n\right)}{63360}miles}$

Explanation of Solution

Given information:

The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are $4$ inches, $2$ inches, and $14$  inches, respectively. A cyclist pedals at a rate of $1$ revolution per second.

  

Use your result from part $\left(a\right)$ to write a function for the distance $d$ (in miles) a cyclist travels in terms of the number $n$ of revolutions of the pedal sprocket.

Calculation:

As the cyclist is pedaling at the rate of $1radian/\sec ond$ makes $1revolution$ . When pedal sprocket complete one revolution the wheel sprocket complete two revolutions.

The pedal sprocket turns, $2\pi \times 1=2\pi radian/\sec ond$ ,

As each revolution generates $2\pi radians$ .

Pedal sprockets turns = $2\pi radians$

Wheel sprockets turns = $4\pi radians$

Central angle is traced bicycle wheel is $\theta =4\pi n,for,t=1$

Now arc length travelled by bicycle wheel in $n$ revolution is given by,

  $\begin{array}{l}s=r\theta \\ s=14\times \left(4\pi n\right)inches\\ =56\pi ninches\end{array}$

The distance travelled in one second,

  $\begin{array}{l}d=\left(56\pi n\right)inches\\ iinch=\frac{1}{63360}miles\\ d=\frac{\left(56\pi n\right)}{63360}miles\end{array}$

  $\boxed{\frac{\left(56\pi n\right)}{63360}miles}$

Hence, the function for the distance $d$ is, $\boxed{\frac{\left(56\pi n\right)}{63360}miles}$

c.

To determine

Compare the function with the function from part $\left(b\right)$ .

Answer to Problem 68E

  $\boxed{\text{equal}.}$

Explanation of Solution

Given information:

The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are $4$ inches, $2$ inches, and $14$  inches, respectively. A cyclist pedals at a rate of $1$ revolution per second.

  

Write a function for the distance $d$ (in miles) a cyclist travels in terms of the time $t$ (in seconds). Compare this function with the function from part $\left(b\right)$ .

Calculation:

As per part $\left(a\right)$ and $\left(b\right)$ ,

The distance travelled in one second,

  $\begin{array}{l}d=\left(56\pi n\right)inches\\ iinch=\frac{1}{63360}miles\\ d=\frac{\left(56\pi n\right)}{63360}miles\end{array}$

  $\boxed{\frac{\left(56\pi n\right)}{63360}miles}$ or

  $\begin{array}{l}d=r\theta \\ =\frac{56\pi }{63360}miles\end{array}$

  $\begin{array}{l}linear,speed=\frac{s}{t}\\ =\frac{56\pi }{63360}miles/\sec \\ dis\tan ce,travelled=linear,speed\times time\\ =\frac{56\pi }{63360}miles\times t\\ d=\boxed{\frac{56\pi }{63360}t,miles}\end{array}$

Hence, both the functions are $\boxed{\text{equal}.}$ .