Chapter 4.1, Problem 68E

a.

To determine

Findthe speed of the bicycle in thegiven measurement.

Answer to Problem 68E

Thespeed in the given measurements are $\text{14}\text{.66}\text{feets/second}$ and $\text{9}\text{.996}\text{miles/hour}$ .

Explanation of Solution

Given:

The radii of the socket, the wheel sprocket, and the wheel of the bicycle are $4$ inches, $2$ inches and $14$ inches respectively. The pedaling rate of the cyclist is $1 \text{revolutions/second}$ .

  

Calculation:

Circumference of the socket $=2\pi r=2\pi \times 4\text{inches}=8\pi \text{inches}$

Circumference of the sprocket $=2\pi r=2\pi \times 2\text{inches}=4\pi \text{inches}$

Circumference of the wheel $=2\pi r=2\pi \times 14\text{inches}=28\pi \text{inches}$

Now,

Speed of the bicycle

  $\begin{array}{l}=28\pi \text{inches}\times \frac{8\pi \text{inches}}{4\pi \text{inches}}\times 1 \text{revolutions/second}\\ =28\pi \times 2\text{revolutions/second}\\ \text{=175}\text{.929}\text{inches/second}\end{array}$

Measurement in feet/second

  $\begin{array}{l}\text{175}\text{.929}\text{inches/second}\\ \text{=175}\text{.929}\times \frac{1}{12}\text{feets/second}\\ \text{=14}\text{.66}\text{feets/second}\end{array}$

Measurement in miles/hour

  $\begin{array}{l}\text{175}\text{.929}\text{inches/second}\\ \text{=175}\text{.929}\times 0.0568182\text{miles/hour}\\ \text{=9}\text{.996}\text{miles/hour}\end{array}$

Hence thespeed inthe given measurements are $\text{14}\text{.66}\text{feets/second}$ and $\text{9}\text{.996}\text{miles/hour}$ .

b.

To determine

Find the function for the distance $d$ for $n$ number of revolution of the pedal sprocket using result of ‘a’.

Answer to Problem 68E

The function is $d=n\times \frac{7n\pi }{7920}\text{miles}$ .

Explanation of Solution

Given:

Given:

The radii of the socket, the wheel sprocket, and the wheel of the bicycle are $4$ inches, $2$ inches and $14$ inches respectively. The pedaling rate of the cyclist is $1 \text{revolutions/second}$ .

  

Calculation:

Circumference of the socket $=2\pi r=2\pi \times 4\text{inches}=8\pi \text{inches}$

Circumference of the sprocket $=2\pi r=2\pi \times 2\text{inches}=4\pi \text{inches}$

Circumference of the wheel $=2\pi r=2\pi \times 14\text{inches}=28\pi \text{inches}$

Now,

Speed of the bicycle

  $\begin{array}{l}=28\pi \text{inches}\times \frac{8\pi \text{inches}}{4\pi \text{inches}}\times 1 \text{revolutions/second}\\ =28\pi \times 2\text{revolutions/second}\\ \text{=175}\text{.929}\text{inches/second}\end{array}$

Measurement in feet/second

  $\begin{array}{l}\text{175}\text{.929}\text{inches/second}\\ \text{=175}\text{.929}\times \frac{1}{12}\text{feets/second}\\ \text{=14}\text{.66}\text{feets/second}\end{array}$

Measurement in miles/hour

  $\begin{array}{l}\text{175}\text{.929}\text{inches/second}\\ \text{=175}\text{.929}\times 0.0568182\text{miles/hour}\\ \text{=9}\text{.996}\text{miles/hour}\end{array}$

Distance travelled $=$ number of revolution of the pedal sprocket $\times$ distance travelled per revolution.

  $\begin{array}{l}d=n\times \frac{105\pi }{33}\text{miles/hour}\\ \Rightarrow d=n\times \frac{105\pi }{33\times 3600}\text{miles/second}\\ \Rightarrow d=n\times \frac{7\pi }{7920}\text{miles}\\ \Rightarrow d=n\times \frac{7n\pi }{7920}\text{miles}\end{array}$

Hence thefunction is $d=n\times \frac{7n\pi }{7920}\text{miles}$ .

c.

To determine

Find the function for the distance $d$ in terms of time $t$ .

Answer to Problem 68E

The function for the distance $d$ in terms of time $t$ is $d\left(t\right)=\frac{7n\pi t}{7920}\text{miles}$ .

Explanation of Solution

Given:

Given:

The radii of the socket, the wheel sprocket, and the wheel of the bicycle are $4$ inches, $2$ inches and $14$ inches respectively. The pedaling rate of the cyclist is $1 \text{revolutions/second}$ .

  

Calculation:

Circumference of the socket $=2\pi r=2\pi \times 4\text{inches}=8\pi \text{inches}$

Circumference of the sprocket $=2\pi r=2\pi \times 2\text{inches}=4\pi \text{inches}$

Circumference of the wheel $=2\pi r=2\pi \times 14\text{inches}=28\pi \text{inches}$

Now,

Speed of the bicycle

  $\begin{array}{l}=28\pi \text{inches}\times \frac{8\pi \text{inches}}{4\pi \text{inches}}\times 1 \text{revolutions/second}\\ =28\pi \times 2\text{revolutions/second}\\ \text{=175}\text{.929}\text{inches/second}\end{array}$

Measurement in feet/second

  $\begin{array}{l}\text{175}\text{.929}\text{inches/second}\\ \text{=175}\text{.929}\times \frac{1}{12}\text{feets/second}\\ \text{=14}\text{.66}\text{feets/second}\end{array}$

Measurement in miles/hour

  $\begin{array}{l}\text{175}\text{.929}\text{inches/second}\\ \text{=175}\text{.929}\times 0.0568182\text{miles/hour}\\ \text{=9}\text{.996}\text{miles/hour}\end{array}$

Distance travelled $=$ number of revolution of the pedal sprocket $\times$ distance travelled per revolution.

  $\begin{array}{l}d=n\times \frac{105\pi }{33}\text{miles/hour}\\ \Rightarrow d=n\times \frac{105\pi }{33\times 3600}\text{miles/second}\\ \Rightarrow d=n\times \frac{7\pi }{7920}\text{miles}\\ \Rightarrow d=n\times \frac{7n\pi }{7920}\text{miles}\end{array}$

Hence thefunction is $d=n\times \frac{7n\pi }{7920}\text{miles}$ .

Use formula $d=rt$ .

  $d=n\times \frac{7n\pi }{7920}\text{miles}$

Cyclist pedal rate $=1\text{revolution}/\text{second}$

  $d\left(t\right)=\frac{7n\pi t}{7920}\text{miles}$

Hence the function for the distance $d$ in terms of time $t$ is $d\left(t\right)=\frac{7n\pi t}{7920}\text{miles}$ .