Chapter 4.1, Problem 46E

Solution

(a)

To prove : $S=\frac{WR}{100}\text{ millimeters}=\frac{WR}{2540}\text{ inches}$

Given information :

The numbers in the “tire type” give the size of the tire in the $P-$ metric system. Each number is of the form $W/R-D$ , where $W$ is the width of the tire in millimeters, $R/100$ is the ratio of the sidewall ( $S$ ) of the tire to its width $W$ , and $D$ is the diameter (in inches) of the wheel without the tire.

Vehicle	Tyre Type	Tyre Diameter (inches)
Chevy Spark EV	$188/55-15$	$23.0$
Nissan Leaf SL	$215/50-17$	$25.5$
Tesla S	$245/45-19$	$27.7$

  

Concept used:

  $1\text{ inch}=25.4\text{ millimeter}$

Proof :

It is given that $\frac{R}{100}$ is the ratio of the sidewall ( $S$ ) of the tire to its width $W$ .

So,

  $\begin{array}{c}\frac{R}{100}=\frac{S}{W}\\ S=\frac{WR}{100}\text{ millimeter}\\ S=\frac{WR}{100}÷25.4\text{ inches }\left[1\text{ inch}=25.4\text{ mm}\right]\\ S=\frac{WR}{100}\times \frac{1}{25.4}\text{ inches}\\ S=\frac{WR}{2540}\text{ inches}\end{array}$

Hence, it is proved that $S=\frac{WR}{100}\text{ millimeters}=\frac{WR}{2540}\text{ inches}$ .

(b)

To derive : A formula for the tire diameter that involves only the variables $D,W,\text{ and }R$ , if the diameter of tire is $D+2S$ .

The required formula is $D+2S=D+\frac{WR}{50}\text{ millimeters}=D+\frac{WR}{1270}\text{ inches}$ .

Given information :

The numbers in the “tire type” give the size of the tire in the $P-$ metric system. Each number is of the form $W/R-D$ , where $W$ is the width of the tire in millimeters, $R/100$ is the ratio of the sidewall ( $S$ ) of the tire to its width $W$ , and $D$ is the diameter (in inches) of the wheel without the tire.

Vehicle	Tyre Type	Tyre Diameter (inches)
Chevy Spark EV	$188/55-15$	$23.0$
Nissan Leaf SL	$215/50-17$	$25.5$
Tesla S	$245/45-19$	$27.7$

Calculation :

The diameter of tire is $D+2S$ . From the given information,

  $\begin{array}{c}\frac{R}{100}=\frac{S}{W}\\ S=\frac{WR}{100}\text{ millimeter}\end{array}$

So,

  $\begin{array}{c}D+2S=D+2\left(\frac{WR}{100}\text{ millimeters}\right)\\ =D+\frac{WR}{50}\text{ millimeters}\end{array}$

As $D$ is the diameter (in inches), so

  $\begin{array}{c}D+2S=D+2\left(\frac{WR}{2540}\text{ inches}\right)\\ =D+\frac{WR}{1270}\text{ inches}\end{array}$

Therefore, the required formula is $D+2S=D+\frac{WR}{50}\text{ millimeters}=D+\frac{WR}{1270}\text{ inches}$ .

(c)

To calculate : The tire diameter for the 2013 Cadillac Escalade, which comes with $\frac{265}{65}-18$ tires.

The tire diameter is $\text{31}\text{.56 inches}$ .

Given information :

The numbers in the “tire type” give the size of the tire in the $P-$ metric system. Each number is of the form $W/R-D$ , where $W$ is the width of the tire in millimeters, $R/100$ is the ratio of the sidewall ( $S$ ) of the tire to its width $W$ , and $D$ is the diameter (in inches) of the wheel without the tire.

Vehicle	Tyre Type	Tyre Diameter (inches)
Chevy Spark EV	$188/55-15$	$23.0$
Nissan Leaf SL	$215/50-17$	$25.5$
Tesla S	$245/45-19$	$27.7$

Calculation :

From the given information,

  $\begin{array}{l}W=265\\ D=18\\ R=65\end{array}$

From the above part, the diameter of the tire is $D+\frac{WR}{1270}\text{ inches}$ .

The diameter of the car will be,

  $\begin{array}{l}=D+\frac{WR}{1270}\text{ inches}\\ \text{=}18+\frac{265\times 65}{1270}\text{ inches}\\ \text{=}18+\frac{17225}{1270}\text{ inches}\\ \approx \text{18+13}\text{.56 inches}\\ \approx \text{31}\text{.56 inches}\end{array}$

Therefore, the tire diameter is $\text{31}\text{.56 inches}$ .