Chapter 4.1, Problem 29E

(a)

To determine

To writea linear model that represents the world record (in minutes) for the men’s mile as a function of the number of the years since $1960$ .

Answer to Problem 29E

The linear model is $y=-0.005x+3.91$ .

Explanation of Solution

Given:

Men’s mile in $1960$$=$$3.91$ minutes

Men’s mile in $1980$$=$$3.81$ minutes

Formula Used:

The equation of a line is written as $y=mx+b$ where m is the slope and $b$ is the $y$ -intercept.

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

Given:

Men’s mile in $1960$$=$$3.91$ minutes

Men’s mile in $1980$$=$$3.81$ minutes

Let the men’s mile in $1960$ be $x=0$

Thus, we have two points as $\left(0,3.91\right)$ and $\left(20,3.81\right)$

Thus, slope $=$$m$$=$$\frac{y_2-y_1}{x_2-x_1}$

  $m$$=$$\frac{3.81-3.91}{20-0}$

  $m$$=$$-0.005$

As per the above definition, we have equation of the line as

  $y_1=m{x}_1+b$

  $\begin{array}{l}3.91=\left(-0.005\times 0\right)+b\\ b=3.91\end{array}$

Thus, the equation of the line −

  $\begin{array}{l}y=\left(-0.005\times x\right)+3.91\\ y=-0.005x+3.91\end{array}$

Thus, the equation of the line is $y=-0.005x+3.91$ .

Conclusion:

Hence, the linear model is $y=-0.005x+3.91$ .

(b)

To determine

To estimate the record time in $2000$ and predict the record time in $2020$ .

Answer to Problem 29E

The record time in $2000$ is $3.71$ minutes and record time in $2020$ is $3.61$ minutes.

Explanation of Solution

Given:

Equation of the line is $y=-0.005x+3.91$ .

Formula Used:

The equation of a line is written as $y=mx+b$ where m is the slope and $b$ is the $y$ -intercept.

  $m=\frac{y_2-y_1}{x_2-x_1}$

Calculation:

Given:

Equation of the line is $y=-0.005x+3.91$ .

Thus, when $x=40$

  $\begin{array}{l}y=\left(-0.005\times 40\right)+3.91\\ y=3.71\end{array}$

And when $x=60$

  $\begin{array}{l}y=\left(-0.005\times 60\right)+3.91\\ y=3.61\end{array}$

Thus, the record time in $2000$ is $3.71$ minutes and record time in $2020$ is $3.61$ minutes.

Conclusion:

Hence, the record time in $2000$ is $3.71$ minutes and record time in $2020$ is $3.61$ minutes.