Chapter 4, Problem 12P

a.

To determine

Use the $Ln\mathrm{Re}g$ command on your calculator to find a logarithmic model for these production ?gures.

Answer to Problem 12P

  $C=-7154.888128+(1061.006551)In t$

Explanation of Solution

Given information:

A logarithmic model is a function of the form

  $y=a+binx$

Many relationships between variables in the real world can be modeled by this type of function. The table and the scatter plot show the coal production (in metric tons) from a small mine in northern British Columbia.

  

Calculation:

  $\begin{array}{cc}Year(t)& Metric,tons,of,coal(C)\\ 1950& 882\\ 1960& 889\\ 1970& 894\\ 1980& 899\\ 1990& 905\\ 2000& 909\end{array}$

Find the logrithem model for these data.

Use $TI-84$ calculator.

Press on $\boxed{MATH}+\boxed{STAT}$ then choose option $\boxed{EDIT}$ and then press on $1$ .

  

Enter the values in the table as shown below:

  

Press on $\boxed{MATH}+\boxed{STAT}$ then choose option $\boxed{CALC}$ and then press on $9$ .

  

Press on $\boxed{ENTER}$ .

  

Hence, the required logarithemic model is $C=-7154.888128+(1061.006551)In t$ .

b.

To determine

Use the model to predict coal production from this mine in $2010$ .

Answer to Problem 12P

  $915.010989$

Explanation of Solution

Given information:

A logarithmic model is a function of the form

  $y=a+binx$

Many relationships between variables in the real world can be modeled by this type of function. The table and the scatter plot show the coal production (in metric tons) from a small mine in northern British Columbia.

  

Calculation:

  $\begin{array}{cc}Year(t)& Metric,tons,of,coal(C)\\ 1950& 882\\ 1960& 889\\ 1970& 894\\ 1980& 899\\ 1990& 905\\ 2000& 909\end{array}$

Find the coal production when $t=2010$ .

Substitute $t=2010$ in the logarithmic model then solve for $t$ .

  $\begin{array}{l}C=-7154.888128+(1061.006551)In t\\ C=-7154.888128+(1061.006551)In\left(2010\right)\\ =-7154.888128+(1061.006551)\left(7.605890001\right)\\ =-7154.888128+8069.899117\\ =915.010989\end{array}$

Hence, the required coal production at time $t=2010$ is $915.010989$ .