Chapter 4.3, Problem 12Q

Solution

To Find: The decay factor and reduction percentage. Also, graph the obtained model and determine its domain and range. Calculate the number of object T that was sold in the year 1999.

  $\begin{array}{l}\text{Decay factor}=0.85,\text{ Reduction percentage}=15\%,\text{Domain}=\left\{x:x\in N\cup \left\{0\right\}\right\}\text{,}\\ \text{Range}=\left\{y\in R:y>0\right\},\text{ and }19.36\text{ millions}\end{array}$

Given information:

The number of object T’s that were sold from the year 1997 to 2001 can be estimated by the model given below:

  $n\left(t\right)=26.8{\left(0.85\right)}^t$

Here, $n$ is in millions and $t$ represents the number of year since 1997.

Concept used:

• Exponential decay formula: $y=a{\left(1-r\right)}^t$ . Here, $a$ is the initial value, $\left(1-r\right)$ is the decay factor, $r$ is decay rate and $t$ is the time period.
• The graph of a function can be drawn by finding the points that lies on the curve of the given function.
• The domain of an exponential function is the set of all real numbers.

Calculation:

Compare the given model with the exponential decay formula:

  $1-r=0.85$

Thus, the decay factor is $0.85$ and the reduction percentage is,

  $\begin{array}{c}\text{Reduction in }\%=\left(1-0.85\right)\times 100\%\\ =\left(0.15\right)\times 100\%\\ =15\%\end{array}$

To make the graph of the given model, substitute different values of $t$ in the given function and find the respective value of $n(t)$ as follows:

Substitute $t=0$ into $n\left(t\right)=26.8{\left(0.85\right)}^t$ :

  $\begin{array}{c}n\left(t\right)=26.8{\left(0.85\right)}^0\\ =26.8\end{array}$

Similarly, find the values of $n\left(t\right)$ at $t=-2,2,6\text{ and 16}$ and list them in the table given below:

  $\begin{array}{cccccc}t& -2& 0& 2& 6& 16\\ n\left(t\right)& 37.09& 26.8& 19.36& 10.11& 1.99\end{array}$

Now, draw each point of form $\left(t,n\left(t\right)\right)$ in a rectangular coordinate system and connect them with a smooth curve as follows:

  

From the graph, it is clear that the given model is defined for all real values of $t$ , but as $t$ represents the number of years; therefore, $t$ should be a whole number. Thus, the domain of the given model is the set of whole numbers.

Further, observe that the line $n\left(t\right)=0$ is the asymptote of the given model; therefore, the range of the given function is the set of all positive real numbers.

Now, to find the number of object T’s that was sold in the year 1999, observe that the year 1999 comes 2 years after the year 1997. Thus, the respective value of $t$ is 2 years. From the table and the graph, it is clear that $t=2$ corresponds to $n\left(t\right)=19.36$ . Thus, in the year 1999, approximately $19.36$ millions of object T’s were sold.

Conclusion:

Hence, the Decay factor and the reduction percentage for the given model are $0.85\text{ and 15\%}$ , respectively. The domain of the given model is the set $\left\{x:x\in N\cup \left\{0\right\}\right\}$ , and the range of the given function is the set $\left\{y\in R:y>0\right\}$ . In the year 1999, approximately $19.36$ millions of object T’s were sold.