Chapter 7 - Algebra: Graphs, Functions, and Linear Systems - Chapter 7 Test - Page 485: 30

See below:

(a) The given linear function is: \[f\left( x \right)=0.03x+0.63\] The general equation of slope intercept form is: \[f\left( x \right)=mx+c\] Compare the general equation with given equation. Therefore, the slope of the linear equation is 0.03, and slope of the linear function represents the increase in cost to make a penny per year increase after 1982. (b) The given functions are: \[\begin{align} & f\left( x \right)=0.03x+0.63 \\ & f\left( x \right)=0.72{{\left( 1.03 \right)}^{x}} \\ \end{align}\] But the data from bar graph, and scatter plot as below: In year 1982, the cost was \[0.86\text{¢}\]. In year 1992, the cost was \[0.83\text{¢}\]. In year 2002, the cost was \[0.89\text{¢}\]. In year 2006, the cost was \[1.21\text{¢}\]. In year 2012, the cost was \[2.00\text{¢}\]. In year 1982, \[x=0\]. The cost given by linear function is: \[\begin{align} & f\left( x \right)=0.03x+0.63 \\ & =0.03\times 0+0.63 \\ & =0.63 \end{align}\] The cost given by exponential function is: \[\begin{align} & f\left( x \right)=0.72{{\left( 1.03 \right)}^{x}} \\ & =0.72\times {{\left( 1.03 \right)}^{0}} \\ & =0.72 \end{align}\] In year 1992, \[x=10\]. The cost given by linear function is: \[\begin{align} & f\left( x \right)=0.03x+0.63 \\ & =0.03\times 10+0.63 \\ & =0.3+0.63 \\ & =0.93 \end{align}\] The cost given by exponential function is: \[\begin{align} & f\left( x \right)=0.72{{\left( 1.03 \right)}^{10}} \\ & =0.72\times {{\left( 1.03 \right)}^{10}} \\ & =0.9676 \end{align}\] In year 2002, \[x=20\]. The cost given by linear function is: \[\begin{align} & f\left( x \right)=0.03x+0.63 \\ & =0.03\times 20+0.63 \\ & =0.6+0.63 \\ & =1.23 \end{align}\] The cost given by exponential function is: \[\begin{align} & f\left( x \right)=0.72{{\left( 1.03 \right)}^{20}} \\ & =0.72\times {{\left( 1.03 \right)}^{20}} \\ & =1.3004 \end{align}\] In year 2006, \[x=24\]. The cost given by linear function is: \[\begin{align} & f\left( x \right)=0.03x+0.63 \\ & =0.03\times 24+0.63 \\ & =1.35 \end{align}\] The cost given by exponential function is: \[\begin{align} & f\left( x \right)=0.72{{\left( 1.03 \right)}^{24}} \\ & =0.72\times {{\left( 1.03 \right)}^{24}} \\ & =1.4636 \end{align}\] In year 2012, \[x=30\]. The cost given by linear function is: \[\begin{align} & f\left( x \right)=0.03x+0.63 \\ & =0.03\times 30+0.63 \\ & =0.9+0.63 \\ & =1.53 \end{align}\] The cost given by exponential function is: \[\begin{align} & f\left( x \right)=0.72{{\left( 1.03 \right)}^{30}} \\ & =0.72\times {{\left( 1.03 \right)}^{30}} \\ & =1.7476 \end{align}\] Therefore, from the above calculation, it can be seen that, cost by exponential function is increasing more rapidly than linear function, so exponential function is a better model. (c) The given functions are: \[\begin{align} & f\left( x \right)=0.03x+0.63 \\ & f\left( x \right)=0.72{{\left( 1.03 \right)}^{x}} \\ \end{align}\] But the data from bar graph, and scatter plot are as below: In year 1982, the cost was \[0.86\text{¢}\]. In year 1992, the cost was \[0.83\text{¢}\]. In year 2002, the cost was \[0.89\text{¢}\]. In year 2006, the cost was \[1.21\text{¢}\]. In year 2012, the cost was \[2.00\text{¢}\]. In year 2012, \[x=30\]. The cost given by linear function is: \[\begin{align} & f\left( x \right)=0.03x+0.63 \\ & =0.03\times 30+0.63 \\ & =0.9+0.63 \\ & =1.53 \end{align}\] The cost given by exponential function is: \[\begin{align} & f\left( x \right)=0.72{{\left( 1.03 \right)}^{30}} \\ & =0.72\times {{\left( 1.03 \right)}^{30}} \\ & =1.7476 \\ & \approx 1.75 \end{align}\] From the above calculation, it can be seen that, neither of these two functions, cost values match with this scatter plot or bar graph data. The cost of penny in exponential function data is the better one, because it is increasing rapidly. Also, it can be seen that this observation matches with previous data. Therefore, the cost given by linear function is\[1.53\text{¢}\], the cost given by exponential function is\[1.75\text{¢}\]. None of the model serves as the good model. The cost of penny in exponential function data is the better one, and it can be seen that this observation matches with previous data.