For question a), we have to write the equation of the cost (c) as a function of the minutes (m), so:
[tex]c_A=0.05\cdot m+20+20[/tex]
In the equation above the term 0.05*m represent the 5 cents for minute then we have to sum the $20 for 250 texts and $20 of service fee.
Before we draw the lines, we can solve the question c). If a customer wants to spend $75 monthly we can recommen him the plan wich more minutes for that cost, so we need to calculate the minutes for each plan:
[tex]\begin{gathered} \text{For Plan A:} \\ c_A=75=0.05\cdot m+20+20 \\ 0.05\cdot m=75-20-20=35 \\ m=\frac{35}{0.05}=700 \\ \text{For Plan B:} \\ c_B=75=0.1\cdot(m-100)+15+20 \\ 0.1\cdot(m-100)=75-15-20=40 \\ m-100=\frac{40}{0.1}=400 \\ m=400+100=500 \end{gathered}[/tex]
The customer should choose the Plan A, because it has more minutes and more texts for $75.
For point b), we can evaluate each equation in two differents m-values and found the pairs (m, c) to graph the lines, so:
[tex]\begin{gathered} \text{For Plan A, we can choose m=100 and m=500}\colon \\ m=100\Rightarrow c_{}=0.05\cdot100+20+20=45 \\ m=500\Rightarrow c=0.05\cdot500+20+20=65 \\ P_{1A}=(100,45),P_{2A}=(500,65) \end{gathered}[/tex][tex]\begin{gathered} \text{For Plan B, we can choose m=100 and m=500:} \\ m=100\Rightarrow c=0.1\cdot(100-100)+15+20=35 \\ m=500\Rightarrow c=0.1\cdot(500-100)+15+20=75 \\ P_{1B}=(100,35),P_{2B}=(500,75) \end{gathered}[/tex]
And the graphs are:
In the Graphs we can see the lines intercept in m=300 and evluating the equations in that value the cost is $55.
