Question

A cellphone company offers two talk and text plans. The company charges a monthly service fee of $20 for either plan the customer chooses: Customers that choose Talk and Text Plan A are charged five cents a minute and twenty dollars for 250 texts. Customers that choose Talk and Text Plan B are charged ten cents a minute (first 100 minutes free) and fifteen dollars for 200 texts. The equation c = .10(m – 100) + 15 + 20 can be used to represent how much a customer would spend monthly for the minutes used. ** Express Plan A as an equation where c equals the cost and m equals the minutes used. a. Graph each Talk and Text plan to determine when both plans cost the same.

Answer 1

For question a), we have to write the equation of the cost (c) as a function of the minutes (m), so:

`c_A=0.05\cdot m+20+20`

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:

`\begin{gathered} \text{For Plan A:} \\ c_A=75=0.05\cdot m+20+20 \\ 0.05\cdot m=75-20-20=35 \\ m=(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=(40)/(0.1)=400 \\ m=400+100=500 \end{gathered}`

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:

`\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}`
`\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}`

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.