By your cell phone contract, you pay a monthly fee plus some money for each minute you use the phone during the month. In one month, you spent 290 minutes on the phone, and paid $22.25. In another month, you spent 360 minutes on the phone, and paid $24.00.
Let x be the number of minutes you talk over the phone in a month, and let y be your cell phone bill for that month. Use a linear equation to model your monthly bill based on the number of minutes you talk over the phone.
a. This linear model's slope-intercept equation is_____________.
b. If you spent 140 minutes over the phone in a month, you would pay________________ .
c. If in a month, you paid $26.25 of cell phone bill, you must have spent ______________minutes on the phone in that month.

Respuesta :

Answer:

(a) [tex]y = 0.025x+15[/tex]

(b) 18.5 minutes

(c) 450.4 minutes

Step-by-step explanation:

Given

[tex]290\ minutes = \$22.25[/tex]

[tex]360\ minutes = \$24.00[/tex]

Solving (a): Determine the linear equation

First, we need to calculate the slope (m)

[tex]m = \frac{y_2 - y_1}{x_2 - x_1}[/tex]

Where

[tex](x_1,y_1) = (290,22.25)[/tex]

[tex](x_2,y_2) = (360,24.00)[/tex]

So, we have:

[tex]m = \frac{24.00 - 22.25}{360 - 290}[/tex]

[tex]m = \frac{1.75}{70}[/tex]

Multiply through by 100

[tex]m = \frac{1.75 * 100}{70 * 100}[/tex]

[tex]m = \frac{175}{7000}[/tex]

Next, is to calculate the equation of the line using:

[tex]y - y_1 = m(x - x_1)[/tex]

Recall that:

[tex](x_1,y_1) = (290,22.25)[/tex]

[tex]m = \frac{175}{7000}[/tex]

[tex]y - 22.25 = \frac{175}{7000}(x - 290)[/tex]

[tex]y - 22.25 = \frac{175x}{7000} - \frac{175}{7000} * 290[/tex]

[tex]y - 22.25 = \frac{175x}{7000} - \frac{50750}{7000}[/tex]

Add 22.25 to both sides

[tex]y = \frac{175x}{7000} - \frac{5075}{700} + 22.25[/tex]

[tex]y = \frac{175x}{7000} + \frac{-5075 + 15575}{700}[/tex]

[tex]y = \frac{175x}{7000} + \frac{10500}{700}[/tex]

[tex]y = 0.025x+15[/tex]

(b) Solve for y when x = 140

[tex]y = 0.025x+15[/tex]

Substitute 140 for x

[tex]y = 0.025 * 140 + 15[/tex]

[tex]y = 3.5 + 15[/tex]

[tex]y = 18.5[/tex]

(c) Solve for x when y = 26.25

[tex]y = 0.025x+15[/tex]

Substitute 26.25 for y

[tex]26.25 = 0.025x + 15[/tex]

Solve for 0.025x

[tex]0.025x = 26.26 - 15[/tex]

[tex]0.025x = 11.26[/tex]

Solve for x

[tex]x = 11.26/0.025[/tex]

[tex]x = 450.4\ minutes[/tex]