Question

The credit remaining on a phone card (in dollars) is a linear function of the total calling time made with the card (in minutes). The remaining credit after 40 minutes of calls is 25.20, and the remaining credit after 68 minutes of calls is 21.84 . What is the remaining credit after 75 minutes of calls?

Answer 1

Answer:

The remaining credit after 75 minutes of calls is $20.9515.

Step-by-step explanation:

The amount of credit remaining on a phone card can be modeled by the following linear function:

$C(t) = C(0) - at$

In which C(t) is the amount remaining after t minutes, C(0) is the initial amount and a is the cost of each minute.

We take what the problem gives us, and find an expression for C(t), solving a system of equations.

The remaining credit after 40 minutes of calls is 25.20

This means that $C(40) = 25.20$

So

$C(t) = C(0) - at$

$25.40 = C(0) - 40a$

The remaining credit after 68 minutes of calls is 21.84.

This means that $C(68) = 21.84$

So

$C(t) = C(0) - at$

$21.84 = C(0) - 68a$

So we have to solve the following system of equations:

$25.40 = C(0) - 40a$

$21.84 = C(0) - 68a$

From the first equation, we have that:

$C(0) = 25.40 + 40a$

Replacing in the second one:

$21.84 = C(0) - 68a$

$21.84 = 25.40 + 40a - 68a$

$28a = 3.56$

$a = \frac{3.56}{28}$

$a = 0.1271$

We also know that

$C(0) = 25.40 + 40a = 25.40 + 40*0.1271 = 30.484$

So

$C(t) = C(0) - at$

$C(t) = 30.484 - 0.1271t$

What is the remaining credit after 75 minutes of calls?

This is C(75).

$C(t) = 30.484 - 0.1271t$

$C(75) = 30.484 - 0.1271*75 = 20.9515$

The remaining credit after 75 minutes of calls is $20.9515.