Application problem with a linear function: Finding a coordinate given two points

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 36 minutes of calls is $19.96, and the remaining credit after 63 minutes of calls is $16.18. 

What is the remaining credit after 80 minutes of calls?



Answer 1

Hi, Kristen,

Karina has answered the question correctly. I'd simply like to add a couple of other perspectives that might help.

Karina calculated the slope of the line between the two know data points. Slope is defined as the rise over the run, so you can see why she set up the calculation m = y₂ - y₁ /(x₂ - x₁) [y is the "rise" and x is the "run." I calculated -0.14 from the fraction.

So we know the equation looks like y = -0.14x + b. "b" is the y intercept, where the line crosses the x axis at "0." It is a useful number to know, since it represents the original value of the card (t = 0 means no calls were made). By inputting one set of data (let's use 36 minutes with $19.96 left on the card, we can calculate b:

$19.96 = (-0.14 $/min)*(36 min) + b

$19.96 = (-$5.04) + b

b = $25.00

The original value on the card is $25.

Calculating the value after 80 minutes is then easy:

V = ($-0.14/min)*(80min) + $25

V = $13.80

Karina's answers are good, so pick the approach that works best for you.

Bob

Answer 2

Hope this helps...

With the given information that:

AFTER 36 minutes, the remaining credit is $19,96 and AFTER 63 minutes, the remaining credit is $16.18

tells you that if you were to graph these points with the Y-axis representing the CREDITS and the X-axis representing TIME...and connected the points, the line will slope downward meaning the function has a negative slope. And you can tell this by the fact that with time the credits available decreases.

(Sorry can't show you a graph of the points)

But if you calculated the rate of change or SLOPE of the line, it would be:

Using points (36, 19.96) and (63, 16.18)

m = y₂ - y₁ /(x₂ - x₁) = (19.96 - 16.18)/(36 - 63) = -3.78 / 27

So you can use the POINT-SLOPE formula to find what y equals when x = 80mins and use the one of the given points in the equation...

I will use the last point given (63, 16.18)

y - y₁ = m ( x - x₁)

y - 16.18 = -(^3.27/₂₇)(x - 63)

Now set x = 80 and clear up the RHS before isolating y on the LHS (i.e., brining the 16.18 over to RHS)

y - 16.18 = -(^3.27/₂₇)(80 - 63)

y - 16.18 = -(^3.27/₂₇)(17)

y - 16.18 = -2.38

y = -2.38 + 16.18

y = 13.80

So AFTER 80mins the remaining credits are $ 13.80