A cloud storage service charges a one-time activation fee and a fixed monthly charge. The total amount paid, , in dollars, after months is a linear function of . If a customer has paid 86 dollars after 4 months and 257 dollars after 13 months, which equation gives in terms of ?

Solution available with step-by-step explanation

SAT Linear Functions Question 6 | Free SAT Prep | Aniko

00:00

Question 6·Hard·Linear Functions

0 / 140

A cloud storage service charges a one-time activation fee and a fixed monthly charge. The total amount paid, $T$ , in dollars, after $m$ months is a linear function of $m$ . If a customer has paid 86 dollars after 4 months and 257 dollars after 13 months, which equation gives $T$ in terms of $m$ ?

For linear function word problems with two data points, immediately treat the given situations as coordinate points $(x,y)$ , where $x$ is the input (here, months) and $y$ is the output (total cost). Use the slope formula to get the rate of change, plug that into $y = mx + b$ , then substitute one point to solve for $b$ . Finally, quickly verify your equation by checking it matches all given data; on multiple-choice questions, you can also plug the given $x$ -values into each option to see which one fits every condition.

Hints

Click me!

Use the two payment amounts as points on a line

Think of $(m,T)$ as coordinates: one when $m=4$ and one when $m=13$ . What are those two points?

Find the monthly cost from the two points

Use the slope formula $\frac{T_2 - T_1}{m_2 - m_1}$ with your two points to find how much the total increases each month.

Use slope-intercept form

Once you know the monthly charge (slope), write $T$ in the form $T = (\text{monthly charge})m + b$ , then plug in one point to solve for $b$ , the activation fee.

Check your equation with both points

After you find an equation, substitute $m=4$ and $m=13$ to make sure it gives 86 and 257, respectively.

Desmos Guide

Enter the data points

In Desmos, add two points by typing (4,86) on one line and (13,257) on another line. They will appear as points on the graph.

Graph each answer choice

On separate lines, type each choice as a function of $m$ and $T$ , for example T = 19m + 10, T = 10m + 19, etc. Desmos will draw four lines.

Compare the lines to the data points

Look at which line passes exactly through both points $(4,86)$ and $(13,257)$ . The equation of that line is the correct model for $T$ in terms of $m$ .

Step-by-step Explanation

Translate the situation into coordinate points

The total cost $T$ is a linear function of the number of months $m$ .

After 4 months, the customer has paid 86 dollars. This gives the point $(m,T) = (4,86)$ .
After 13 months, the customer has paid 257 dollars. This gives the point $(m,T) = (13,257)$ .

These are two points on the line representing $T$ as a function of $m$ .

Find the monthly charge (the slope)

For a linear function, the slope is

\text{slope} = \frac{\Delta T}{\Delta m} = \frac{T_2 - T_1}{m_2 - m_1}.

Using the two points $(4,86)$ and $(13,257)$ :

\text{slope} = \frac{257 - 86}{13 - 4} = \frac{171}{9} = 19.

So the customer is paying 19 dollars per month. In the equation for a line $T = (\text{slope})\cdot m + b$ , the number 19 will multiply $m$ .

Find the activation fee (the y-intercept)

Now write the linear equation in the form $T = 19m + b$ , where $b$ is the one-time activation fee (the value of $T$ when $m=0$ ).

Use one of the known points to solve for $b$ . Using $(4,86)$ :

86 = 19(4) + b.

Compute $19\cdot 4$ and then solve for $b$ .

Solve for the intercept and write the final equation

From the previous step:

\begin{aligned} 86 &= 19\cdot 4 + b \\ 86 &= 76 + b \end{aligned}

Subtract 76 from both sides:

\begin{aligned} 86 - 76 &= b \\ 10 &= b \end{aligned}

So the activation fee is 10 dollars, and the equation for the total cost is

T = 19m + 10.

This matches answer choice A.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation