A small startup tracks its cumulative operating costs, in thousands of dollars, during its first year. The cost , months after January 1, is modeled by What does the constant term represent in this model?

Solution available with step-by-step explanation

SAT Linear Functions Question 54 | Free SAT Prep | Aniko

00:00

Question 54·Hard·Linear Functions

0 / 140

A small startup tracks its cumulative operating costs, in thousands of dollars, during its first year. The cost $C(m)$ , $m$ months after January 1, is modeled by

C(m)=84.2-3.6\,(m-9)

What does the constant term $84.2$ represent in this model?

For questions about what a constant in a linear model represents, first identify any expression like $(m - a)$ : the constant added or subtracted inside the parentheses tells you the input $a$ that makes that term zero. Plug that input into the function to see the exact output value the constant corresponds to, then translate that input and output back into the real-world meaning (date, time, cost, etc.). This prevents the common mistake of automatically calling every constant the “starting value” or “y-intercept” without checking the form of the equation.

Hints

Click me!

Focus on when the changing part is zero

Look at the expression $-3.6(m-9)$ . For what value of $m$ does this entire term become 0?

Relate that $m$ -value to the function output

Once you know the value of $m$ that makes $(m-9)=0$ , plug that $m$ into $C(m)$ and see what the function equals.

Connect the $m$ -value to a date

$m$ is months after January 1. What exact date is $m=9$ ? What does $C(9)$ represent in context?

Desmos Guide

Enter the cost model

Type C(m) = 84.2 - 3.6*(m - 9) into Desmos. This will graph the cumulative cost as a function of $m$ .

Evaluate the function at the key $m$ -value

Either click on the graph and add a point at m = 9, or type C(9) in a new line. Note the output value of $C(9)$ and then think about what date $m=9$ corresponds to and what that output represents in the context of the problem.

Step-by-step Explanation

Translate the variables into the real-world situation

The function $C(m)$ gives the cumulative operating cost (in thousands of dollars) $m$ months after January 1.

So:

$m=0$ is January 1,
$m=1$ is February 1,
...
$m=9$ is 9 months after January 1 (we will use this soon).

Identify when the constant 84.2 is the output of the function

The model is

C(m) = 84.2 - 3.6(m-9).

The part $-3.6(m-9)$ changes with $m$ , but 84.2 is fixed. To see what 84.2 represents, find the value of $m$ that makes the changing part zero:

m - 9 = 0 \\[0.7em] m = 9.

If $m=9$ , then $-3.6(m-9) = -3.6(0) = 0$ , so

C(9) = 84.2.

This means 84.2 is the cumulative cost (in thousands of dollars) when $m=9$ .

Convert $m=9$ into a calendar date

Since $m$ is the number of months after January 1, $m=9$ is 9 months after January 1:

1 month after Jan 1 → Feb 1
2 months after Jan 1 → Mar 1
...
9 months after Jan 1 → Oct 1

So $C(9) = 84.2$ means the projected cumulative cost on October 1 is 84.2 thousand dollars.

Therefore, the constant 84.2 represents the projected cumulative cost on October 1.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation