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?
A linear model written with an offset like gives its listed amount when the input reaches : the change term becomes zero. Find that input, match it to the calendar, and evaluate the function in Desmos to confirm. The number outside the parentheses isn't necessarily the starting value.
Hints
- Hint 1
In a linear model, the rate is multiplied by how far the input has moved from a reference input. Which input makes the parentheses equal zero? At that input, the rate contributes nothing.
- Hint 2
The input counts months after January 1. Month is January 1, and month is February 1. Which date goes with the input that makes the parentheses zero?
- Hint 3
A function value such as means the predicted cumulative cost at month . Evaluate the function at the input you found, then describe the result as a cost in thousands of dollars, not a monthly rate.
Step-by-step
Find when the change term vanishes
Step 1Find the input tied to the constant
The rate of change is multiplied by . At month , , so that whole change term vanishes. The is tied to month , not automatically to the starting month.
- Step 2
Match the input to a date
The problem counts months after January 1. January 1 is , so is October 1. This is the date attached to the constant.
- Step 3
Confirm the cost at that date
Type the given function in Desmos, then type . Desmos displays . Since measures cumulative cost in thousands of dollars, represents the projected cumulative cost on October 1. Choice C.