Question 642·Hard·Algebra

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A remote weather station models the snow depth, in centimeters, on a mountain peak using the function $S$ below, where $t$ is the number of days after March 5.

S(t)=82.4-3.6\,(t-4)

What does the value $82.4$ represent in this model?

A linear model written with a term like $t-a$ pairs its fixed value with input $a$ : that's where the changing term becomes zero. Find that input, then evaluate the function in Desmos to check its output. Finally, use the story's starting date to turn the input into a calendar date. Don't assume the fixed value belongs to day zero.

Hints

Hint 1
In a linear model, the changing term disappears when the parentheses equal zero. Which input makes $t-4$ zero? That input is paired with the fixed snow depth.
Hint 2
The input $t$ counts days after March 5, not calendar dates. Once you've found the input paired with the fixed depth, count that many days forward from March 5.

Step-by-step

Find the day paired with the fixed depth
Step 1
Find the input that cancels the change
In this linear model, $t-4$ becomes zero when $t=4$ . That makes the entire changing term disappear, so the fixed depth belongs to input $4$ , not automatically to input $0$ .
Step 2
Check the depth at that input
Type $S(t)=82.4-3.6(t-4)$ in Desmos, then type $S(4)$ . Desmos shows $S(4)=82.4$ . An output of $S$ is a snow depth, so $82.4$ centimeters is the depth at input $4$ .
Step 3
Convert the input to a date
The input $t$ counts days after March 5. Four days after March 5 is March 9, so $82.4$ represents the snow depth on March 9. In a shifted linear model, the fixed value belongs to the input that makes the parentheses zero. Choice B.

Question 642·Hard·Algebra

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