A library added 1,200 identical copies of a popular book to its collection. Each month afterward, an average of 35 of these copies are damaged and removed from the collection, and no additional copies are added. After how many months will exactly 815 copies of the book remain in the collection?
A linear model fits a collection that loses the same number of copies each month. Multiply the monthly loss by the number of months: starting amount minus total loss equals the amount remaining. Set that amount equal to the requested count, then use a Desmos regression to find the month. Adding the loss would model growth instead.
Hints
- Hint 1
A rate tells you how much changes per unit of time. If is the number of months, multiply the copies removed each month by to get the total removed.
- Hint 2
The books that remain are the starting copies minus the copies removed. Write that relationship first, then set it equal to the requested number of copies.
Step-by-step
Model the copies remaining
Step 1Find the total removed after any number of months
Let be the number of months after the books were added. The library removes an average of copies each month, so the total removed after months is copies.
- Step 2
Set the remaining copies equal to the target
The library starts with copies and adds no more. Subtract the removed copies from that starting count, then set the result equal to the 815 copies remaining:
The minus sign matters because damaged copies leave the collection.
- Step 3
Solve for the number of months
Type in Desmos, replacing with . This regression finds a value for instead of graphing it. Under PARAMETERS, Desmos shows . Since counts months, exactly copies remain after 11 months. Choice C.