A shelf filled by objects with variable thicknesses leads to a compound inequality: the total length must fall between the smallest and largest possible totals. Use a Desmos regression for each endpoint to find the fewest and most objects that fit. Watch the direction: thicker objects mean fewer objects, and “at least” and “at most” include their endpoints.
Hints
- Hint 1
If each book had the smallest allowed thickness, how much shelf space would books take? If each had the largest allowed thickness, how much would they take? The shelf’s length must lie between those totals.
- Hint 2
To fit the fewest books, use the thickest allowed books. Find the count at that endpoint first, then use the thinnest allowed books to find the greatest possible count.
Step-by-step
Bound the total thickness
Step 1Put the shelf length between the two totals
At centimeters each, books take centimeters; at centimeters each, they take centimeters. Since the books exactly fill the -centimeter shelf, its length must be between those totals: .
- Step 2
Find the fewest books
Use the thickest allowed books to find the smallest count. Type in Desmos, where is the count at this endpoint. Desmos reports under PARAMETERS. Thicker books mean fewer books; thinner books mean more books. So .
- Step 3
Find the greatest count and combine the bounds
For the greatest count, use the thinnest allowed books. Keep the first Desmos line and add , where is the count at this endpoint. Desmos reports under PARAMETERS, so . Every whole-number count between the endpoints works by using books of thickness centimeters. The shelf can hold books, inclusive. Choice A.