Find the box tops!
Geometry
Level
2
- Anna has one really large box with dimensions 50 in x 30 in x 20 in made of paper without a top/lid.
- For some reason, she decides that she wants many smaller rectangular prisms with the very specific dimensions 1 in x 2 in x 3 in, so she cuts up the big box into many smaller pieces, and uses them to make as many new boxes as she can.
- All of these small boxes also are missing tops.
What is the maximum number of small boxes she can make?
Details and assumptions:
- I intentionally left out the size of the top of the big box! Given the three possibilities, you can assume that the lid's size allows for the maximum number of small boxes to be formed.
- I also left out the size of the top of the small box, so also figure out which size of a small box's lid will allow for the maximum number of small boxes.
- Assume she cuts up the pieces and tapes them together to make new boxes. No overlapping of pieces.
The answer is 350.
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1 solution
We want to get the largest number of small boxes possible from this large box, so we want to maximise the surface area of the big box and minimise the surface area of the small boxes.
Three possible lid sizes for the large box: 30 x 50, 30 x 20, 50 x 20. To maximise the surface area of the large box, we want to subtract the smallest possible lid area which is 30 x 20. So the surface area of the large box is 2(30 x 50) + (30 x 20) + 2(50 x 20) = 5600.
To minimise the surface area of the small boxes, we want the largest possible lid size to be subtracted from the total surface area. Three possible lid sizes for small boxes: 1 x 2, 2 x 3, 1 x 3. The largest size is given by 2 x 3. So, the surface area of one small box is 2(1 x 2) + (2 x 3) + 2(1 x 3) = 16.
Finally, divide the large box surface area by the small box surface area to get the maximum number of small boxes that can be made using the large box's materials: 5 6 0 0 ÷ 1 6 = 3 5 0 boxes