You want to make a rectangular box, open at the top, by cutting the same size square corners out of a rectangular sheet of cardboard and then folding up the sides. The cardboard measures 10 in. by 12 in. What are the dimensions of the box that will have greatest volume if the possible corner cuts measure 1 in., 2 in., 3 in., or 4 in.?
well, the volume is width * length * height (x)
v = (10-2x)(12-2x)(x)
now plug in x=1,2,3,4 and see which gives the greatest volume
a.9 in. by 12 in. by 1 in.
b.3 in. by 6 in. by 4 in.
c.7 in. by 10 in. by 2 in.
d.5 in. by 8 in. by 3 in.
c?
Nope. Think about it. Since the two original dimensions are 8x10, after the cuts, the base of the box will be
8x10
6x8
4x6
2x4
You cannot even get (c) by cutting equal squares from all corners.
is it a?
To determine the dimensions of the box that will have the greatest volume, we can first calculate the volume for each possibility and compare them.
1. For a 1-inch corner cut:
- The length of the resulting rectangular base will be 10 - 2(1) = 8 inches.
- The width of the resulting rectangular base will be 12 - 2(1) = 10 inches.
- The height of the box will be 1 inch (since each corner cut is 1 inch).
- Therefore, the volume of the box will be 8 × 10 × 1 = 80 cubic inches.
2. For a 2-inch corner cut:
- The length of the resulting rectangular base will be 10 - 2(2) = 6 inches.
- The width of the resulting rectangular base will be 12 - 2(2) = 8 inches.
- The height of the box will be 2 inches (since each corner cut is 2 inches).
- Therefore, the volume of the box will be 6 × 8 × 2 = 96 cubic inches.
3. For a 3-inch corner cut:
- The length of the resulting rectangular base will be 10 - 2(3) = 4 inches.
- The width of the resulting rectangular base will be 12 - 2(3) = 6 inches.
- The height of the box will be 3 inches (since each corner cut is 3 inches).
- Therefore, the volume of the box will be 4 × 6 × 3 = 72 cubic inches.
4. For a 4-inch corner cut:
- The length of the resulting rectangular base will be 10 - 2(4) = 2 inches.
- The width of the resulting rectangular base will be 12 - 2(4) = 4 inches.
- The height of the box will be 4 inches (since each corner cut is 4 inches).
- Therefore, the volume of the box will be 2 × 4 × 4 = 32 cubic inches.
By comparing the volumes calculated for each possibility, we find that the 2-inch corner cut yields the largest volume, which is 96 cubic inches.