A container needs to package hazardous chemicals in a special plastic rectangular prism that holds 80 cubic feet. Find the whole number dimensions of the container that would use the least amount of plastic.
My answer is 5ft, 2ft, and 4ft
How about 5 by 4 by 4?
I learned that surface area is pb(h)+2ba
so 5x4x4 would be
18(4)+2(20)
72+40
112
and the 10x4x2 would be
28(2)+2(40)
56+80
136
therefore a 5x4x4 box would have the least surface area
In order to use the least amount of plastic, you need to find the surface dimensions of this box.
If your box is 10 by 4 by 2 --
2(10*4) + 2(4 * 2) + 2(2*2) = ?
What are the surface dimensions of a box 5 by 4 vy 4?
4 ft, 4 ft, and 5 ft
That's only 40 cubic feet.
Let the dimensions of the box by x by y by z
xyz = 80
z = 80/(xy)
"least amount of plasic" implies least surface area
SA = 2xy + 2xz + 2yz
= 2xy + 2x(80/(xy)) + 2y(80/(xy))
= 2xy + 160/y + 160/x
now we know that xyz = 80
and 80 = (2)(2)(2)(2)(5)
combinations of x y z and SA
2 2 20 168
2 4 10 136
2 8 5 132
4 2 10 -- same as 2 4
4 4 5 112
So there are 5 different combinations we can use to get the volume of 80, but
to have the least surface area would be
4 by 4 by 5, as Ms Sue had suggested.