Question 974810: A man living in the Northwest wants to mail a Christmas tree to a friend living in the South. He plans to bundle the tree tightly and pack it in a long box with a square end. The mailing service told him that the length plus the girth of the package cannot exceed 130 inches. If the Christmas tree is 5'10" tall, what is the maximum width and height for his package? (Note: The girth is the distance around the package as shown)
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Length =L
width= G, which is a square, so 4G is the girth.
L+ G=130
Tree is 70 inches long.
I am going to assume nobody is using the diagonal of the package.
L=130-4G
I am also assuming that the question has a mistake, and one wants the minimum width and height for the package, since the maximum has no bound.
The easy answer is that 4G, the girth, is 60 inches.
If I use the hypotenuse packing, then the floor of the package is a right triangle with hypotenuse z^2, and the sides L^2 and G^2
Then z^2=L^2 + G^2
But the hypotenuse across the box, from lower to upper, is another right triangle, with one of the legs the original hypotenuse and the other leg G.
That is [z^2+ G^2]=4900; (L^2+G^2)+G^2=4900 ;;; L^2 + 2G^2=4900
We know that L=130-4G; L^2=16900-1040G +16G^2
Substituting, 16900-1040G +16G^2+2G^2=4900
18G^2-1040G+12000=0
9G^2-520G+6000=0
x=(1/18) [520 +/- sqrt (270400-216000) ;;; discriminant is 233.2
x=(1/18) * 753.2 ;; (1/18) (286.8)= 15.933
x=41.844, which won't work
Minimum box girth is G=63.73; all 4 sides are 15.933 inches.
Length is 66.268
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