SOLUTION: A 20-inch square piece of metal is to be used to make an open-top box by cutting equal-sized squares from each corner and folding up the sides. The length, width, and height of the

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Question 276740: A 20-inch square piece of metal is to be used to make an open-top box by cutting equal-sized squares from each corner and folding up the sides. The length, width, and height of the box are each to be less than 14 inches. What size squares should be cut out to produce a box with volume 550 cubic inches?
What size squares should be cut out to produce a box with largest possible volume?
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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A 20-inch square piece of metal is to be used to make an open-top box by cutting equal-sized squares from each corner and folding up the sides.
The length, width, and height of the box are each to be less than 14 inches. What size squares should be cut out to produce a box with volume 550 cubic inches?
:
Let x = side of the squares to be cut out of the metal
:
Since the piece of metal is square, the base will be square also
Dimensions of the base (20-2x) by (20-2x), the height will be x
therefore
(20-2x)^2 = 550%2Fx
(20-2x) = sqrt%28550%2Fx%29
plot these two equations
y = 20-2x
and
y = sqrt%28550%2Fx%29
:
+graph%28+300%2C+200%2C+-4%2C+10%2C+-4%2C+18%2C+sqrt%28550%2Fx%29%2C+20-2x%29+
Note there are two points of intersection
x=2.35, y=15.3; not this one y > 14"
and
x=4.43, y=11.15; this one will be Ok
:
We can say 4.43" squares to be cut out of each corner of the metal
;
;
Check the vol
(20-2(4.43))^2 * 4.43 = 549.76 ~ 550
:
:
What size squares should be cut out to produce a box with largest possible volume?
Graph this equation, volume is on the y axis
y = x(20-2x)^2
+graph%28+300%2C+200%2C+-4%2C+10%2C+-100%2C+700%2C+x%2A%2820-2x%29%5E2%29+
max volume: x=3.33", y=592.6 cu/in
;
We can say 3.33" squares should be cut out each corner for max volume