SOLUTION: You want to make a vertical file by bending the long side of an 8x14 inch sheet of metal along two sides to make a U shape. Express the volume of the file as a function of the hei

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Question 1720: You want to make a vertical file by bending the long side of an 8x14 inch sheet of metal along two sides to make a U shape. Express the volume of the file as a function of the height. Find the domain and range of the resulting function. How tall should the file be in order to maximize the volume that it can hold?
Answer by khwang(438) (Show Source):
You can put this solution on YOUR website!
Let the height ofthe U shape be h inches and the width be w inches.
Since 2h+ w = 14, so w = 14 -2h.
The volume V of the U shape is V(h) = 8 hw = 8 h(14 -2h) = 16 h(7-h).

Since,0 < 2h < 14, the domain of the function V(h) is (0,7).
Note that V(h) = 16 h(7-h) = -16[h^2-7h+ (7/2)^2] + 49*4
= -16(h- 7/2)^2 + 196 <= 196
Clearly, V(h)>0, so the range of the function V is (0, 196].
since -16(h- 7/2)^2 <=0, we see that when h = 7/2 (inches),
the volume has max value 196 cubic inches.

Kenny