SOLUTION: 1)You are designing a rain gutter made from a piece of sheet metal 1 foot by 5 feet. The gutter will be formed by turning up two sides. You want the gutter to have the greatest v

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Question 424413: 1)You are designing a rain gutter made from a piece of sheet metal 1 foot by 5 feet. The gutter will be formed by turning up two sides. You want the gutter to have the greatest volume possible.
a.How much should you turn up?




b.What is the maximum volume?
Answer by Gogonati(855) About Me  (Show Source):
You can put this solution on YOUR website!
Solution: Denote the base of the gutter x ft, then the length of the turning up
will be:
%281-x%29%2F2 Therefore the shape of gutter is a parallelepiped base rectangle with dimensions x, (1-x)/2 and altitude 5 ft.
The volume of the gutter is: V=5%2Ax%2A%281-x%29%2F2

V=2.5%2Ax-2.5%2Ax%5E2 The graph of this function is
a downward parabola and its vertex is the maximum.
x=%28-2.5%2F-5%29=1%2F2
y=2.5%2A%281%2F2%29-2.5%2Ax%5E2 y =5/8 cubic feet.
Answer: We need to turn up (1- 1/2)/2= (1/2)/2= 1/4 ft (quoter feet) that the volume of gutter will be maximum: V = 5/8 cubic feet.