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Question 37399: There is no book for this problem. This is a problem asked by the professor. Please help.
1) An open-top box is to be constructed from a 4 by 6 foot rectangular cardboard by cutting out equal squares at each corner and the folding up the flaps. Let x denote the length of each side of the square to be cut out.
a) Find the function V that represents the volume of the box in terms of x.
Answer:
b) Graph this function and show the graph over the valid range of the variable x..
Show Graph here.
c) Using the graph, what is the value of x that will produce the maximum volume?
Answer.
Answer by fractalier(6550)   (Show Source):
You can put this solution on YOUR website! Well this is a bit difficult to show you without a diagram and graph, but here goes...
Since we are cutting a length x out of both ends of both the length and the width, the dimensions of the base of the box will be (6 - 2x) by (4 - 2x). It will have a height of x. Therefore the function for the volume is
V(x) = x(6 - 2x)(4 - 2x) = 4x^3 - 20x^2 + 24x
I can't show you the graph but you know x must be less than two.
To find the maximum volume, we take the derivative of V(x) and set it equal to zero...
dV/dx = 12x^2 - 40x + 24 = 0
Now solve for x
Use the quadratic formula to get
x = (5 ± rad(7)) / 3
But only one root works (as mentioned above), so the answer here is
x = (5 - rad(7)) / 3 or about .785 feet
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