Question 38205: 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! Again,
If we cut out x from all four corners, the new length will be 6 - 2x and the new width will be 4 - 2x, with the height of the box being just x.
Now Volume V = lwh, so we have
V = x(6 - 2x)(4 - 2x)
I cannot graph it for you, but you can see that x must be more than zero and less than 2.
Using early calculus I can show you how to find the maximum area, but without a way to show you it's graph, that's the best I can do...
We need to maximize the function
V(x) = x(6 - 2x)(4 - 2x)
We do that by taking its derivative and setting it equal to zero, then solving for x...here goes...
V(x) = x(6 - 2x)(4 - 2x)
V(x) = 24x - 20x^2 + 4x^3
V'(x) = 24 - 40x + 12x^2 thus
12x^2 - 40x + 24 = 0
3x^2 - 10x + 6 = 0
This isn't factorable so we use the quadratic formula and get
x = (10 ± 2sqrt(7)) / 6 = (5 ± sqrt(7)) / 3
The positive root is too big, so x must be
x = (5 - sqrt(7)) / 3 or about .785
Now plug that in to V(x) to get the maximum volume...
V(.785) = .785(6 - 2(.785))(4 - 2(.785)) = 8.45 cubic feet
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