Question 1079254: You have been asked to design an open rectangular box with square base no top and a volume of 171.5 cubic feet. What are the dimensions of the box that can be made with the least amound of cardboard? Please help I keep getting this question wrong
Found 2 solutions by jim_thompson5910, josmiceli:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let's set up some variables
L = length of the square base
H = height of box
V = volume of the box
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The volume of the box is simply the result of multiplying the area of the base (L*L = L^2) by the height of the box, so,
volume of box = (area of base)*(height)
The given volume is 171.5 cubic feet, which means V = 171.5. Let's replace the V with 171.5 to get
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now solve for H
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Now onto dealing with how much cardboard we need. This is essentially the same as asking for the surface area but minus the top part of the box (since this part is missing to allow an open box). The surface area of the box is
SA = 2*(length*width + length*height + width*height)
SA = 2*(L*L + L*H + L*H)
SA = 2*(L^2 + 2LH)
SA = 2L^2 + 4LH
But now subtract off one copy of L^2 to get rid of the top of the box
SA - L^2 = 2L^2 + 4LH - L^2
SA - L^2 = L^2 + 4LH
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Using that last equation, let's plug in . You should notice that the H variables are eliminated and there's only L left over.
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The function we now have is  or essentialy 
Use calculus or a calculator to find the minimum of this function. I'll assume that you aren't in calculus so I'll go with the calculator option.
What you do is graph the function with a proper window so you can see the basic shape of it. This is what the graph looks like
Note: ignore the portion where x < 0. It doesn't make sense to have negative lengths
As you can see, the function has one piece that looks like a parabola on the right side. It's not really a parabola, but let's go with it since this term is learned fairly early in algebra. The lowest part of the parabola is the part we're interested in. This is known as the local minimum or relative minimum. Using a TI calculator I get the local min to be (7,147). Here's how I got it
Step 1) Enter the equation into y1 of the "y=" menu
Step 2) Adjust the window so that: xmin = -30, xmax = 30, ymin = -50, ymax = 200. There are other possible window configurations but this works out fine.
Step 3) Hit the blue (or I think some calculators it's a yellow button?) button that has "2ND" on it, then hit the TRACE button. Scroll down to #3, which is the "Minimum"
Step 4) Move the slider to that the point is to the left of the local min. It doesn't have to be exact. Make sure that you're on the parabola portion. This will input the left/lower bound
Step 5) Move the slider to the right making sure to pass through the local min. Hit enter. This sets the right/upper bound.
Step 6) Move the slider closer to the local min. Make a guess where it is. This doesn't have to be exact. Hit enter to finish up and the calculator will spit out (7,147)
Note: You can use online calculators like Wolfram Alpha to achieve the same result
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So after all that, the local min is (7,147)
What does this mean? Well recall that any point on the curve is (x,y) and y = f(x). So if f(x) = x^2 + 686/x then this represents all the possible ways to plug in a given length L for x, and have it spit out the amount of material needed f(x)
Minimizing the function f(x) is the same as minimizing the material needed, simply because this is how we forced and defined things to be set up. If x = L = 7 feet, then f(x) = 147 square feet of material will be needed, which is the lowest possible amount.
If L = 7, then H is...
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Let's wrap things up: The lowest point of the function is (x,y) = (7,147) where we only consider x > 0 to make sure the lengths aren't negative.
So this means that the local min is (7,147) and the least amount of material needed is 147 square feet of cardboard. So the length of the base is L = 7, which means that this square base has length of 7 feet and width of 7 feet. The height of the box is 3.5 feet
Final Answer: This box has dimensions of 7 feet by 7 feet by 3.5 feet
Answer by josmiceli(19441)  (Show Source):
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