SOLUTION: Can you please help me find the function that gives the surface area, S, of the sides of a box in terms of x (ignoring the bottom and top since it is an open box)using the formula

Question 541226: Can you please help me find the function that gives the surface area, S, of the sides of a box in terms of x (ignoring the bottom and top since it is an open box)using the formula S=f(x)? I have been trying for the entire weekend and cannot get it! Thanks
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Your problem is lacking some information. Ask yourself, "What does x represent?"
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I'll describe how to look at this problem, and maybe this will give you some clues on how to solve it.
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Here goes. Is the open bottom of the box a square or a rectangle? If it is a rectangle, then it has a length (call it L) and a width (call it W) and by convention, L is thought of as being longer than W.
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The box has a height. Normally each side has the same height, so I'll assume this is the case for this box. Call this height H.
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Now suppose that you are going to build this box (open on the bottom and open on the top), and you are going to make it from a board that you buy (Home Depot?). Your plan is to buy the board wide enough so that when you stand it up on its edge, its width is equal to the height of the box you are building. The question then becomes, how long does the board have to be so that you can cut it exactly into the four pieces you need for sides.
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For example, if you were going to build a box 1 foot high and you wanted the sides to be 3 feet long and 2 feet wide (if it had a bottom, the dimensions of the bottom would be 3 ft by 2 ft) you would use a board 1 ft wide and long enough to cut into 4 side pieces ... 2 pieces 3 ft long and 2 pieces 2 ft long. That would mean that the board you need would have to be 10 ft long (2 pieces at 3 ft each is 6 ft and 2 pieces at 2 ft each is 4 more feet for a total of 6 + 4 = 10). So you buy a board 1 ft wide and 10 ft long to make the sides of the open box. The surface area of that board is found by multiplying its 1 ft width by its 10 ft length to find that the total surface area of the sides you are making will be 10 square ft.
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Now let's look at what we did in summary. First we decided on the height (H) that we wanted the box to be. Then we decided on the dimensions of the bottom (or the top) of the box (L and W). We selected the dimensions L and W and then added them up (isn't that the same as the perimeter of the bottom of the box?). Once we have the perimeter of the bottom of the box and the height of the box, then we multiplied those two figures to get the surface area of the sides.
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The one thing that you will note is that there are 3 unknown dimensions and they are H, L, and W.
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The resulting formula for finding the surface area of the sides of the box would be:
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S = H * P
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where P is the perimeter (total distance) around the bottom of the box. And we can replace P by its equivalent sum of 2 times the length plus 2 times the width of the bottom to get:
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S = H * (2L + 2W)
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You could reduce this number of unknown dimensions by saying that you wanted the bottom of the box to be a square. That would mean that the length of the bottom of the box was equal to the width, so that you could replace W by L to get:
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S = H * (2L + 2L)
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and by adding the two terms in the parentheses, you get:
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S = H * 4L
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and doing the multiplication this gives:
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S = 4*H*L
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And you could further reduce the number of dimensions by saying that you wanted the box to be a cube, that is that you want the height of the sides to equal the length of each of the 4 sides of the square bottom. In the above equation, this would allow you to replace H by L to get:
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S = 4*L*L = 4*L^2
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And, since L just is the dimension of any side of the cube shaped box, you could replace it with x and say:
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S = 4*x^2
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in which S is the surface area of the sides of a box with an open top and bottom, and x is the dimension of any side of the sides (length, width, or height). But, a caution. This last formula only holds for a cube. The most general formula for the surface area of the sides is:
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S = H * (2L + 2W)
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In which you select or identify the dimensions of the bottom and the height.
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I hope this helps you to figure out the answer to your problem. If not, please provide further information and post your problem again so one of the tutors can help you.
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Have a good weekend (what's left of it).
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