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Okay, so here we have this open top box
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and we know it's volume, and our goal is
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to find the surface area as a function of a
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side link. So we know that the base is
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a square, so we could label the base with
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side length A and A and we don't know the
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height of the box, so we can just call
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it H. Now we do know the volume is
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too. So we know volume is length times,
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width, times, height. So that would be
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a times a times age, a square times age
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is too. So perhaps will end up using that
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relationship as we go. Our goal is to find
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the surface area and because there's no top, we
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have only five services. We have the base,
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which is a times A, and we have four
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of these sides which are eight times h. So
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the surface area, which I'm going to call capital
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all college A s since we're already using an A
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for something else surface area is s is going to
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be the some of these areas. So a squared
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for the area of the base, plus four times
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a times age for the four sides. Now we
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need this to be a function of just the side
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, not the side and the height. So here's
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where that other equation comes into play. Let's take
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that other equation we got from volume and isolate H
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. We get H equals two over a squared and
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we can substitute that into the equation we're working with
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to get rid of the H. So now we
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have s equals a squared plus four, a times
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two over a squared. Now that's gonna need a
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little bit of simplifying. So here's that formula again
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, and we want to simplify it. So we're
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going to go ahead and multiply the four a by
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the two over a squared and that gives us eight
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over a. That should be a plus. Okay
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, now, this would be perfectly fine. However
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, if you want to keep simplifying and get a
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common denominator, you could multiply the first term by
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a over a. We have a cubed over a
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plus eight over a and that gives us a cubed
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plus eight over a. I don't feel strongly that
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that's necessary. I would be happy to leave it
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like this, but these are equivalent. All right
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, What about the domain? So what could a
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be a represented the length of aside? And so
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it has to be positive. Other than that,
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we don't know of any restrictions on it.