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Today we're looking at maximizing function and we're given a practical problem where we have a sheet of aluminium and we know this is 20 inches wide and from here to here is 20 and then what we do is we fold up the corners so we get a shape like this a square missing the top edge and now we know that its length is still 20 however we have three sides still it now.
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So we know that these sides are equal and we're going to call them both of y and then we have this length here we're going to call x and we know that we must have the x plus two lots of y must be our 20 so that's our restriction.
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Now we want to maximize the surface area which is this area here.
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Now we see this is given by x times y.
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And so our surface area, a is equal to times y.
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That's just the red area.
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And so this is a function we want to maximize.
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We want the maximum possible surface area.
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So what we do is we're going to rearrange this equation in terms of x.
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So we have x equals 20 minus to y and we're going to substitute that into here.
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So our equation for area just depends on one of our length.
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And so what this becomes is a equal, but x is 20 minus 2y, and then we can expand this, 20 y minus 2 y squared.
02:20
And then what we're going to do is we're going to factorize it...