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Calculus/Marginal Amounts
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Question
1. You are designing a rectangular poster to contain 256 in squared of printing with a 3 in margin at the top and bottom and a 2 in margin at each side. What overall dimensions will minimize the amount of paper used?
2. A rectangle with its base on the x-axis is to be inscribed under the graph of y=2-x squared. Find the height of the rectangle if the area is the largest possible area.
1. It is known that 16² = 256, so we need an area that is 16x16 with a border.
There are 2" on either side, so add 16+2+2 to get the width.
The top is 3 nd the bottom is 2, so 16+3+2 is the height.
2. If the function is y = 2 - x² ....
The base is centered at the origin with x units on either side, so the width is 2x.
The height is y, which is 2 - x².
The total area is then 2x(2-x²) = 4x - 2x³.
I will say that this if A(x), for area.
From here, it can be seen that A'(x) = 4 - 6x².
Set that to 0, solve for x, and put back in A to find the area.
Note that x would be a plus or minus,
but only the plus makes sense since we are talking about a physical width.
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Scotto
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