You can
put this solution on YOUR website!Since you are cutting x out of each corner, the length of the box must be 10 - 2x, and the width of the box must be 5 - 2x, and the height is x. So the volume must be given by:
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The domain of this function is determined by the physical limitations. In the first place, if x = 0, then there is no box, so one condition on the domain is that x > 0. On the other hand, if x = 2.5, there also is no box because the width (5 - 2x) would be zero, so the other condition on the domain is x < 2.5. So the entire expression for the domain is
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For the range, on one end we know that the box must have some volume, so x > 0. On the other end, the maximum value of the range is equal to the maximum Volume. Therefore, we have to solve the third part of the problem now.
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The volume function can be expressed as:
This function will have a local maximum at the point in the range where the first derivative is zero.
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, which is > 2.5, so invalid
, is the x coordinate of the local maximum
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I'll let you simplify that mess, but it is the upper value of the range, and the answer to the third part of the problem.
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Surface Area Problem:
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The surface area of the flat piece of cardboard is 5 X 10 or 50 sq inches. But you are cutting out 4 pieces each x * x, so the box surface area is
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Again the domain must be
because if x is anywhere outside of that interval, there is no box.
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The top end of the range is x < 50, and the bottom end is 25 which is the limit of S as x approaches 2.5.
