document.write( "Question 111234: A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 5 in. by 10 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure.
\n" ); document.write( "(a) Express the volume V of the box as a function of x, then determine the range and the domain of this function.
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\n" ); document.write( "(b) Express the surface area S of the box as a function of x, then determine the range and the domain of this function. \r
\n" ); document.write( "\n" ); document.write( "(c) Find the value of x that gives the largest volume of the box \n" ); document.write( "
Algebra.Com's Answer #81174 by solver91311(24713)\"\" \"About 
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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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\n" ); document.write( "\"V=x%2810-2x%29%285-2x%29\"
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\n" ); document.write( "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 \"0%3Cx%3C2.5\".
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\n" ); document.write( "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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\n" ); document.write( "The volume function can be expressed as:
\n" ); document.write( "\"V=4x%5E3-30x%5E2%2B50x\"
\n" ); document.write( "This function will have a local maximum at the point in the range where the first derivative is zero.
\n" ); document.write( ":
\n" ); document.write( "\"dV%2Fdx=12x%5E2-60x%2B50\"
\n" ); document.write( "\"12x%5E2-60x%2B50=0\"
\n" ); document.write( "\"6x%5E2-30x=-25\"
\n" ); document.write( "\"x%5E2-5x=-25%2F6\"
\n" ); document.write( "\"x%5E2-5x%2B25%2F4=-25%2F6%2B25%2F4\"
\n" ); document.write( "\"%28x-5%2F2%29%5E2=25%2F12\"
\n" ); document.write( "\"x=5%2F2%2B-sqrt%2825%2F12%29\"
\n" ); document.write( "\"x=5%2F2%2B%285%2Asqrt%283%29%2F6%29\", which is > 2.5, so invalid
\n" ); document.write( "\"x=5%2F2-%285%2Asqrt%283%29%2F6%29\", is the x coordinate of the local maximum
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\n" ); document.write( "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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\n" ); document.write( "Surface Area Problem:
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\n" ); document.write( "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 \"S=50-4x%5E2\".
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\n" ); document.write( "Again the domain must be \"0%3Cx%3C2.5\" because if x is anywhere outside of that interval, there is no box.
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\n" ); document.write( "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.
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