document.write( "Question 968969: what dimensions would a rectangular prism have if its surface area was 160 fee square. Based on the answer what would the figure's volume be. \n" ); document.write( "

Algebra.Com's Answer #592070 by KMST(5328) $\"\"$ $\"About$

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Let $\"X\"$ , $\"Y\"$ , and $\"Z\"$ be the dimensions (in feet) of the rectangular prism.
\n" ); document.write( "Let's say $\"X%3C=Y%3C=Z\"$ .
\n" ); document.write( "

\n" ); document.write( "The surface area of the base of that {box) is $\"YZ\"$ square feet.
\n" ); document.write( "The surface area of the top of that {box) is $\"YZ\"$ square feet.
\n" ); document.write( "The surface area of the sides of that {box) is $\"2XY%2B2XZ\"$ square feet.
\n" ); document.write( "The total surface area of the base of that {box) in square feet is
\n" ); document.write( "is $\"2XY%2B2XZ%2B2YZ=2%28XY%2BXZ%2BYZ%29\"$ .
\n" ); document.write( "We are told that the total surface area is $\"160\"$ square feet, so we know that
\n" ); document.write( " $\"2%28XY%2BXZ%2BYZ%29=160\"$ <---> $\"XY%2BXZ%2BYZ=160%2F2\"$ <---> $\"XY%2BXZ%2BYZ=80\"$ .
\n" ); document.write( "We can give values to $\"X\"$ and $\"Y\"$ and solve for $\"Z\"$ .
\n" ); document.write( "Then we can calculate the volume of that box (in cubic feet) as
\n" ); document.write( " $\"Volume=X%2AY%2AZ\"$ , by multiplying together $\"X\"$ , $\"Y\"$ , and $\"Z\"$ .
\n" ); document.write( "
\n" ); document.write( "Let's say that $\"system%28X=1%2C%22and%22%2CY=2%29\"$ .
\n" ); document.write( "Then, substituting those values, we get
\n" ); document.write( " $\"1%2A2%2B1%2AZ%2B2%2AZ=80\"$ --> $\"2%2BZ%2B2Z=80\"$ --> $\"2%2B3Z=80\"$ --> $\"3Z=80-2\"$ --> $\"3Z=78\"$ --> $\"Z=78%2F3\"$ --> $\"Z=26\"$ .
\n" ); document.write( " $\"Volume=1%2A2%2A26=52\"$ .
\n" ); document.write( "
\n" ); document.write( "Let's say that $\"system%28X=1%2C%22and%22%2CY=8%29\"$ .
\n" ); document.write( "Then, substituting those values, we get
\n" ); document.write( " $\"1%2A8%2B1%2AZ%2B8%2AZ=80\"$ --> $\"8%2B9Z=80\"$ --> $\"9Z=80-8\"$ --> $\"9Z=72\"$ --> $\"Z=72%2F8\"$ --> $\"Z=8\"$ .
\n" ); document.write( " $\"Volume=1%2A8%2A8=64\"$ .
\n" ); document.write( "
\n" ); document.write( "Let's say that $\"system%28X=2%2C%22and%22%2CY=2%29\"$ .
\n" ); document.write( "Then, substituting those values, we get
\n" ); document.write( " $\"2%2A2%2B2%2AZ%2B2%2AZ=80\"$ --> $\"4%2B4Z=80\"$ --> $\"4Z=80-4\"$ --> $\"4Z=76\"$ --> $\"Z=76%2F4\"$ --> $\"Z=19\"$ .
\n" ); document.write( " $\"Volume=2%2A2%2A19=76\"$ .
\n" ); document.write( "
\n" ); document.write( "If we start with $\"system%28X=2%2C%22and%22%2CY=4%29\"$ , we find
\n" ); document.write( " $\"system%28Z=12%2C%22and%22%2CVolume=96%29\"$ .
\n" ); document.write( "
\n" ); document.write( "If we start with $\"system%28X=2%2C%22and%22%2CY=5%29\"$ , we find
\n" ); document.write( " $\"system%28Z=10%2C%22and%22%2CVolume=100%29\"$ .
\n" ); document.write( "
\n" ); document.write( "If we start with $\"system%28X=4%2C%22and%22%2CY=4%29\"$ , we find
\n" ); document.write( " $\"system%28Z=8%2C%22and%22%2CVolume=128%29\"$ .
\n" ); document.write( "
\n" ); document.write( "Other choices yield measurements that are not whole numbers.
\n" ); document.write( "For example, if we start with $\"system%28X=5%2C%22and%22%2CY=5%29\"$ , we find
\n" ); document.write( " $\"system%28Z=5.5%2C%22and%22%2CVolume=137.5%29\"$ ,
\n" ); document.write( "and if we wanted a cube, with $\"X=Y=Z\"$ , we would have
\n" ); document.write( " $\"X%2AY%2BX%2AZ%2BY%2AZ=80\"$ ---> $\"X%2AX%2BX%2AX%2BX%2AX=80\"$ ---> $\"X%5E2%2BX%5E2%2BX%5E2=80\"$ ---> $\"3X%5E2=80\"$ ---> $\"X%5E2=80%2F3\"$ ---> $\"Z=Y=X=sqrt%2880%2F3%29=4sqrt%285%2F3%29=4sqrt%2815%29%2F3=about5.164\"$
\n" ); document.write( "and

\n" ); document.write( "
\n" ); document.write( "NOTE:
\n" ); document.write( "The way to answer the question depends on the math level of the class where this question was asked.
\n" ); document.write( "Was it asked in elementary school? In a college advanced calculus class? somewhere in between.
\n" ); document.write( "I do not believe this is a question from a multivariate calculus class, but intuition would tell us that the rectangular prism with greatest volume will be a cube. \n" ); document.write( "

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