document.write( "Question 720885: find two positive numbers with a product of 185 and the sum is a minimum. round of your answer in 4 decimal places. \n" ); document.write( "

Algebra.Com's Answer #442051 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
I hope you know some Calculus because I can;t think of any other way to solve this. If you don't know Calculus then re-post the question and say that the solution cannot use Calculus.

\n" ); document.write( "If the two numbers are x and y, then the fact that their product is 185 would translate into:
\n" ); document.write( "x*y = 185
\n" ); document.write( "Solving for y this would give us:
\n" ); document.write( "y = 185/x

\n" ); document.write( "The equation for the sum of x and y would be:
\n" ); document.write( "s = x + y (where \"s\" is the sum)
\n" ); document.write( "Substituting the expression we found for y gives us:
\n" ); document.write( " $\"s+=+x+%2B+185%2Fx\"$
\n" ); document.write( "To find a minimum sum we will find the first derivative...
\n" ); document.write( " $\"ds%2Fdx+=+1+%2B+%28-185%29%2Fx%5E2\"$
\n" ); document.write( "...set it to zero...
\n" ); document.write( " $\"0+=+1+%2B+%28-185%29%2Fx%5E2\"$
\n" ); document.write( "... and solve for x:
\n" ); document.write( " $\"x+=+sqrt%28185%29\"$ or $\"x+=+-sqrt%28185%29\"$
\n" ); document.write( "Since x (and y) must be positive numbers, we reject the negative solution. Using $\"x+=+sqrt%28185%29\"$ and solving for y gives us $\"y+=+sqrt%28185%29\"$

\n" ); document.write( "There are various ways to check if this is a minimum and not a maximum. One way is to pick a different pair of positive numbers whose product is 185. For example, 1 and 185. The sum of these two is 186. Since $\"sqrt%28185%29+%3C+14\"$ the sum of two of them will be less than 28, far less than the sum of 1 and 185.

\n" ); document.write( "So $\"x+=+sqrt%28185%29\"$ and $\"y+=+sqrt%28185%29\"$ does indeed represent the pair of positive \n" ); document.write( "

\n" );