document.write( "Question 725967: Okay, got 3 problems that are giving me problems! All 3 involve maximizing and minimizing. I know the answers, I just don't know how to do the work. I must show it algebraically, not just by guessing. Please help me with set up!!\r
\n" ); document.write( "\n" ); document.write( "1) What is the maximum product of two numbers that have a sum of 26? (I know the answer is 169, 13x13, but how do I show this by solving an equation?)\r
\n" ); document.write( "\n" ); document.write( "2) What is the minimum product of two numbers that differ by 8? What are the numbers? \r
\n" ); document.write( "\n" ); document.write( "3)A rectangular compost container is to be formed in a corner of a fenced yard, with 8ft. of chicken wire completing the other two sides of the rectangle. If the chicken wire is 3 ft. high, what dimensions of the base will maximize the container's volume? \r
\n" ); document.write( "\n" ); document.write( "Thanks in advance! \n" ); document.write( "
Algebra.Com's Answer #444473 by fcabanski(1391)\"\" \"About 
You can put this solution on YOUR website!
1. x+y=26 thus y = 26-x. Maximize x*y, which is the same as x(26-x) = \"26x-x%5E2\" That's a parabola. You can use calculus to find the maximum, or you can find the x-coordinate of the vertex of the parabola = -b/2a where a is the coefficient of the squared term and b is the coefficient of the variable term.


\n" ); document.write( "a=-1, b=26 so -b/2a = -(26/-2) = 13.


\n" ); document.write( "When x=13 y= 26-13=13. The numbers are 13 and 13.

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\n" ); document.write( "\n" ); document.write( "Using calculus set the derivative - 0. 26-2x = 0 ---> -2x = -26 ---> x = 13.


\n" ); document.write( "2. y=x+8. Minimize xy, which is minimize x(x+8) = \"x%5E2+%2B+8x\".


\n" ); document.write( "a=1, b=8. -b/2a = -8/2 = -4. y = -4+8=4. The numbers are 4 and -4.


\n" ); document.write( "Using calculus set 2x+8=0---> 2x=-8--->x=-4. y=-4+8 = 4.


\n" ); document.write( "3. Volume is length*width*height. Call length x and width y. The height is 3. So maximize x*y*3 when x+y=8 or y=8-x.


\n" ); document.write( "x*(8-x)*3 = \"24x+-3x%5E2\"


\n" ); document.write( "a=-3 and b=24. The x-coordinate of the vertex is -(24/-6) = 4.


\n" ); document.write( "y=8-4 = 4. 4x4 maximizes the volume.


\n" ); document.write( "Using calculus set 24-6x=0 --->-6x=-24 --->x=4 and y=8-4=x.
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Hope the solution helped. Sometimes you need more than a solution. Contact fcabanski@hotmail.com for online, private tutoring, or personalized problem solving (quick for groups of problems.) \n" ); document.write( "

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