document.write( "Question 86040: I have a homeowrk question that states: find the maximum revenue for the revenue function R(x)=490x-0.07x^2. How do I work this problem out? I have tried everything. \n" ); document.write( "
Algebra.Com's Answer #62168 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
\"R%28x%29=490x-0.07x%5E2\"\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\"R%28x%29=490x-%287%2F100%29x%5E2\" Make into \"0.07\" into a fraction\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\"R%28x%29=-%287%2F100%29x%5E2%2B490x\" Rearrange the terms\r
\n" ); document.write( "\n" ); document.write( "Now lets convert this to vertex form to find the max revenue\r
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "\n" ); document.write( "
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

\n" ); document.write( "
\n" ); document.write( " \"y=%28-7%2F100%29+x%5E2%2B490+x%2B0\" Start with the given equation
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y-0=%28-7%2F100%29+x%5E2%2B490+x\" Subtract \"0\" from both sides
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y-0=%28-7%2F100%29%28x%5E2-7000x%29\" Factor out the leading coefficient \"%28-7%2F100%29\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Take half of the x coefficient \"-7000\" to get \"-3500\" (ie \"%281%2F2%29%28-7000%29=-3500\").
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Now square \"-3500\" to get \"12250000\" (ie \"%28-3500%29%5E2=%28-3500%29%28-3500%29=12250000\")
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y-0=%28-7%2F100%29%28x%5E2-7000x%2B12250000-12250000%29\" Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of \"12250000\" does not change the equation
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y-0=%28-7%2F100%29%28%28x-3500%29%5E2-12250000%29\" Now factor \"x%5E2-7000x%2B12250000\" to get \"%28x-3500%29%5E2\"
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y-0=%28-7%2F100%29%28x-3500%29%5E2-%28-7%2F100%29%2812250000%29\" Distribute
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y-0=%28-7%2F100%29%28x-3500%29%5E2%2B857500\" Multiply
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y=%28-7%2F100%29%28x-3500%29%5E2%2B857500%2B0\" Now add \"0\" to both sides to isolate y
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"y=%28-7%2F100%29%28x-3500%29%5E2%2B857500\" Combine like terms
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Now the quadratic is in vertex form \"y=a%28x-h%29%5E2%2Bk\" where \"a=-7%2F100\", \"h=3500\", and \"k=857500\". Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Check:
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Notice if we graph the original equation \"y=%28-7%2F100%29x%5E2%2B490x%2B0\" we get:
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-7%2F100%29x%5E2%2B490x%2B0%29\" Graph of \"y=%28-7%2F100%29x%5E2%2B490x%2B0\". Notice how the vertex is (\"3500\",\"857500\").
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " Notice if we graph the final equation \"y=%28-7%2F100%29%28x-3500%29%5E2%2B857500\" we get:
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " \"graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-7%2F100%29%28x-3500%29%5E2%2B857500%29\" Graph of \"y=%28-7%2F100%29%28x-3500%29%5E2%2B857500\". Notice how the vertex is also (\"3500\",\"857500\").
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( " So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "

\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the maximum revenue that can be generated is $857,500 (the y-coordinate of the vertex) by selling 3,500 units (the x-coordinate of the vertex) \n" ); document.write( "
\n" );