document.write( "Question 472258: Solve the problem.\r
\n" ); document.write( "\n" ); document.write( "The price p and the quantity x sold of a certain product obey the demand equation
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\n" ); document.write( "\"+p+=+%28%28-1%29%2F2%29+x+%2B+160+\" , 0 ≤ x ≤ 320.\r
\n" ); document.write( "\n" ); document.write( "What price should the company charge to maximize revenue?\r
\n" ); document.write( "\n" ); document.write( "$80\r
\n" ); document.write( "\n" ); document.write( "$120\r
\n" ); document.write( "\n" ); document.write( "$96\r
\n" ); document.write( "\n" ); document.write( "$40
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Algebra.Com's Answer #323812 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
the equation is:\r
\n" ); document.write( "\n" ); document.write( "\"+p+=+%28%28-1%29%2F2%29+x+%2B+160+\"
\n" ); document.write( "0 <= x <= 320
\n" ); document.write( "the revenue equation would be r = p*x
\n" ); document.write( "r is the revenue
\n" ); document.write( "p is the price per unit
\n" ); document.write( "x is the number of units sold.
\n" ); document.write( "since p is equivalent to (-1/2)*x+160, we can replace p in the equation to get:
\n" ); document.write( "r = ((-1/2)*x+160)*x
\n" ); document.write( "this becomes:
\n" ); document.write( "r = (-1/2)*x^2 + 160*x
\n" ); document.write( "if we replace r with y, we can graph this equation.
\n" ); document.write( "the graph is shown below:
\n" ); document.write( "\"graph%28600%2C600%2C-50%2C350%2C-10000%2C50000%2C%28-1%2F2%29%2Ax%5E2%2B160%2Ax%29\"
\n" ); document.write( "from the graph, it appears that revenue will peak at somewhere between 150 and 200.
\n" ); document.write( "we can use the max/min formula of x = -b/2a to find the x value of the max/min point.
\n" ); document.write( "our equation is:
\n" ); document.write( "y = (-1/2)*x^2 + 160*x
\n" ); document.write( "a = (-1/2)
\n" ); document.write( "b = 160
\n" ); document.write( "x = -b/2a becomes -160/(-1) which becomes x = 160.
\n" ); document.write( "this appears to be the maximum revenue point.
\n" ); document.write( "when x = 160, y = (-1/2)*(160)^2 + 160*160 = $12,800
\n" ); document.write( "when x = 150, y = (-1/2)*(150)^2 + 160*150 = $12,750
\n" ); document.write( "when x = 170, y = (-1/2)*(170)^2 + 160*170 = $12,750
\n" ); document.write( "numbers support the graph.
\n" ); document.write( "your answer is:
\n" ); document.write( "x = 160.
\n" ); document.write( "since p is a function of x, this makes p = (-1/2)*x + 160 which becomes:
\n" ); document.write( "p = (-1/2)*160 + 160 = -80 + 160 = $80.
\n" ); document.write( "When p = $80 and x = 160, the revenue is 80*160 = $12,800.
\n" ); document.write( "the company should charge $80 per unit in order to maximize revenue, based on the equations given.\r
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