document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1110766>Question 1110766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The price p and the quantity x sold of a certain product obey the demand equation below.<BR>\n" );
document.write( "x equals negative 2 p plus 40 comma 0 less than or equals p less than or equals 20x=&#8722;2p+40,     0&#8804;p&#8804;20 <BR>\n" );
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document.write( "Express the revenue R as a function of x.<BR>\n" );
document.write( "&#8203;(b)<BR>\n" );
document.write( "What is the revenue if 44 units are&#8203; sold?<BR>\n" );
document.write( "&#8203;(c)<BR>\n" );
document.write( "What quantity x maximizes&#8203; revenue? What is the maximum&#8203; revenue?<BR>\n" );
document.write( "&#8203;(d)<BR>\n" );
document.write( "What price should the company charge to maximize&#8203; revenue?<BR>\n" );
document.write( "&#8203;(e)<BR>\n" );
document.write( "What price should the company charge to earn at least &#8203;$192 in&#8203; revenue? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #725813 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=725813\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i think that what you wanted to say is:\r<BR>\n" );
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document.write( "price p and quantity x obey the demand equation below:\r<BR>\n" );
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document.write( "x = -2p + 40, where 0 <= p <= 20\r<BR>\n" );
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document.write( "0 <= p <= 20\r<BR>\n" );
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document.write( "your revenue equation would be r = price * quantity = p * x.\r<BR>\n" );
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document.write( "r is the revenue<BR>\n" );
document.write( "x is the number of units sold.<BR>\n" );
document.write( "p is the price per unit.\r<BR>\n" );
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document.write( "the equation is r = p * x\r<BR>\n" );
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document.write( "you are given that x = -2p + 40\r<BR>\n" );
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document.write( "solve for p to get p = 20 - .5x\r<BR>\n" );
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document.write( "in the revenue equation of r = p * x, replace p with 20 - .5x to get:\r<BR>\n" );
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document.write( "r = (20 - .5x) * x\r<BR>\n" );
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document.write( "simplify to get r = 20x - .5x^2.\r<BR>\n" );
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document.write( "set the revenue equal to 0 and you get 0 = 20x - .5x^2\r<BR>\n" );
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document.write( "rearrange the terms to get this equation in standard quadratic form to get: \r<BR>\n" );
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document.write( "-.5x^2 + 20x = 0\r<BR>\n" );
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document.write( "in this form:\r<BR>\n" );
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document.write( "a = coefficient of x^2 term = -.5<BR>\n" );
document.write( "b = coefficient of x term = 20<BR>\n" );
document.write( "c = constant term = 0.\r<BR>\n" );
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document.write( "the maximum value of this quadratic equation is when x = -b/2a.\r<BR>\n" );
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document.write( "solve for x to get x = -20 / -1 = 20.\r<BR>\n" );
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document.write( "when x = 20, the original equation of r = 20x - .5x^2 becomes r = 20*20 - .5*20^2 which becomes r = 400 - 200 which results in r = 200.\r<BR>\n" );
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document.write( "your maximum revenue is 200 when x = 20\r<BR>\n" );
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document.write( "since r = x * p, then you can solve for p to get p - r/x, which becomes:\r<BR>\n" );
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document.write( "200/20 = 10\r<BR>\n" );
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document.write( "the price per shirt would have to be 10 in order to get a revenue of 200 when the quantity sold is 20.\r<BR>\n" );
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document.write( "20 * 10 = 200\r<BR>\n" );
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document.write( "you were previously given that x = -2p + 40\r<BR>\n" );
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document.write( "when x = 20, the equation becomes 20 = -2p + 40\r<BR>\n" );
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document.write( "subtract 40 from both sides to get -20 = -2p\r<BR>\n" );
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document.write( "divide both sides by -2 to get p = 10.\r<BR>\n" );
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document.write( "the price of 10 when the quantity sold is 20 confirms the solution that tells you that the maximum revenue is 200 when the quantity sold is 20.\r<BR>\n" );
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document.write( "answers to your questions are shown below:\r<BR>\n" );
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document.write( "(a)\r<BR>\n" );
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document.write( "Express the revenue R as a function of x.\r<BR>\n" );
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document.write( "r = 20x - .5x^2\r<BR>\n" );
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document.write( "(b)\r<BR>\n" );
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document.write( "What is the revenue if 44 units are sold?\r<BR>\n" );
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document.write( "when x = 44, r = 20x - .5x^2 becomes:\r<BR>\n" );
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document.write( "r = 20*44 - .5 * (44)^2 which becomes:\r<BR>\n" );
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document.write( "r = -88\r<BR>\n" );
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document.write( "since r = p * x, p would have had to be -2.\r<BR>\n" );
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document.write( "this solution is invalid, since your problem stated that 0 <= p <= 20.\r<BR>\n" );
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document.write( "therefore 44 units could not be sold, since, by the equation, x = 44 results in p = -2 which is out of the range of the permissible values of p.\r<BR>\n" );
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document.write( "if you go back to the original equation that says x = -2p + 40, then:\r<BR>\n" );
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document.write( "when p = 0, x = 40 and when p = 20, x = 0\r<BR>\n" );
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document.write( "therefore 0 <= p <= 20 translates to 0 <= x <= 40\r<BR>\n" );
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document.write( "x = 44 is out of the permissible range of of both p and x.\r<BR>\n" );
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document.write( "(c)\r<BR>\n" );
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document.write( "What quantity x maximizes revenue? What is the maximum revenue?\r<BR>\n" );
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document.write( "when x = 20, r = 200.\r<BR>\n" );
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document.write( "this is when the maximum revenue occurs.\r<BR>\n" );
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document.write( "(d)\r<BR>\n" );
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document.write( "What price should the company charge to maximize revenue?\r<BR>\n" );
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document.write( "they should charge 10 dollars per unit.\r<BR>\n" );
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document.write( "(e)\r<BR>\n" );
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document.write( "What price should the company charge to earn at least $192 in revenue?\r<BR>\n" );
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document.write( "your equation is r = 20x - .5x^2\r<BR>\n" );
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document.write( "if you want the revenue to be greater than 192, then r must be > 192.\r<BR>\n" );
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document.write( "this means that 20x - .5x^2 must be greater than 192.\r<BR>\n" );
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document.write( "your equation becomes an inequality that says 20x - .5x^2 > 192\r<BR>\n" );
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document.write( "if you subtract 192 from both side of this inequality, you get:\r<BR>\n" );
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document.write( "20x - .5x^2 - 192 > 0\r<BR>\n" );
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document.write( "to find the 0 points, set 20x - .5x^2 - 192 = 0 and solve for x.\r<BR>\n" );
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document.write( "you will get x = 16 or x = 24.\r<BR>\n" );
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document.write( "that's when 20x - .5x^2 - 192 = 0\r<BR>\n" );
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document.write( "since 20x - .5x^2 represents the revenue, then that's when the revenue - 192 = 0 which means that's when the revenue = 192, because, when you add 192 to both sides of 20x - .5x^2 - 192 = 0, you get 20x - .5x^2 = 192, which is the same as saying revenue = 192.\r<BR>\n" );
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document.write( "so when x = 16 or when x = 24, the revenue is equal to 192.\r<BR>\n" );
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document.write( "to find out what interval the revenue is greater than 192, then pick values less than 16, greater than 24, between 16 and 24.\r<BR>\n" );
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document.write( "you will find that the revenue is greater than 192 when the quantity of x is > 16 and < 24.\r<BR>\n" );
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document.write( "the graph of the original revenue equation shows that to be true.\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src= \"http://theo.x10hosting.com/2018/022804.jpg\" alt=\"$$$\" > </SPAN>\n" );
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