document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=202700>Question 202700</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Suppose you have a lemonade stand, and when you charge $2 per cup of lemonade you sell 120 cups.  But when you raise your price to $3 you only sell 60 cups.  \r<BR>\n" );
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document.write( "a.    Write an equation for the number of cups you sell as a function of the price you charge.  \r<BR>\n" );
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document.write( "b.    Denote \"C\" for number of cups, and \"P\" for the price you charge.\r<BR>\n" );
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document.write( "c.    Assume the function is linear.\r<BR>\n" );
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document.write( " Continuing our lemonade stand question:\r<BR>\n" );
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document.write( "a.    We all know that total revenue (TR) is a function of the price we charged (P) multiplied by the item quantity sold (in our case – Cups), i.e., TR = Price * Cups\r<BR>\n" );
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document.write( "b.    Please write the equation for your TR by inputting your answer from the function you have calculated in question #2.\r<BR>\n" );
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document.write( "c. What price would maximize your TR?\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #152881 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=152881\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you can graph this as a linear equation by taking the y value and making it the price and taking the x value and making it the quantity of cups sold.<BR>\n" );
document.write( "for the equation I will be using y = m*x + b which I will be substituting and changing to c = m*p + b down below after all the work has been done.<BR>\n" );
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document.write( "formula for a straight line is:<BR>\n" );
document.write( "y = m*x + b<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "m = slope = delta y / delta x = (y2 - y1) / (x2 - x1)<BR>\n" );
document.write( "you have two points.<BR>\n" );
document.write( "first point:<BR>\n" );
document.write( "when price is 2, quantity sold is 120<BR>\n" );
document.write( "when price is 3, quantity sold is 60<BR>\n" );
document.write( "x is the price<BR>\n" );
document.write( "y is the quantity sold<BR>\n" );
document.write( "x1 = 2, y1 = 120<BR>\n" );
document.write( "x2 = 3, y2 = 60<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "(y2-y1)/(x2-x1) = (60-120)/(3-2) = -60/1 = -60<BR>\n" );
document.write( "your slope = m = (-60)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "you know the slope; now you want to find the y intercept.<BR>\n" );
document.write( "take any point on the graph that you know.<BR>\n" );
document.write( "(x1,y1) should do.<BR>\n" );
document.write( "(x1,y1) = (2,120) from above.<BR>\n" );
document.write( "plug it into the equation y = m*x + b to get:<BR>\n" );
document.write( "120 = -60*2 + b<BR>\n" );
document.write( "solve for b<BR>\n" );
document.write( "120 = -120 + b<BR>\n" );
document.write( "add 120 to both sides to get:<BR>\n" );
document.write( "b = 240<BR>\n" );
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document.write( "you now have the equation for the straight line.<BR>\n" );
document.write( "m = -60<BR>\n" );
document.write( "y intercept = 240<BR>\n" );
document.write( "equation is:<BR>\n" );
document.write( "y = -60*x + 240<BR>\n" );
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document.write( "you test this to be true by substituting known values and look for the answer you expect.<BR>\n" );
document.write( "when y = 2, x = 120<BR>\n" );
document.write( "in the equation, y = -60*x + 240, substitute 120 for y and 2 for x to get:<BR>\n" );
document.write( "120 = -60*(2) + 240<BR>\n" );
document.write( "which becomes<BR>\n" );
document.write( "120 = -120 + 240<BR>\n" );
document.write( "which becomes<BR>\n" );
document.write( "120 = 120<BR>\n" );
document.write( "which is true so the equation is good.<BR>\n" );
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document.write( "your equation is good, but they wanted you to let c = number of cups sold and p = price, so you need to substitute c for y and p for x to get:<BR>\n" );
document.write( "c = -60*p + 240<BR>\n" );
document.write( "that's your equation.<BR>\n" );
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document.write( "Total Revenue is a function of price * quantity.<BR>\n" );
document.write( "our linear formula calculated quantity as a function of price.<BR>\n" );
document.write( "If Revenue is to be calculated, we can used this formula (I think).<BR>\n" );
document.write( "Our formula for quantity sold is:<BR>\n" );
document.write( "c = -60*p + 240<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we should be able to use this same formula to solve for price as follows:<BR>\n" );
document.write( "c - 240 = -60*p<BR>\n" );
document.write( "p = -(c-240)/60 = (240-c)/60<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we now have a formula for c, and a formula for p.<BR>\n" );
document.write( "let r = total revenue = price * quantity<BR>\n" );
document.write( "since price = p and quantity = c, we have:<BR>\n" );
document.write( "r = p*c<BR>\n" );
document.write( "we know that c = -60p + 240<BR>\n" );
document.write( "we also know that p = (240-c)/60<BR>\n" );
document.write( "our formula becomes:<BR>\n" );
document.write( "r = p*c = (-60p+240)*(240-c)/60<BR>\n" );
document.write( "we multiply both sides of this equation by 60 to get:<BR>\n" );
document.write( "60*r = (-60p+240)*(240-c)/60<BR>\n" );
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document.write( "we solve this eqauation to get:<BR>\n" );
document.write( "60*r = -14400p + 60p*c + 57600 - 240c<BR>\n" );
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document.write( "we divide both sides of this equation by 60 to get:<BR>\n" );
document.write( "r = -240p + p*c + 960 - 4c<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since c = -60p + 240, we substitute for c by replacing it with -60p + 240 in the equation to get:<BR>\n" );
document.write( "r = -240p + p*(-60p+240) + 960 - 4(-60p+240)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "r = -240p - 60p^2 + 240p + 960 +240p - 960<BR>\n" );
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document.write( "this reduces to:<BR>\n" );
document.write( "r = -60p^2 + 240p<BR>\n" );
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document.write( "our equation for revenue is r = -60p^2 + 240p<BR>\n" );
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document.write( "to find the roots, we set it equal to 0.<BR>\n" );
document.write( "-60p^2 + 240p = 0<BR>\n" );
document.write( "multiply both sides of the equation by (-1) to get:<BR>\n" );
document.write( "60p^2 - 240p = 0<BR>\n" );
document.write( "divide both sides of the equation by 60 to get:<BR>\n" );
document.write( "p^2 - 4p = 0<BR>\n" );
document.write( "factor out the p to get:<BR>\n" );
document.write( "p*(p-4) = 0<BR>\n" );
document.write( "the roots are:<BR>\n" );
document.write( "p = 0<BR>\n" );
document.write( "or <BR>\n" );
document.write( "p = 4<BR>\n" );
document.write( "these would be the points where revenue would = 0 which means that the graph crosses the x axis.<BR>\n" );
document.write( "from the equation you see that when p = 0 r = 0.<BR>\n" );
document.write( "from the equation you see that when p = 4, r = 0<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since this is a quadratic equation, we can find the maximum/minimnum point by taking -b/2a.<BR>\n" );
document.write( "standard form of quadratic equation is ax^2 + bx + c = 0<BR>\n" );
document.write( "our equation is p^2 - 4p = 0 (the p stands for x).<BR>\n" );
document.write( "this makes:<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = -4<BR>\n" );
document.write( "c = 0<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "-b/2a = 4/2 = 2<BR>\n" );
document.write( "our equation will be at a maximum or at a minimum when p = 2.<BR>\n" );
document.write( "it turns out to be a maximum, so we have maximum revenue with p = 2.<BR>\n" );
document.write( "when p = 2, c = 120.<BR>\n" );
document.write( "a graph of the equation is shown below.<BR>\n" );
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document.write( "the equation we are graphing is the final equation for revenue which was:<BR>\n" );
document.write( "r = -60p^2 + 240p<BR>\n" );
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document.write( "to graph the equation, we set<BR>\n" );
document.write( "x = p = price of lemonade<BR>\n" );
document.write( "and we set the y = c = quantity of cups of lemonade sold<BR>\n" );
document.write( "values on the x-axis becomes the price of lemonade and values on the y-axis become the quantity of cups of lemonade sold.<BR>\n" );
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document.write( "this problem was not tremendously difficult to solve, but it was tricky.  <BR>\n" );
document.write( "you had to keep very close attention to the numerous sign changes that occurred.<BR>\n" );
document.write( "you needed to know the formula for a straight line.<BR>\n" );
document.write( "you needed to know the formula for a quadratic equation.<BR>\n" );
document.write( "you needed to know the formula for the maximum/minimum point of a quadratic equation.<BR>\n" );
document.write( "you needed to know that you could get the formula of p from the formula for c, or that you could get the formula of c from the formula for p.<BR>\n" );
document.write( "you needed to know that you could substitute c for p, or p for c, in the final equation to eliminate one of the unknowns.<BR>\n" );
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