document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168072>Question 1168072</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A coffee shop currently sells 410 lattes a day at $2.75 each. They recently tried raising the by price by $0.25 a latte, and found that they sold 30 less lattes a day.\r<BR>\n" );
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document.write( "a) Assume that the number of lattes they sell in a day, N, is linearly related to the sale price, p (in dollars). Find an equation for  <BR>\n" );
document.write( "N as a function of p\r<BR>\n" );
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document.write( "N(p) = ?<BR>\n" );
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document.write( "b) Revenue (the amount of money the store brings in before costs) can be found by multiplying the cost per cup times the number of cups sold. Again using p as the sales price, use your equation from above to write an equation for the revenue,  <BR>\n" );
document.write( "R as a function of p.\r<BR>\n" );
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document.write( "R(p)= ?<BR>\n" );
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document.write( "c) The store wants to maximize their revenue (make as much money as possible). Find the value of p that will maximize the revenue (round to the nearest cent).\r<BR>\n" );
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document.write( "p = ?\r<BR>\n" );
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document.write( "which will give a maximum revenue of $ ?\r<BR>\n" );
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document.write( "I'm in real need of resolving this problem. Please help me!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #792692 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=792692\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;My solution and answers are inserted in your text.\r<BR>\n" );
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document.write( "a)  Assume that the number of lattes they sell in a day, N, is linearly related to the sale price, p (in dollars). \r\n" );
document.write( "    Find an equation for N as a function of p\r\n" );
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document.write( "        They lose 30 customers for each raising the price of $0.25.   It means that the rate of losing customers \r\n" );
document.write( "        is  30*4 = 120 per dollar raising. So\r\n" );
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document.write( "            N(p) = 410 + 120*(2.75 - p),      (1)\r\n" );
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document.write( "        where \"p\" is the new price.   Part (a) is completed.\r\n" );
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document.write( "b)  Revenue (the amount of money the store brings in before costs) can be found by multiplying the cost per cup \r\n" );
document.write( "    times the number of cups sold. Again, using p as the sales price, use your equation from above to write an equation \r\n" );
document.write( "    for the revenue, R as a function of p.\r\n" );
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document.write( "        Revenue is the product of the price by the number of customers\r\n" );
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document.write( "            R(p)= p*N(p) = p*(410 + 120*(2.75 - p)),    (2)\r\n" );
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document.write( "        according to (1).\r\n" );
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document.write( "        So, the equation is obtained and part (b) is completed.\r\n" );
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document.write( "c)  The store wants to maximize their revenue (make as much money as possible). \r\n" );
document.write( "    Find the value of p that will maximize the revenue (round to the nearest cent).\r\n" );
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document.write( "        The revenue, as I showed you in part (b) (formula (2)) is this quadratic function\r\n" );
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document.write( "            R(p) = p*(410 + 120*(2.75 - p)) = -120*p^2 + 740p.    (3)\r\n" );
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document.write( "        This quadratic function has a negative leading coefficient at p^2; so the parabola is open downward and has a maximum.\r\n" );
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document.write( "        For any such parabola, its maximum is achieved at  p = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-b%2F%282a%29 ALIGN=MIDDLE  ALT=\"-b%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r\n" );
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document.write( "            where \"a\" is the coefficient at the quadratic term and \"b\" is the coefficient at the linear term.\r\n" );
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document.write( "        In your case,  a = -120,  b = 740,  therefore the optimum value of p is  \r\n" );
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document.write( "            <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5Bopt%5D ALIGN=MIDDLE  ALT=\"p%5Bopt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-740%2F%282%2A%28-120%29%29 ALIGN=MIDDLE  ALT=\"-740%2F%282%2A%28-120%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=740%2F240 ALIGN=MIDDLE  ALT=\"740%2F240\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37%2F12 ALIGN=MIDDLE  ALT=\"37%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 3.08 dollars (rounded to two decimals).   <U>ANSWER</U>\r\n" );
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document.write( "        The optimum price is 3.08 dollars per latte, and the maximum revenue is then  -120*3.08^2 + 740*3.08 = 1140.83 dollars.\r\n" );
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document.write( "        Compare it with the current revenue of  410*2.75 = 1127.50 dollars.\r\n" );
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document.write( "    See the plot below.\r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28+400%2C+400%2C+-2%2C+6%2C+-100%2C+1200%2C%0D%0A++++++++++x%2A%28410+%2B+120%2A%282.75+-+x%29%29+++++++%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "                Plot y = x*(410 + 120*(2.75 - x))\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "---------------\r<BR>\n" );
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document.write( "You may find this technique slightly complicated, &nbsp;if you learn the subject for the first time.\r<BR>\n" );
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document.write( "It is very normal for the beginner students. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;* * * &nbsp;&nbsp;ONLY &nbsp;PRACTICING &nbsp;MAY &nbsp;HELP &nbsp;&nbsp;* * * \r<BR>\n" );
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document.write( "To give more practice for such students, &nbsp;I prepared lessons in this site.\r<BR>\n" );
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document.write( "One group of lessons explain how to find a minimum/maximum of a quadratic function. These lessons are\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>\r<BR>\n" );
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document.write( "They will help you to develop your skills in finding a maximum/minimum of a quadratic function to the &nbsp;AUTOMATIC &nbsp;LEVEL.\r<BR>\n" );
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document.write( "Another lesson presents applications of this technique to problems similar to yours\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/Using-quadratic-functions-to-solve-problems-on-maximizing-profit.lesson>Using quadratic functions to solve problems on maximizing revenue/profit</A>\r<BR>\n" );
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document.write( "Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lessons are the part of this textbook under the topic \"<U>Finding minimum/maximum of quadratic functions</U>\". \r<BR>\n" );
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document.write( "Save the link to this online textbook together with its description\r<BR>\n" );
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document.write( "Free of charge online textbook in ALGEBRA-I<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r<BR>\n" );
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document.write( "to your archive and use it when it is needed.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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