document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176522>Question 1176522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1.  You sell peanuts at UNK athletic events to make some extra money.  When peanuts are sold for $1.00 per bag, approximately 600 bags are sold at each event.  You tried to raise the price to $1.25 and found that the quantity demanded dropped to 560 bags.  The startup cost was $500 and the cost to you per bag of peanuts is $0.50. \r<BR>\n" );
document.write( "\n" );
document.write( "A.   Assume that the demand function is linear and write a function that models the profit from selling x bags of peanuts at an event.\r<BR>\n" );
document.write( "\n" );
document.write( "B.   What number of bags and what price per bag will get you a maximum profit?\r<BR>\n" );
document.write( "\n" );
document.write( "C.   Does your answer to part B agree with what you \"thought\" it would be?  Why or why not?\r<BR>\n" );
document.write( "\n" );
document.write( "Even telling me what kind of question this is so I can find other examples would be helpful, thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850572 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850572\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
document.write( "\n" );
document.write( "**A. Model the Profit Function**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Find the Demand Function (p(x))**\r<BR>\n" );
document.write( "\n" );
document.write( "    * We have two points: (600, 1.00) and (560, 1.25)<BR>\n" );
document.write( "    * Find the slope (m):<BR>\n" );
document.write( "        * m = (1.25 - 1.00) / (560 - 600) = 0.25 / -40 = -0.00625<BR>\n" );
document.write( "    * Use point-slope form (y - y1 = m(x - x1)):<BR>\n" );
document.write( "        * p - 1.00 = -0.00625(x - 600)<BR>\n" );
document.write( "        * p = -0.00625x + 3.75 + 1.00<BR>\n" );
document.write( "        * p(x) = -0.00625x + 4.75\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Find the Revenue Function (R(x))**\r<BR>\n" );
document.write( "\n" );
document.write( "    * Revenue = price * quantity<BR>\n" );
document.write( "    * R(x) = x * p(x)<BR>\n" );
document.write( "    * R(x) = x(-0.00625x + 4.75)<BR>\n" );
document.write( "    * R(x) = -0.00625x² + 4.75x\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Find the Cost Function (C(x))**\r<BR>\n" );
document.write( "\n" );
document.write( "    * Cost = startup cost + cost per bag * quantity<BR>\n" );
document.write( "    * C(x) = 500 + 0.50x\r<BR>\n" );
document.write( "\n" );
document.write( "4.  **Find the Profit Function (P(x))**\r<BR>\n" );
document.write( "\n" );
document.write( "    * Profit = Revenue - Cost<BR>\n" );
document.write( "    * P(x) = R(x) - C(x)<BR>\n" );
document.write( "    * P(x) = (-0.00625x² + 4.75x) - (500 + 0.50x)<BR>\n" );
document.write( "    * P(x) = -0.00625x² + 4.25x - 500\r<BR>\n" );
document.write( "\n" );
document.write( "**B. Maximize Profit**\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **Find the Vertex of the Profit Function**\r<BR>\n" );
document.write( "\n" );
document.write( "    * The profit function is a quadratic, so its maximum occurs at the vertex.<BR>\n" );
document.write( "    * The x-coordinate of the vertex is given by x = -b / 2a, where a = -0.00625 and b = 4.25.<BR>\n" );
document.write( "    * x = -4.25 / (2 * -0.00625)<BR>\n" );
document.write( "    * x = -4.25 / -0.0125<BR>\n" );
document.write( "    * x = 340\r<BR>\n" );
document.write( "\n" );
document.write( "2.  **Find the Price per Bag**\r<BR>\n" );
document.write( "\n" );
document.write( "    * p(x) = -0.00625x + 4.75<BR>\n" );
document.write( "    * p(340) = -0.00625(340) + 4.75<BR>\n" );
document.write( "    * p(340) = -2.125 + 4.75<BR>\n" );
document.write( "    * p(340) = 2.625\r<BR>\n" );
document.write( "\n" );
document.write( "3.  **Find the Maximum Profit**\r<BR>\n" );
document.write( "\n" );
document.write( "    * P(x) = -0.00625x² + 4.25x - 500<BR>\n" );
document.write( "    * P(340) = -0.00625(340)² + 4.25(340) - 500<BR>\n" );
document.write( "    * P(340) = -0.00625(115600) + 1445 - 500<BR>\n" );
document.write( "    * P(340) = -722.5 + 1445 - 500<BR>\n" );
document.write( "    * P(340) = 222.5\r<BR>\n" );
document.write( "\n" );
document.write( "**Answers for B:**\r<BR>\n" );
document.write( "\n" );
document.write( "* Number of bags: 340<BR>\n" );
document.write( "* Price per bag: $2.625 (or $2.63)\r<BR>\n" );
document.write( "\n" );
document.write( "**C. Agreement with Intuition**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Initial Intuition:** You might have thought that raising the price would always decrease the number of bags sold and potentially lower the profit.<BR>\n" );
document.write( "* **Actual Result:** The analysis shows that there's an optimal price point that maximizes profit. In this case, raising the price significantly above the initial $1.00 leads to a lower quantity sold but a higher profit.<BR>\n" );
document.write( "* **Explanation:** This is because the higher price per bag more than compensates for the reduced quantity sold, up to a certain point. The profit function is a parabola, and the vertex represents the optimal balance between price and quantity.\r<BR>\n" );
document.write( "\n" );
document.write( "**Why the \"thought\" might not be what the answer is:**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Linear Demand Assumption:** The linear demand function is a simplification. Real-world demand might not be perfectly linear.<BR>\n" );
document.write( "* **Cost Structure:** The constant cost per bag and fixed startup cost simplify the model. Real costs might be more complex.<BR>\n" );
document.write( "* **Consumer Behavior:** The model assumes rational consumer behavior. In reality, factors like brand loyalty, perceived value, and competitor pricing can influence demand.\r<BR>\n" );
document.write( "\n" );
document.write( "In conclusion, the mathematical analysis provides a more precise and optimal solution than relying on initial intuition alone.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );