document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171702>Question 1171702</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The profit P(x) that company earn for selling x number of toy cars can be modelled by<BR>\n" );
document.write( "P(x) =25x^2 +1000 -3000.\r<BR>\n" );
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document.write( "1. Write an inequality that models the problem if the profit P has to be at least php 5000\r<BR>\n" );
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document.write( "2. what could be the range of possible values of x\r<BR>\n" );
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document.write( "3. how many toy cars must be sold for a profit of at least php 5000\r<BR>\n" );
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document.write( "4. how many toy cars must be sold to obtain the maximum profit\r<BR>\n" );
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document.write( "5. how much is the maximum profit<BR>\n" );
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 #850877 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=850877\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**Given Profit Function:**\r<BR>\n" );
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document.write( "* P(x) = 25x² + 1000x - 3000\r<BR>\n" );
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document.write( "**1. Inequality for Profit at Least PHP 5000:**\r<BR>\n" );
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document.write( "* P(x) ≥ 5000<BR>\n" );
document.write( "* 25x² + 1000x - 3000 ≥ 5000<BR>\n" );
document.write( "* 25x² + 1000x - 8000 ≥ 0\r<BR>\n" );
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document.write( "**2. Range of Possible Values of x:**\r<BR>\n" );
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document.write( "* To find the range, we need to solve the inequality 25x² + 1000x - 8000 ≥ 0.<BR>\n" );
document.write( "* First, simplify by dividing by 25: x² + 40x - 320 ≥ 0<BR>\n" );
document.write( "* Find the roots of the quadratic equation x² + 40x - 320 = 0 using the quadratic formula:<BR>\n" );
document.write( "    * x = [-b ± √(b² - 4ac)] / 2a<BR>\n" );
document.write( "    * x = [-40 ± √(40² - 4(1)(-320))] / 2(1)<BR>\n" );
document.write( "    * x = [-40 ± √(1600 + 1280)] / 2<BR>\n" );
document.write( "    * x = [-40 ± √2880] / 2<BR>\n" );
document.write( "    * x = [-40 ± 24√5] / 2<BR>\n" );
document.write( "    * x = -20 ± 12√5<BR>\n" );
document.write( "* Approximate the roots:<BR>\n" );
document.write( "    * x₁ = -20 - 12√5 ≈ -46.83<BR>\n" );
document.write( "    * x₂ = -20 + 12√5 ≈ 6.83<BR>\n" );
document.write( "* Since x represents the number of toy cars sold, it must be non-negative. Therefore, we only consider the positive root.<BR>\n" );
document.write( "* The inequality x² + 40x - 320 ≥ 0 is satisfied when x ≤ -46.83 or x ≥ 6.83.<BR>\n" );
document.write( "* Since x must be non-negative, the range of possible values for x is x ≥ 6.83.<BR>\n" );
document.write( "* However, since we are dealing with toy cars, x must be a whole number.<BR>\n" );
document.write( "* Therefore the range of x is x ≥ 7.\r<BR>\n" );
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document.write( "**3. Number of Toy Cars for Profit of at Least PHP 5000:**\r<BR>\n" );
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document.write( "* Since x ≥ 6.83, the company must sell at least 7 toy cars to achieve a profit of at least PHP 5000.\r<BR>\n" );
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document.write( "**4. Number of Toy Cars for Maximum Profit:**\r<BR>\n" );
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document.write( "* To find the maximum profit, we need to find the vertex of the parabola P(x) = 25x² + 1000x - 3000.<BR>\n" );
document.write( "* The x-coordinate of the vertex is given by x = -b / 2a, where a = 25 and b = 1000.<BR>\n" );
document.write( "* x = -1000 / (2 * 25) = -1000 / 50 = -20<BR>\n" );
document.write( "* Since the coefficient of x² is positive (25), the parabola opens upwards, meaning it has a minimum, not a maximum. There must be an error with the problem. I will assume the coefficent of x² is negative.<BR>\n" );
document.write( "* Let us assume the profit function is P(x) = -25x² + 1000x - 3000.<BR>\n" );
document.write( "* x = -1000 / (2 * -25) = -1000 / -50 = 20<BR>\n" );
document.write( "* Therefore, 20 toy cars must be sold to obtain the maximum profit.\r<BR>\n" );
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document.write( "**5. Maximum Profit:**\r<BR>\n" );
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document.write( "* Using the corrected profit function P(x) = -25x² + 1000x - 3000, substitute x = 20:<BR>\n" );
document.write( "* P(20) = -25(20)² + 1000(20) - 3000<BR>\n" );
document.write( "* P(20) = -25(400) + 20000 - 3000<BR>\n" );
document.write( "* P(20) = -10000 + 20000 - 3000<BR>\n" );
document.write( "* P(20) = 7000\r<BR>\n" );
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document.write( "* The maximum profit is PHP 7000.\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "1.  **Inequality:** 25x² + 1000x - 8000 ≥ 0<BR>\n" );
document.write( "2.  **Range of x:** x ≥ 7<BR>\n" );
document.write( "3.  **Toy Cars for Profit ≥ PHP 5000:** 7<BR>\n" );
document.write( "4.  **Toy Cars for Maximum Profit (corrected function):** 20<BR>\n" );
document.write( "5.  **Maximum Profit (corrected function):** PHP 7000<BR>\n" );
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