document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191404>Question 1191404</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Profit per tree grown and sold by a tree grower depends upon the height of a tree at the<BR>\n" );
document.write( "time of sale. Taking ha as a tree height in metres, the profit per tree in thousands shillings<BR>\n" );
document.write( "is approximated by : P(h) = (10+2h) - 0.1h<BR>\n" );
document.write( "(a) What tree height provides maximum profit per tree?<BR>\n" );
document.write( "(b) What is the maximum profit per tree?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849162 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=849162\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the tree height that maximizes profit and the maximum profit:\r<BR>\n" );
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document.write( "**(a) Finding the Tree Height for Maximum Profit:**\r<BR>\n" );
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document.write( "1. **Recognize the Function:** The profit function P(h) = (10 + 2h) - 0.1h² is a quadratic function in the form of P(h) = a + bh - ch², where a=10, b=2 and c=0.1. Since the coefficient of the h² term (-0.1) is negative, the parabola opens downward, meaning there's a maximum point.\r<BR>\n" );
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document.write( "2. **Find the Vertex:** The vertex of a parabola gives the maximum (or minimum) value of the function. The h-coordinate of the vertex is given by:\r<BR>\n" );
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document.write( "   h = -b / (2a)\r<BR>\n" );
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document.write( "   In our case, a= -0.1 and b=2\r<BR>\n" );
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document.write( "   h = -2 / (2 * -0.1)<BR>\n" );
document.write( "   h = -2 / -0.2<BR>\n" );
document.write( "   h = 10\r<BR>\n" );
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document.write( "Therefore, a tree height of 10 meters provides the maximum profit.\r<BR>\n" );
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document.write( "**(b) Calculating the Maximum Profit:**\r<BR>\n" );
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document.write( "Substitute the value of h (10 meters) back into the profit function P(h):\r<BR>\n" );
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document.write( "P(10) = (10 + 2 * 10) - 0.1 * 10²<BR>\n" );
document.write( "P(10) = (10 + 20) - 0.1 * 100<BR>\n" );
document.write( "P(10) = 30 - 10<BR>\n" );
document.write( "P(10) = 20\r<BR>\n" );
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document.write( "Since the profit is in thousands of shillings, the maximum profit per tree is 20,000 shillings.\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "(a) A tree height of 10 meters provides the maximum profit per tree.<BR>\n" );
document.write( "(b) The maximum profit per tree is 20,000 shillings.<BR>\n" );
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