document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=596252>Question 596252</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The radius of a circle as a function of time is defined by the equation, r(t)=4t^2+3t+1. Determine the rate of change in the area of the circle when dr/dt=11. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #377655 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=377655\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{dr}{dt} = 8t + 3 = 11 \Rightarrow t = 1\"> so when dr/dt = 11, t = 1.\r<BR>\n" );
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document.write( "The area of the circle as a function of time is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A = \pi (r(t))^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE A = \pi (4t^2 + 3t + 1)^2\">\r<BR>\n" );
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document.write( "We differentiate both sides with respect to t:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{dA}{dt} = 2\pi (4t^2 + 3t + 1)(8t + 3)\">\r<BR>\n" );
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document.write( "Replace t = 1 to obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{dA}{dt} = 2\pi (8)(11) = 176 \pi\"> (units squared per unit of time) </SPAN>\n" );
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