document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1202240>Question 1202240</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The volume of water in a tank at a particular time (measured in<BR>\n" );
document.write( "seconds) is given by V (t) = 5t(2 − t) m3<BR>\n" );
document.write( ". Find the rate of change of the<BR>\n" );
document.write( "volume of water in the tank from first principles (using the definition of the rate<BR>\n" );
document.write( "of change).<BR>\n" );
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document.write( "The term first principles in this context refers to the limit definition of the derivative.\r<BR>\n" );
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document.write( "Let's first determine V(t+h).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V(t) = 5t(2-t)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V(t) = 10t-5t^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V(t+h) = 10(t+h)-5(t+h)^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V(t+h) = 10(t+h)-5(t^2+2th+h^2)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V(t+h) = 10t+10h-5t^2-10th-5h^2\">\r<BR>\n" );
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document.write( "Then we can compute the derivative to get the instantaneous rate of change.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large V'(t) = \frac{dV}{dt}= \text{instantaneous rate of change of the volume}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle V'(t) = \lim_{h \to 0} \frac{V(t+h)-V(t)}{h}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle V'(t) = \lim_{h \to 0} \frac{(10t+10h-5t^2-10th-5h^2)-(10t-5t^2)}{h}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle V'(t) = \lim_{h \to 0} \frac{10t+10h-5t^2-10th-5h^2-10t+5t^2}{h}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle V'(t) = \lim_{h \to 0} \frac{10h-10th-5h^2}{h}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle V'(t) = \lim_{h \to 0} \frac{h(10-10t-5h)}{h}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle V'(t) = \lim_{h \to 0} (10-10t-5h)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle V'(t) = 10-10t-5*0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \displaystyle V'(t) = 10-10t\">\r<BR>\n" );
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document.write( "The instantaneous rate of change of the volume is V'(t) = 10-10t<BR>\n" );
document.write( "The units for the rate of change is \"cubic meters per second\".\r<BR>\n" );
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document.write( "Extra info: <BR>\n" );
document.write( "1 cubic meter = 1000 liters = 1 kiloliter<BR>\n" );
document.write( "1 meter = 3.2808 feet approximately<BR>\n" );
document.write( "1 cubic meter = (3.2808)^3 = 35.313 cubic feet approximately<BR>\n" );
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