document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170843>Question 1170843</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Some biologists model the number of species S in a fixed area A (such as an island) by the species-area relationship<BR>\n" );
document.write( "log(S) = log(c) + k log(A)<BR>\n" );
document.write( "where c and k are positive constants that depend on the type of species and habitat.\r<BR>\n" );
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document.write( "(a) Solve the equation for S.<BR>\n" );
document.write( "S =	<BR>\n" );
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document.write( "(b) Using part (a), if k = 2 and the area is doubled, then by what magnitude is the number of species increased? </SPAN>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "We'll use these log rules<BR>\n" );
document.write( "log(A)+log(B) = log(A*B)<BR>\n" );
document.write( "B*log(A) = log(A^B)<BR>\n" );
document.write( "which we'll refer to as rule 1 and rule 2 respectively.\r<BR>\n" );
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document.write( "log(S) = log(c) + k*log(A)<BR>\n" );
document.write( "log(S) = log(c) + log(A^k) .... use rule 2<BR>\n" );
document.write( "log(S) = log(c*A^k) .... use rule 1\r<BR>\n" );
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document.write( "Since the log on the left equals the log on the right, this means the arguments (ie the stuff inside) for each log function must be equal.\r<BR>\n" );
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document.write( "In other words, log(A) = log(B) implies that A = B.\r<BR>\n" );
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document.write( "The equation<BR>\n" );
document.write( "log(S) = log(c*A^k)<BR>\n" );
document.write( "turns into<BR>\n" );
document.write( "S = c*A^k\r<BR>\n" );
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document.write( "Answer:  S = c*A^k\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "The area is doubled. So A turns into 2A.\r<BR>\n" );
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document.write( "We'll replace A with 2A like so<BR>\n" );
document.write( "S = c*A^k<BR>\n" );
document.write( "S = c*(2A)^k\r<BR>\n" );
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document.write( "Now plug in k = 2<BR>\n" );
document.write( "S = c*(2A)^k<BR>\n" );
document.write( "S = c*(2A)^2<BR>\n" );
document.write( "S = c*(4A^2)\r<BR>\n" );
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document.write( "Plug k = 2 into the original equation<BR>\n" );
document.write( "S = c*A^k<BR>\n" );
document.write( "S = c*A^2\r<BR>\n" );
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document.write( "Divide the two results<BR>\n" );
document.write( "[ c*(4A^2) ] / [ c*A^2 ]<BR>\n" );
document.write( "(4A^2)/(A^2)<BR>\n" );
document.write( "4<BR>\n" );
document.write( "The 'c's cancel and the 'A^2's cancel.\r<BR>\n" );
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document.write( "For k = 2, if we double the area from A to 2A, then the number of species has increased by a factor of 4. It turns out this works for any value of k as well. \r<BR>\n" );
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document.write( "Answer: 4\r<BR>\n" );
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document.write( "</font>\r<BR>\n" );
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document.write( " </SPAN>\n" );
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