document.write( "Question 268809: if the U.S grows at an annual rate of 7.5%, how long will take to reach a population of 400 000 000?(the population, now, is 300 000 000) \n" ); document.write( "
Algebra.Com's Answer #196977 by jsmallt9(3758)\"\" \"About 
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(NOTE: There is an error in the solution provided by another tutor.)

\n" ); document.write( "This problem tells us that the population grows by 7.5% each year. After one year the population will be 7.5% bigger. After a second year the population will be 7.5% bigger than the population after the first year, which includes the 7.5% increase. So this is like compound interest where interest is earned on previously earned interest. (This is the error in the other solution. It treats the problem like simple interest.) So we will use an equation similar to the one used for compund interest:
\n" ); document.write( "\"P%28t%29+=+P%5B0%5D%281+%2B+r%29%5Et\" where \"P%5B0%5D\" is the initial population and r is the rate of increase. This makes the equation for this problem:
\n" ); document.write( "\"P%28t%29+=+300000000%281+%2B+0.075%29%5Et\"
\n" ); document.write( "or
\n" ); document.write( "\"P%28t%29+=+300000000%281.075%29%5Et\"
\n" ); document.write( "To find when the population will be 400,000,000:
\n" ); document.write( "\"400000000+=+300000000%281.075%29%5Et\"
\n" ); document.write( "Now we solve for t. Divide both sides by 300,000,000:
\n" ); document.write( "\"4%2F3+=+%281.075%29%5Et\"
\n" ); document.write( "Find the logarithm of each side. Use a base that your calculator \"knows\" (usually base 10 or base e (ln)). I'll use base 10:
\n" ); document.write( "\"log%28%284%2F3%29%29+=+log%28%28%281.075%29%5Et%29%29\"
\n" ); document.write( "Now we use a property of logarithms, \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\", to move the exponent out in front. (This gets the variable out of the exponent and this property of logarithms is the reason we use logarithms on this equation.) Using this property on our equation gives us:
\n" ); document.write( "\"log%28%284%2F3%29%29+=+t%2Alog%28%281.075%29%29\"
\n" ); document.write( "Divide both sides by \"log%28%281.075%29%29\":
\n" ); document.write( "\"log%28%284%2F3%29%29%2Flog%28%281.075%29%29+=+t\"
\n" ); document.write( "This is an exact answer. For a decimal approximation we get out our calculators:
\n" ); document.write( "\"log%28%281.3333333333333%29%29%2Flog%28%281.075%29%29+=+t\"
\n" ); document.write( "\"0.1249387366083000%2F0.0314084642516241+=+t\"
\n" ); document.write( "\"3.9778683735496349+=+t\"
\n" ); document.write( "So it will take almost 4 years for the population to reach 400,000,000. \n" ); document.write( "
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