document.write( "Question 1190364: Carbon-14 is used to determine the age of artifacts. It has a half-life of 5700 years. This means that
\n" ); document.write( "in 0 years 100%, of the original amount of carbon-14 remains in the artifact. In 5700 years 50% of
\n" ); document.write( "the original amount of carbon-14 remains.
\n" ); document.write( "a) Suppose that 40% of the original carbon remains in a piece of pottery.
\n" ); document.write( "Estimate the age of the pottery.
\n" ); document.write( "b) Suppose that 75% of the original carbon remains in a piece of pottery.
\n" ); document.write( "Estimate the age of the pottery.
\n" ); document.write( "c) Suppose that 2% of the original carbon remains in a piece of pottery.
\n" ); document.write( "Estimate the age of the pottery.
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Algebra.Com's Answer #822005 by ikleyn(52803)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "Carbon-14 is used to determine the age of artifacts. It has a half-life of 5700 years. This means that
\n" ); document.write( "in 0 years 100%, of the original amount of carbon-14 remains in the artifact. In 5700 years 50% of
\n" ); document.write( "the original amount of carbon-14 remains.
\n" ); document.write( "a) Suppose that 40% of the original carbon remains in a piece of pottery.
\n" ); document.write( "Estimate the age of the pottery.
\n" ); document.write( "b) Suppose that 75% of the original carbon remains in a piece of pottery.
\n" ); document.write( "Estimate the age of the pottery.
\n" ); document.write( "c) Suppose that 2% of the original carbon remains in a piece of pottery.
\n" ); document.write( "Estimate the age of the pottery.
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document.write( "You are given that half-life of C-14 is 5700 years.\r\n" );
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document.write( "So, you can write the decay function in the form\r\n" );
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document.write( "    m(t) = \"m%280%29%2A%281%2F2%29%5E%28t%2F5700%29\",    (1)\r\n" );
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document.write( "where m(0) is the original mass, m(t) is the current mass and t is the time, in years.\r\n" );
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document.write( "To answer all questions that follow, divide both sides of the decay equation (1) by m(0). You will get\r\n" );
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document.write( "    \"m%28t%29%2Fm%280%29\" = \"%281%2F2%29%5E%28t%2F5700%29\".        (2)\r\n" );
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document.write( "Now let's answer the questions, one after another.\r\n" );
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document.write( "(a)  You are given \"m%28t%29%2Fm%280%29\" = 40% = 0.4.\r\n" );
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document.write( "     So, equation (2) gives you\r\n" );
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document.write( "         \"%281%2F2%29%5E%28t%2F5700%29\" = 0.4.\r\n" );
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document.write( "     Take logarithm base 10 of both sides\r\n" );
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document.write( "         \"%28t%2F5700%29%2Alog%28%280.5%29%29\" = log(0.4)\r\n" );
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document.write( "         t = \"%285700%2Alog%28%280.4%29%29%29%2Flog%28%280.5%29%29\" = 7535  years.    ANSWER\r\n" );
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document.write( "(b)  You are given \"m%28t%29%2Fm%280%29\" = 75% = 0.75.\r\n" );
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document.write( "     So, equation (2) gives you\r\n" );
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document.write( "         \"%281%2F2%29%5E%28t%2F5700%29\" = 0.75.\r\n" );
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document.write( "     Take logarithm base 10 of both sides\r\n" );
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document.write( "         \"%28t%2F5700%29%2Alog%28%280.5%29%29\" = log(0.75)\r\n" );
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document.write( "         t = \"%285700%2Alog%28%280.75%29%29%29%2Flog%28%280.5%29%29\" = 2366  years.    ANSWER\r\n" );
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document.write( "(c)  You are given \"m%28t%29%2Fm%280%29\" = 2% = 0.02.\r\n" );
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document.write( "     So, equation (2) gives you\r\n" );
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document.write( "         \"%281%2F2%29%5E%28t%2F5700%29\" = 0.02.\r\n" );
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document.write( "     Take logarithm base 10 of both sides\r\n" );
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document.write( "         \"%28t%2F5700%29%2Alog%28%280.5%29%29\" = log(0.02)\r\n" );
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document.write( "         t = \"%285700%2Alog%28%280.02%29%29%29%2Flog%28%280.5%29%29\" = 32170  years.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "To solve the problem, you do not need make tons of unnecessary calculations, that @Theo makes in his post.\r
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\n" ); document.write( "\n" ); document.write( "When you know the half-life, the way which I show you is the standard, shortest and straightforward way to solve the problem.\r
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\n" ); document.write( "\n" ); document.write( "There is a group of lessons in this site, which covers many similar problems on Carbon dating\r
\n" ); document.write( "\n" ); document.write( "    - Using logarithms to solve real world problems,  and\r
\n" ); document.write( "\n" ); document.write( "    - Radioactive decay problems \r
\n" ); document.write( "\n" ); document.write( "    - Carbon dating problems \r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Logarithms\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this online textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-I
\n" ); document.write( "https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson\r
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