Page 28 - Now, divide by 5730 to solve for k. k = ln (1/2)...

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Huynh, Kazyak, Rees, Ufner Page 28, #46 The information given to us states that Vermeer supposedly painted a picture between 300 and 400 years ago. It contains 99.5% of its carbon-14, and its half-life is 5,730 years. The question asks us to determine whether the picture is real or not, and to explain our reasoning. P(t) = P o e kt We use this equation because the exponential decay is continuous. ‘T’ represents the amount of time (in years) that the picture has decayed. Our goal is to solve for ‘k’. ½ P o = P o e 5730k We do not need to find the value of P o since they cancel out. ½ = e 5730k To bring 5730k down, and get rid of e, we take the ln of both sides. ln (1/2) = 5730k
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Unformatted text preview: Now, divide by 5730 to solve for k. k = ln (1/2) . /. 5730 = -1.21e-4 Now that we know the value of k, we insert 99.5% into the equation to find out how many years it takes for the pictures carbon-14 to decay 99.5%. .995 = e (-1.21e-4)t Again, we need to take the ln of both sides to get ride of e. ln (.995) = (-1.21e-4 )t Divide by -1.21e-4 to solve for t, time in years. t = ln (.995) . /. -1.21e-4 = 41.426 We concluded that the picture is fake. We know this because it only takes 41.426 years for the carbon-14 to decay to 99.5%, but they claim it is over 300 years old. If it were truly over 300 years old, there would be much less carbon-14 present....
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