7.5.18-eg-1

# 7.5.18-eg-1 - ² = 80 k and so k = ln(1 2 80 =-ln 2 80 2...

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Example A radioactive substance has a half-life of 80 days. How long will it take for 15 grams of the substance to decay to 10 grams? We use the exponential decay model y ( t ) = y 0 e kt where y is the amount of the substance (in grams) at time t (in days), y 0 is the initial amount of the substance and k is the decay constant. 1. Find the decay constant k . Since the half-life of the substance is 80 days, we have 1 2 y 0 = y 0 e 80 k ln ± 1 2
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Unformatted text preview: ² = 80 k, and so k = ln(1 / 2) 80 =-ln 2 80 . 2. Use the model to determine the answer. Using the exact value of k throughout the remainder of the problem, the model becomes y ( t ) = y e-ln 2 80 t . Given y = 15, our goal is to ﬁnd t when y ( t ) = 10. Thus 10 = 15 e-ln 2 80 t ln ± 2 3 ² =-ln 2 80 t and so t = ln(2 / 3)-ln2 80 ≈ 46 . 797 days...
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