Answer:
Step-by-step explanation:
The exponential equation for the fraction remaining after x years can be written as ...
y = (1/2)^(x/30)
A) For x=1, the fraction remaining is ...
y = (1/2)^(1/30) ≈ 0.97716 = 1 - 0.0228
Of the original amount, 0.0228 decays each year.
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B) The continuous decay rate is the natural log of the growth factor, so is ...
ln(0.97716) = -0.0231
The continuous decay rate is 0.0231 of the present amount (per year).
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C) For y=.10 (1/10 of the original amount) we find x to be ...
.1 = .5^(x/30)
ln(.1) = (x/30)ln(.5) . . . . . take the natural log
30ln(0.1)/ln(0.5) = x ≈ 100 . . . years
It will take 100 years for a 10-gram sample to decay to 1 gram.
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D) As x goes to infinity, y goes to zero.
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The relationship between growth rate and growth factor is ...
growth factor = 1 + growth rate
When the growth rate is negative, it is called a decay rate.
