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Element X decays radioactively with a half life of 9 minutes. If there are 960 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 40 grams?

y=a(.5)^t/h




User Sonlexqt
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1 Answer

5 votes

Answer:

It will take 41.3 minutes for the element to decay to 40 grams

Explanation:

The amount of element after t minute is given by the following equation:


x(t) = x(0)e^(-rt)

In which x(0) is the initial amount and r is the rate that it decreases.

Element X decays radioactively with a half life of 9 minutes.

This means that
x(9) = 0.5x(0). We use this to find r. So


x(t) = x(0)e^(-rt)


0.5x(0) = x(0)e^(-9r)


e^(-9r) = 0.5


\ln{e^(-9r)} = ln(0.5)


-9r = ln(0.5)


9r = -ln(0.5)


r = -(ln(0.5))/(9)


r = 0.077

So


x(t) = x(0)e^(-0.077t)

There are 960 grams of Element X

This means that
x(0) = 960


x(t) = 960e^(-0.077t)

How long, to the nearest tenth of a minute, would it take the element to decay to 40 grams?

This is t when
x(t) = 40. So


x(t) = 960e^(-0.077t)


40 = 960e^(-0.077t)


e^(-0.077t) = (40)/(960)


\ln{e^(-0.077t)} = \ln{(40)/(960)}


-0.077t = \ln{(40)/(960)}


0.077t = -\ln{(40)/(960)}


t = -\frac{\ln{(40)/(960)}}{0.077}


t = 41.3

It will take 41.3 minutes for the element to decay to 40 grams

User Pyae Phyo Aung
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