Mathematics: Element X decays radioactively with a half life of 12 minutes. If there are 760 grams of element X, how long, to the nearest tenth a minute, would it take the element to decay to 15 grams?

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Element X decays radioactively with a half life

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kayciemarie9486

28.01.2022

Mathematics

The time it'd take for the element to have 15 g of mass is approximately 68 min.

Step-by-step explanation:

The radioactive decay of a substance is given by the following formula:

$mass(t) = mass(0)*e^{-\lambda*t}$

Since the element has a half life of 12 minutes, this means that after this time the mass of the element will be half of it was originally, therefore:

$\frac{mass(0)}{2} = mass(0)*e^{-\lambda*12}$

$\frac{1}{2} = e^{-\lambda*12}$

$ln(\frac{1}{2}) = -12*\lambda\\\lambda = -\frac{ln(0.5)}{12} =0.0577623$

Therefore the mass of the element is given by:

$mass(t) = mass(0)*e^{-0.0577623*t}$

If the initial mass is 760 g and the final mass is 15 g, we have:

$mass(t) = mass(0)*e^{-0.0577623*t}\\\\15 = 760*e^{-0.0577623*t}\\\\e^{-0.0577623*t} = \frac{15}{760}\\\\ln(e^{-0.0577623*t}) = ln(\frac{15}{760})\\\\-0.0577623*t = ln(\frac{15}{760})\\\\t = \frac{ln(\frac{15}{760})}{-0.0577623}\\\\t = 67.9555$

The time it'd take for the element to have 15 g of mass is approximately 68 min.

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Element X decays radioactively with a half life of 12 minutes. If there are 760 grams of element X, how long, to the nearest tenth a minute, would it take the element to decay to 15 grams?

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