Respuesta :
Answer:
It would take 24 minutes for the element to decay to 50 grams
Step-by-step explanation:
The equation for the amount of the element present, after t minutes, is:
[tex]Q(t) = Q(0)e^{-rt}[/tex]
In which Q(X) decays radioactively with a half life of 12 minutes.(0) is the initial amount and r is the rate it decreases.
Half life of 12 minutes
This means that [tex]Q(12) = 0.5Q(0)[/tex]
So
[tex]Q(t) = Q(0)e^{-rt}[/tex]
[tex]0.5Q(0) = Q(0)e^{-12r}[/tex]
[tex]e^{-12r} = 0.5[/tex]
[tex]\ln{e^{-12r}} = \ln{0.5}[/tex]
[tex]-12r = \ln{0.5}[/tex]
[tex]12r = -\ln{0.5}[/tex]
[tex]r = -\frac{\ln{0.5}}{12}[/tex]
[tex]r = 0.05776[/tex]
If there are 200 grams of Element X, how long, to the nearest tenth of a minute, would it take the element to decay to 50 grams?
This is t when Q(t) = 50. Q(0) = 200.
[tex]Q(t) = Q(0)e^{-rt}[/tex]
[tex]50 = 200e^{-0.05776t}[/tex]
[tex]e^{-0.05776t} = 0.25[/tex]
[tex]\ln{e^{-0.05776t}} = \ln{0.25}[/tex]
[tex]-0.05776t = \ln{0.25}[/tex]
[tex]0.05776t = -\ln{0.25}[/tex]
[tex]t = -\frac{\ln{0.25}}{0.05776}[/tex]
[tex]t = 24[/tex]
It would take 24 minutes for the element to decay to 50 grams
