Respuesta :
Answer:
It will take 41.3 minutes for the element to decay to 40 grams
Step-by-step explanation:
The amount of element after t minute is given by the following equation:
[tex]x(t) = x(0)e^{-rt}[/tex]
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 [tex]x(9) = 0.5x(0)[/tex]. We use this to find r. So
[tex]x(t) = x(0)e^{-rt}[/tex]
[tex]0.5x(0) = x(0)e^{-9r}[/tex]
[tex]e^{-9r} = 0.5[/tex]
[tex]\ln{e^{-9r}} = \ln{0.5}[/tex]
[tex]-9r = \ln{0.5}[/tex]
[tex]9r = -\ln{0.5}[/tex]
[tex]r = -\frac{\ln{0.5}}{9}[/tex]
[tex]r = 0.077[/tex]
So
[tex]x(t) = x(0)e^{-0.077t}[/tex]
There are 960 grams of Element X
This means that [tex]x(0) = 960[/tex]
[tex]x(t) = 960e^{-0.077t}[/tex]
How long, to the nearest tenth of a minute, would it take the element to decay to 40 grams?
This is t when [tex]x(t) = 40[/tex]. So
[tex]x(t) = 960e^{-0.077t}[/tex]
[tex]40 = 960e^{-0.077t}[/tex]
[tex]e^{-0.077t} = \frac{40}{960}[/tex]
[tex]\ln{e^{-0.077t}} = \ln{\frac{40}{960}}[/tex]
[tex]-0.077t = \ln{\frac{40}{960}}[/tex]
[tex]0.077t = -\ln{\frac{40}{960}}[/tex]
[tex]t = -\frac{\ln{\frac{40}{960}}}{0.077}[/tex]
[tex]t = 41.3[/tex]
It will take 41.3 minutes for the element to decay to 40 grams
