Respuesta :
Using an exponential function, it is found that 242.5 mg of radium will be left after the 40 days.
What is an exponential function?
The exponential function for the amount of a decaying substance is modeled by:
[tex]A(t) = A(0)e^{-kt}[/tex]
In which:
- A(0) is the initial amount.
- k is the decay rate, as a decimal.
In this problem, the half-life is of 100 days, hence:
[tex]A(100) = 0.5A(0)[/tex]
Which is used to find k.
[tex]A(t) = A(0)e^{-kt}[/tex]
[tex]0.5A(0) = A(0)e^{-100k}[/tex]
[tex]e^{-100k} = 0.5[/tex]
[tex]\ln{e^{-100k}} = \ln{0.5}[/tex]
[tex]-100k = \ln{0.5}[/tex]
[tex]k = -\frac{\ln{0.5}}{100}[/tex]
[tex]k = 0.0069314718[/tex]
Then, the equation is:
[tex]A(t) = A(0)e^{-0.0069314718t}[/tex]
He started with 320 mg of radium, hence [tex]A(0) = 320[/tex], and the equation is:
[tex]A(t) = 320e^{-0.0069314718t}[/tex]
After 40 days, the amount left is:
[tex]A(40) = 320e^{-0.0069314718(40)} = 242.5[/tex]
242.5 mg of radium will be left after the 40 days.
You can learn more about exponential functions at https://brainly.com/question/25537936
