Respuesta :
Using an exponential function, it is found that there will be only 3 milligrams remaining in the patient's system after 201 minutes.
What is an exponential function?
A decaying exponential function is modeled by:
[tex]A(t) = A(0)(1 - r)^t[/tex]
In which:
- A(0) is the initial value.
- r is the decay rate, as a decimal.
In this problem, initially, there are 11 milligrams on the patient's system, hence A(0) = 11. After 70 minutes there are 7 milligrams, hence A(70) = 7, and this is used to find r.
[tex]A(t) = A(0)(1 - r)^t[/tex]
[tex]7 = 11(1 - r)^{70}[/tex]
[tex](1 - r)^{70} = \frac{7}{11}[/tex]
[tex]\sqrt[70]{(1 - r)^{70}} = \sqrt[70]{\frac{7}{11}}[/tex]
[tex]1 - r = \left(\frac{7}{11}\right)^\frac{1}{70}[/tex]
1 - r = 0.99356387084
r = 1 - 0.99356387084
r = 0.00643612916
Hence the equation for the amount after t minutes is:
[tex]A(t) = 11(0.99356387084)^t[/tex]
In will be of 3 mg when A(t) = 3, hence:
[tex]A(t) = 11(0.99356387084)^t[/tex]
[tex]3 = 11(0.99356387084)^t[/tex]
[tex](0.99356387084)^t = \frac{3}{11}[/tex]
[tex]\log{(0.99356387084)^t} = \log{\left(\frac{3}{11}\right)}[/tex]
[tex]t\log{(0.99356387084)} = \log{\left(\frac{3}{11}\right)}[/tex]
[tex]t = \frac{\log{\left(\frac{3}{11}\right)}}{\log{(0.99356387084)}}[/tex]
t = 201
There will be only 3 milligrams remaining in the patient's system after 201 minutes.
More can be learned about exponential functions at https://brainly.com/question/25537936
