The function A(t) = 200e^-0.014t gives the amount of medication, in milligrams, in a patient's bloodstream t minutes after the medication has been injected into the patient's bloodstream.
(a) Find the amount of medication, to the nearest milligram, in the patient's bloodstream after 40 minutes.
? mg
(b) Use a graphing utility to determine how long it will take, to the nearest minute, for the amount of medication in the patient's bloodstream to reach 50 milligrams.
? min
a)
plug 40 into t in the original equation
b)
solve for t analytically
t = 99.021
heres a graph:
http://www.wolframalpha.com/input/?i=200e%5E%28-.014t%29%3D50
To find the amount of medication in the patient's bloodstream after 40 minutes, we need to substitute t = 40 into the function A(t) = 200e^(-0.014t).
(a) To find the amount of medication after 40 minutes, we substitute t = 40 into the function:
A(40) = 200e^(-0.014 * 40)
Calculating this value, we get:
A(40) ≈ 200e^(-0.56) ≈ 162.35
So, to the nearest milligram, the amount of medication in the patient's bloodstream after 40 minutes is approximately 162 mg.
(b) To determine how long it will take for the amount of medication in the patient's bloodstream to reach 50 milligrams, we need to find the value of t that makes A(t) = 50.
We can rearrange the equation to solve for t as follows:
50 = 200e^(-0.014t)
Divide both sides of the equation by 200:
0.25 = e^(-0.014t)
To solve for t, take the natural logarithm (ln) of both sides:
ln(0.25) = ln(e^(-0.014t))
Using the property of logarithms that ln(e^x) = x, we can simplify further:
ln(0.25) = -0.014t
Divide both sides by -0.014:
t = ln(0.25)/(-0.014)
Using a calculator, we find:
t ≈ 151.81
Therefore, it will take approximately 152 minutes (to the nearest minute) for the amount of medication in the patient's bloodstream to reach 50 milligrams.