If a hospital patient is given 50 milligrams of medicine which leaves the bloodstream at $10 \%$ per hour, how many milligrams of medicine will remain in the system after 6 hours? Use the function $A(t)=l e^{r t}$.
Answer (1)
The problem involves exponential decay, where the amount of medicine in the bloodstream decreases over time.
We identify the initial amount l = 50 and the rate of decay r = − 0.1 .
We substitute these values, along with t = 6 , into the formula A ( t ) = l e r t .
We calculate A ( 6 ) = 50 e − 0.6 .
The amount of medicine remaining after 6 hours is approximately 27.44 mg.
Explanation
Understanding the Problem We are given that a patient receives 50 milligrams of medicine, which leaves the bloodstream at a rate of 10% per hour. We need to find out how many milligrams of the medicine remain in the patient's system after 6 hours, using the function A ( t ) = l e r t .
Identifying the Variables In the given function A ( t ) = l e r t , l represents the initial amount of medicine, r is the rate at which the medicine leaves the bloodstream, and t is the time in hours. We are given that l = 50 milligrams and the medicine leaves at 10% per hour, so r = − 0.1 . We want to find the amount remaining after t = 6 hours.
Substituting the Values Now, we substitute the given values into the formula: A ( 6 ) = 50 × e ( − 0.1 ) × 6 = 50 × e − 0.6 .
Calculating the Remaining Amount Calculating the value of e − 0.6 , we have e − 0.6 ≈ 0.5488 . Therefore, A ( 6 ) = 50 × 0.5488 ≈ 27.44 .
Final Answer After 6 hours, approximately 27.44 milligrams of medicine will remain in the patient's system.
Examples
Understanding exponential decay is crucial in various real-world scenarios. For instance, in environmental science, it helps model the decay of pollutants in a lake. If a lake initially contains 1000 tons of pollutants and it decays at a rate of 5% per year, we can use the same exponential decay formula to predict the amount of pollutants remaining after a certain number of years. This helps in making informed decisions about environmental cleanup and conservation efforts. Similarly, in finance, it can model the depreciation of an asset over time.
