Researchers are testing a new type of antibiotic. After administering a dose to a patient, they estimate that the amount of antibiotic in the patient's bloodstream will decrease by 26% every hour. If there are x milligrams of antibiotic in a patient's bloodstream after 2 hours, about how much will there be after 4 hours?
To determine the amount of antibiotic in the patient's bloodstream after 4 hours, we can use the exponential decay formula:
A(t) = A₀ * (1 - r)^t
Where:
A(t) = amount of antibiotic after t hours
A₀ = initial amount of antibiotic
r = decay rate
t = time in hours
In this case, the decay rate is 26% or 0.26, and the amount after 2 hours is x milligrams. So we can write the following equation:
x = A₀ * (1 - 0.26)^2
x = A₀ * (0.74)^2
x = A₀ * 0.5476
Now, we want to find the amount of antibiotic after 4 hours:
A(4) = x * (0.74)^2
A(4) = A₀ * 0.5476 * 0.5476
A(4) = A₀ * 0.29998
Therefore, the amount of antibiotic in the patient's bloodstream after 4 hours will be about 30% of the amount after 2 hours.