Finding the Amount of Quinine in the Body
50 mg dose of quinine is given to a patient with Malaria. Quinine leaves the body at a rate of 6% per hour. Find a formula for the amount, A (in mg), of quinine in the body hours after the dose is given. How long will it take for the body to only have 20 mg of quinine. Please label your answer A and B.
To find a formula for the amount of quinine in the body hours after the dose is given, we can use the exponential decay formula:
A = A₀ * (1 – r)^t
Where: A₀ is the initial amount of quinine (50 mg) r is the decay rate per hour (6% or 0.06, since it leaves the body at a rate of 6% per hour) t is the time in hours
Using this formula, we can calculate the amount of quinine in the body at any given time.
A = 50 * (1 – 0.06)^t
Now, let’s determine how long it will take for the body to have only 20 mg of quinine.
20 = 50 * (1 – 0.06)^t
Divide both sides by 50:
0.4 = (1 – 0.06)^t
Taking the natural logarithm (ln) of both sides:
ln(0.4) = ln[(1 – 0.06)^t]
Applying the logarithmic property:
ln(0.4) = t * ln(1 – 0.06)
Now, divide both sides by ln(1 – 0.06):
t = ln(0.4) / ln(1 – 0.06)
Using a calculator, we can find the value of t:
t ≈ 10.38 hours
Therefore, it will take approximately 10.38 hours for the body to only have 20 mg of quinine.
Alternatively, to find the value of t, we’ll use the formula:
t = ln(0.4) / ln(1 – 0.06)
Calculating it step by step:
ln(0.4) ≈ -0.916
ln(1 – 0.06) ≈ ln(0.94) ≈ -0.0619
Now, we can substitute these values back into the formula:
t = -0.916 / -0.0619 ≈ 14.806
Therefore, it will take approximately 14.806 hours (or about 14 hours and 49 minutes) for the body to only have 20 mg of quinine.
