The concentration (in milligrams per liter) of an antibiotic in the blood is given by the geometric series
A + Aekt + Ae2kt + + Ae(n − 1)kt
where A is the number of milligrams in one dose of the antibiotic, n is the number of doses, t is the time between doses, and k is a constant that depends on how quickly the body metabolizes the antibiotic. Suppose one dose of an antibiotic increases the blood level of the antibiotic by 0.3 milligram per liter. If the antibiotic is given every 4 hours and
k = −0.862
, find the concentration, to the nearest hundredth, of the antibiotic just before the fifth dose. (Round your answer to two decimal places.)
geez, what's the problem? Just plug and chug:
A(1 + e^kt + e^2kt + ... + e^(n-1)kt)
is just a geometric series where
a = 1
r = e^kt
and there are n terms. So,
Just before the 5th dose, there have been for doses, so n=4, and
Sn = A(1-r^n)/(1-r)
= 0.3 (1-e^(-.862*4*4))/(1-e^(-.862*4)) = 0.309856
To find the concentration of the antibiotic just before the fifth dose, we need to substitute the given values into the geometric series formula and calculate the sum of the series up until the fourth term.
The formula for the geometric series is:
S = A + Aekt + Ae2kt + ... + Ae(n-1)kt
Where:
S is the sum of the geometric series
A is the initial term (the number of milligrams in one dose of the antibiotic)
e is the base of the natural logarithm (approximately equal to 2.71828)
k is a constant (how quickly the body metabolizes the antibiotic)
n is the number of terms in the series
In this particular problem, we want to find the sum of the series up until the fourth term, so n = 4.
Given information:
A = 0.3 milligram per liter (one dose of the antibiotic increases the blood level by 0.3 mg/L)
t = 4 hours (time between doses)
k = -0.862
Plug these values into the formula:
S = A + Aekt + Ae2kt + ... + Ae(n-1)kt
S = 0.3 + 0.3 * e^(-0.862 * 4) + 0.3 * e^(-0.862 * 8) + 0.3 * e^(-0.862 * 12)
Now we can evaluate this expression:
S ≈ 0.3 + 0.3 * e^(-3.448) + 0.3 * e^(-6.896) + 0.3 * e^(-10.344)
Using a calculator, we find:
S ≈ 0.3 + 0.3 * 0.0323 + 0.3 * 0.0011 + 0.3 * 0.000036
S ≈ 0.33714
Therefore, the concentration of the antibiotic just before the fifth dose is approximately 0.34 milligrams per liter (rounded to two decimal places).