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 0.4 milligram per liter. If the antibiotic is given every 4 hours and
k = −0.848
, find the concentration, to the nearest hundredth, of the antibiotic just before the fifth dose. (Round your answer to two decimal places.)
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To find the concentration of the antibiotic just before the fifth dose, we need to find the sum of the geometric series up to the fourth term.
The formula for the sum of a geometric series is given by:
S = A(1 - r^n) / (1 - r)
Where:
S = sum of the series
A = first term
r = common ratio
n = number of terms
In this case, the first term (A) is the number of milligrams in one dose of the antibiotic, which is 0.4 milligram per liter.
The common ratio (r) is e^(kt), where t is the time between doses and k is the constant that depends on how quickly the body metabolizes the antibiotic.
The number of terms (n) is 4, since we want to find the concentration just before the fifth dose.
Substituting the given values into the formula, we can calculate the concentration:
S = 0.4(1 - e^(-0.848*4))^4 / (1 - e^(-0.848))
Evaluating this expression using a calculator, we find that the concentration is approximately 0.66 milligram per liter just before the fifth dose. Rounded to the nearest hundredth, the concentration is 0.66 milligram per liter.