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An ill patient is administered 100mg dose of a fever-reducing drug. The total milligrams of the drug in the patient's bloodstream is given by the equation V(t) = 60te-t, where t is measured in hours after the medicine was ingested. a.) When does the amount of drug in the patient's bloodstream reach a maximum? How much of the pill is in the patient's bloodstream at this time? b.) What is the function that describes the rate of change in the amount of the drug in the bloodstream? If this rate is positive, does it mean (overall) that the drug is being absorbed into the bloodstream, or removed from the bloodstream? What if the rate is negative; is the drug entering or exiting the bloodstream? c.) At what time is the rate at which the drug is being absorbed into the bloodstream the greatest? d.) At what time is the rate at which the drug is being removed from the bloodstream the greatest? e.) How much of the drug will remain in the patients system in the long run? That is, what is lim V(t)? t2 t 1 (You may use the fact that lim lim lim.) ttoo et ttoo et t+oo et t- = =

User Linclark
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a.) The amount of drug in the patient's bloodstream reaches a maximum when the derivative of the function V(t) is equal to zero. Taking the derivative of V(t) gives:

V'(t) = 60e^-t - 60te^-t

Setting V'(t) equal to zero and solving for t gives:

60e^-t - 60te^-t = 0
t = 1 hour

Therefore, the amount of drug in the patient's bloodstream reaches a maximum after 1 hour. Plugging t=1 into V(t) gives:

V(1) = 60e^-1 ≈ 22.05 mg

b.) The function that describes the rate of change in the amount of the drug in the bloodstream is the derivative of V(t), which is V'(t) = 60e^-t - 60te^-t. If V'(t) is positive, it means that the drug is being absorbed into the bloodstream. If V'(t) is negative, it means that the drug is being removed from the bloodstream.

c.) To find the time at which the rate at which the drug is being absorbed into the bloodstream is greatest, we need to find the maximum of V'(t). Taking the derivative of V'(t) gives:

V''(t) = 60te^-t - 120e^-t

Setting V''(t) equal to zero and solving for t gives:

t = 2

Therefore, the rate at which the drug is being absorbed into the bloodstream is greatest after 2 hours.

d.) To find the time at which the rate at which the drug is being removed from the bloodstream is greatest, we need to find the minimum of V'(t). Taking the derivative of V'(t) gives:

V''(t) = 60te^-t - 120e^-t

Setting V''(t) equal to zero and solving for t gives:

t = 0

Therefore, the rate at which the drug is being removed from the bloodstream is greatest immediately after the drug is administered.

e.) To find the long-term behavior of V(t), we take the limit as t approaches infinity:

lim V(t) = lim 60te^-t
t->∞ = 0

Therefore, the amount of drug in the patient's system in the long run is zero.
User CCates
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