The concentration of a drug in the body decreases...

The concentration of a drug in the body decreases exponentially after a dosage is given. In one clinical study, adult subjects averaged 14 micrograms/milliliter (mcg/mL) of the drug in their blood plasma 1 hr after a 1000-mg dosage and 3 micrograms/milliliter 4 hr after dosage. a) Find the value k, and write an equation for an exponential function that can be used to predict the concentration of the drug, in micrograms/milliliter, t hours after a 1000-mg dosage. b) Estimate the concentration of the drug 3 hr after a 1000-mg dosage. c) To relieve a fever, the concentration of the drug should go no lower than 4 mcg/mL. After how many hours will a 1000-mg dosage drop to that level? a) k = (Round to three decimal places as needed.)

Transcript

00:01 Hello students, we know that the concentration of a drug in our body decreases exponentially after a dosage is given.

00:08 So here you can see 14 micrograms per milliliter of the drug, which is reduced after 1 hour when 1 ,000 mg dose it is given.

00:22 So after 6 hours, it's reduced to 3 micrograms per milliliter.

00:30 Okay, from this data we have to find out the value of k, which obviously seen in the exponential form, and the concentration of drug after five hours, and how many hours will a thousand mdoseage drop to four my program per millimeter? okay, to start with, we have the general form for the exponential existence that is p .m.

00:56 Raised to kt okay the concentration after t hour d zero is the initial concentration okay so we have concentration is equal to 14 when time is one hour and concentration is equal to three when time is equal to six hour okay concentration three when time is equal to six hours so we can just write 14 is equal to p0 we will be having t0 itself you raise to k t is one and we also have three that's equal to p zero he raised to six k okay okay so now we can divide this two equation that it becomes 14 divided by 3 that's equal to a raised to k divided by a raise to 6k okay that is 14 divided by 3 is equal to 14 divided by 3 is equal to 14 divided by 3 is is equal to a raise to minus 5k.

02:34 Okay, now we can take log on both sides.

02:39 We can take log on both sides.

02:43 So we can have, after taking we can have log 14 divided by 3.

02:53 That's equal to minus 5k.

02:59 Because log of erased to x is equal to x that is k is equal to log 14 divided by 3 divided by negative 5 and that's equal to that's equal to minus 0 .3080 from this scale value we can find the value of 30 that is 14 which is equal to p0 a raised to k here minus 0 .3080.

03:46 From this p 0 will be equal to 14 divided by 8 to the power negative 0 .3080 and that's equal to 19 .17 19 .17...

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