If a person takes a prescribed dose of 10 milligrams of Valium, the amount of Valium in that person’s bloodstream at any time can be modeled

Question

If a person takes a prescribed dose of 10 milligrams of Valium, the amount of Valium in that person’s bloodstream at any time can be modeled with the exponential decay function A ( t ) = 10 e − 0.0188 t where t is in hours. a . How much Valium remains in the person’s bloodstream 12 hours after taking a 10 -mg dose? Round to the nearest tenth of a milligram. mg b . How long will it take 10 mg to decay to 5 mg in a person’s bloodstream? Round to two decimal places. hours c . At what rate is the amount of Valium in a person’s bloodstream decaying 6 hours after a 10 -mg dose is taken. Round the rate to three decimal places.

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Jasmine 1 hour 2021-09-15T22:25:30+00:00 1 Answer 0

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    2021-09-15T22:27:03+00:00

    Answer:

    (a)8.0mg

    (b)36.87 hours

    (c)-0.168

    Step-by-step explanation:

    The amount of Valium in that person’s bloodstream at any time can be modeled with the exponential decay function A ( t ) = 10 e^{ -0.0188 t (t in hours)

    (a)After 12 Hours

    A ( 12) = 10 e^{ -0.0188*12}\\=7.98\\\approx 8.0 mg $(to the nearest tenth of a milligram)

    (b)If A(t)=5mg

    Then:

    5= 10 e^{ -0.0188 t}\\$Divide both sides by 10\\0.5=e^{ -0.0188 t}\\$Take the natural logarithm of both sides\\ln (0.5)=-0.0188 t\\t=ln (0.5) \div (-0.0188)\\$t=36.87 hours (to two decimal places.)

    (c)

    A ( t ) = 10 e^{ -0.0188 t}\\A'(t)=10(-0.0188)e^{ -0.0188 t}\\A'(t)=-0.188e^{ -0.0188 t}\\$At 6 hours, the rate at which the amount of Valium is decaying therefore is:\\A'(6)=-0.188e^{ -0.0188*6}\\$=-0.168 ( to three decimal places)

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