The concentration of a drug in the body decreases exponentially after a dosage is given. In one clinical study, adult subjects averaged 13 micrograms/milliliter (mcg/mL) of the drug in their blood plasma 1 hr after a 1000-mg dosage and 4 micrograms/milliliter 7 hr after dosage. Assume the concentration decreases according to the exponential decay model. 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 5 hr after a 1000-mg dosage. c) To relieve a fever, the concentration of the drug should go no lower than 2 mcg/mL. After how many hours will a 1000-mg dosage drop to that level?View an example | 8 parts remaining one c X after ing to The concentration of a drug in the body decreases exponentially after a dosage is given. In one clinical study, adult subjects averaged 16 micrograms/milliliter (mcg/mL) of the drug in their blood plasma 1 hr after edict a 1000-mg dosage and 3 micrograms/milliliter 3 hr after dosage. Assume the concentration decreases according to the exponential decay model. how 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 2 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? To model the given data, use the exponential decay model, P(t) = Poe ", k > 0, where Po is the initial concentration of a drug at time 0, P(t) is the concentration of a drug after time t, and k is the rate of decrease (as a decimal). a) To find k, identify the values of P(t) and t. Note that Po is unknown. After 1 hour the values of P(t) and t are P(t) = 1 and t = 1 Substitute the values of P(t) and t in the exponential decay function to form the first equation. P(t) = Poe" kt P(1) = Pge- 1k Substitute 1 for t. 16 = Pge-K Substitute 16 for P(1). Simplify the right side. Now identify the values of P(t) and t after 3 hours. P(t) = 3 and t = 3 Substitute the values of P(t) and t in the exponential decay function to form the second equation. P(t) = Poe" kt P(3) = Pge- 3k Substitute 3 for t. 3 = Pge- 3k Substitute 3 for P(3). Divide the two equations to clear Po. 3 = Pperk 3 Pae - 3k 16 = Pge- 3k 16 Pge- k Simplify the right side of the equation using the quotient rule of exponents, = am -n 3 Poe- 3k 16 Poek 3 16 = e- 2k 3 Solve the equation = e" 2k for k. Take the natural logarithm on both sides. 3 16 = Ine - 2k Take the natural logarithm on both sides. 3 16 - 2k Use the fact that In ed = a. 0.837 ~ k Divide both sides by - 2 and simplify, rounding to three decimal places. Therefore, k ~ 0.837. To write an equation, find Po. It is known, that after 1 hr the concentration of the drug is 16 mcg/mL. Substitute P(t) = 16, k = 0.837, t = 1 in the exponential decay function and solve for Po, rounding to one decimal place. P(t) = Pge" kt 16 = Pne- 0.837 . 1 16 = Pge- 0.837 Simplify. 37 = Po Divide both sides by e- 0.837 Therefore, an equation that can be used to predict the concentration of the drug after t hours is P(t) = 37 0.837t Print Continue View an example Textbook Clear all(I) _ ".90.! I. b} To estimate the concentration of the drug after 2 hr. substitute 2 for t in the equation P{t) = 3?: and evaluate. Plt} = Put? "' PIE} = 3?:' 0-337\" 2 PIE} =31; \"1\" Multiply. P(2} = it? Simplify. rounding to one decimal place. Therefore. the concentration of the dmg after 2 hr will be about 6.9 megimL. e] To determine after how many hours a WOO-mg dosage will thee to the level 4 megt'mL. substitute 4 tot Pit:- in the function P\") = an? \"-33" and solve for t. Pit} = Put? \"' 4 = 31.: \"-33\" Substitute. % = r- 0-331": Divide both sides by 3?. In % = h f' \"-33\" Take the natural logarithm on both sides. 4 In 3? ' ' 0-33?\" Use the fact that In e' = a. 3 =t Divide both sides by - (183?. rounding to the nearest integer. Therefore. after about 3 hr the concentration of the drug will have dropped to 4 megt'mL