Please show all work and calculations!!*** The amount of acetaminophen in blood

Question

Please show all work and calculations!!***

The amount of acetaminophen in blood after a dosage behaves as an exponential decay. For a specific blood sample, the best-fit exponential function is

R = (200mg)(0.78)t

where R is the concentration after t hours.

What is the initial amount of acetaminophen taken?

What is the percent rate of decrease in acetaminophen each hour?

Six hours after taking the medication, what is the concentration of acetaminophen?

It is only safe to take a second dosage once the amount of acetaminophen in the blood has reached 25 mg. How many hours after the first dose can a person take a second dose of the medication?

Explanation / Answer

R = (200mg) e0.78t

(i) What is the initial amount of acetaminophen taken?

Answer : t = 0

so R = 200 * e0.78* 0 = 200 mg

(ii) What is the percent rate of decrease in acetaminophen each hour?

Answer : for t hour

R(t) = 200 * ( 0.78)t   Hours

R( t +1) = 200 * ( 0.78) t +1 Hours

so % decrease = [ R(t+1) - R(t)] / R(t) * 100 = [ 200 * ( 0.78) (t +1) - 200 * ( 0.78)t] / 200 * ( 0.78)t * 100

                       = ( 1- 0.78)/ 1 * 100 = 28.205 [ -22/100] = (-22%) = 22% decrease. per hour

(iii) Six hours after taking the medication, what is the concentration of acetaminophen?

R (6) = 200 * (0.78) 6 = 45.04 mg

(iv) How many hours after the first dose can a person take a second dose of the medication?

so to calculate that R <= 25mg

so 25 <= 200 * ( 0.78)t

taking log both side

log (25) <= log (200) + t log ( 0.78)

1.3979 <= 2.30103 - t * 0.1079

t * 0.1079 <= 0.90313

t < = 8.37 so after 8 hours and 22 minutes a person can take a second dose of the medication after first dose.

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