Question

Benadryl is a medicine used to treat runny noses, itching, and other cold and allergy symptoms. Ittakes an average of two to three days for the active ingredient of Benadryl, diphenhydramine, to leave the body. The formula below represents the amount, A, of diphenhydramine left in the body,measured in mg, t hours after taking 50 mg of the medicine.A = 50 (0.9217)^tA) How long does it take for the body to eliminate half of the initial amount of Benadryl? Solve thisalgebraically. Round decimal to two places.

Answer 1

We have to replace A= 25 mg in the equation and solve for t. Doing so, we have:

$\begin{gathered} 25=50(0.9217)^t\text{ } \\ 0.5=(0.9217)^t\text{ (Dividing by 50 on both sides of the equation)} \end{gathered}$

$\begin{gathered} \log _5(0.5)=\log _5(0.9217)^t(\text{ Taking the logarithm base 5 of both sides of the equation)} \\ \log _5(0.5)=t\cdot\log _5(0.9217)\text{ (Using the power rule of logarithms)} \\ -0.431=t\cdot(-0.0507)\text{ (Finding the logarithms for each case)} \\ 8.50=t\text{ (Dividing by }-0.0507\text{ on both sides of the equation)} \end{gathered}$

The answer is 8.50 hours (Rounding to two decimal places)