VCE Methods Differential Calculus Application Task 9

Question 1 [2018 Exam 2 Section B Q2]

A drug, X, comes in 500 milligram (mg) tablets.
The amount, \(b\), of drug X in the bloodstream, in milligrams, \(t\) hours after one tablet is consumed is given by the function

\( b(t) = \frac{4500}{7} \left( e^{(-\frac{t}{5})} - e^{(-\frac{9t}{10})} \right) \)

a. Find the time, in hours, it takes for drug X to reach a maximum amount in the bloodstream after one tablet is consumed. Express your answer in the form \(a \log_e(c)\), where \(a, c \in \mathbb{R}\). 2 marks

The graph of \(y = b(t)\) is shown below for \(0 \le t \le 6\).

b. Find the average rate of change of the amount of drug X in the bloodstream, in milligrams per hour, over the interval [2, 6]. Give your answer correct to one decimal place. 2 marks

c. Find the average amount of drug X in the bloodstream, in milligrams, during the first six hours after one tablet is consumed. Give your answer correct to the nearest milligram. 2 marks

d. Six hours after one 500 milligram tablet of drug X is consumed (Tablet 1), a second identical tablet is consumed (Tablet 2). The amount of drug X in the bloodstream from each tablet consumed independently is shown in the graph below.

i. On the graph above, sketch the total amount of drug X in the bloodstream during the first 12 hours after Tablet 1 is consumed. 2 marks

ii. Find the maximum amount of drug X in the bloodstream in the first 12 hours and the time at which this maximum occurs. Give your answers correct to two decimal places. 2 marks