Problem 44 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Jane took \(100 \mathrm{mg}\) of a drug in the morning and another \(100 \mathrm{mg}\) of the same drug at the same time the following morning. The amount of the drug in her body \(t\) days after the first dosage was taken is given by $$ A(t)=\left\\{\begin{array}{ll} 100 e^{-1.4 t} & \text { if } 0 \leq t<1 \\ 100\left(1+e^{1.4}\right) e^{-1 . A r} & \text { if } t \geq 1 \end{array}\right. $$ What was the amount of drug in Jane's body immediately after taking the second dose? After 2 days?

Step-by-Step Solution

Answer

Immediately after taking the second dose, the amount of the drug in Jane's body was approximately \(147.45 \mathrm{mg}\). After 2 days, the amount of the drug in her body was approximately \(56.03 \mathrm{mg}\).

Step 1: Find the amount of drug immediately after taking the second dose

To find the amount of drug in Jane's body immediately after taking the second dose, we need the value of A(t) when \(t = 1\). Since she took the second dose exactly one day after the first dose, we will use the second part of the piece-wise function: \[A(t) = 100\left(1+e^{1.4}\right)e^{-1.4t}\] Plug in \(t = 1\): \[A(1) = 100\left(1+e^{1.4}\right)e^{-1.4 \times 1}\] Calculate the value: \[A(1) \approx 100\left(1+e^{1.4}\right)e^{-1.4} \approx 147.45 \mathrm{mg}\] So immediately after taking the second dose, the amount of the drug in Jane's body was approximately 147.45 mg.

Step 2: Find the amount of drug after 2 days

To find the amount of drug in Jane's body after 2 days, we need the value of A(t) when \(t = 2\). Since \(t \geq 1\), we will use the second part of the piece-wise function: \[A(t) = 100\left(1+e^{1.4}\right)e^{-1.4t}\] Plug in \(t = 2\): \[A(2) = 100\left(1+e^{1.4}\right)e^{-1.4 \times 2}\] Calculate the value: \[A(2) \approx 100\left(1+e^{1.4}\right)e^{-2.8} \approx 56.03 \mathrm{mg}\] After 2 days, the amount of the drug in Jane's body was approximately 56.03 mg.

Key Concepts

Piecewise Function

A piecewise function is like a mathematical tool that allows us to describe different behaviors of a function over separate intervals. It's particularly useful when one needs to define a function that behaves differently in different scenarios.
In Jane's case, the function describing the amount of drug in her body is piecewise. This means that there are two separate formulas we use depending on whether Jane is within the first day after taking the drug or has already reached the second day.

• For the interval \(0 \leq t < 1\): The function \(A(t) = 100 e^{-1.4 t}\) describes the decay of the drug concentration during the first day.
• For \(t \geq 1\): The formula changes to \(A(t) = 100 (1 + e^{1.4}) e^{-1.4 t}\) to account for the additional dose taken on the second day.

This adaptability of piecewise functions is crucial in accurately modeling situations like Jane's drug intake, where conditions change over time.

Pharmacokinetics

Pharmacokinetics is the field that studies how drugs move through the body over time. It helps to understand how drugs are absorbed, distributed, metabolized, and excreted. In Jane's example, we are focused on how the amount of drug decreases over time, also known as exponential decay.
Drug concentration often decreases in a predictable pattern, described here using an exponential function. The given functions show how rapidly the drug decays with time due to processes the body employs to eliminate it.

• At time \(t=0\), Jane has just taken the drug, so the concentration is high.
• As days pass, the function \(e^{-1.4 t}\) rapidly reduces the drug concentration, indicating how her body breaks down and removes the drug.
• Understanding these dynamics is crucial for determining dosing schedules to maintain therapeutic drug levels without causing toxicity.

Calculus

Calculus is the branch of mathematics that assists us in understanding changes and dynamics. In this context, calculus helps us analyze how the drug concentration decreases over time due to its decay function.

When we talk about exponential decay, we're discussing how a quantity decreases at a rate proportional to its current value. Calculus helps discern and compute the exact amount remaining at any given time. Here, we use it to solve for specific values of the piecewise function for Jane's drug intake.

• The function \(e^{-1.4 t}\) shows exponential decay where the base \(e\) is a constant approximately equal to 2.718.
• When we plug in values like \(t=1\) or \(t=2\), we're engaging in basic calculus operations to find precise amounts of drug remaining in her body.
• These calculations ensure we understand both immediate and longer-term effects of the dosage supplied, critical for safe and effective drug administration.