Problem 9 - Applied Calculus

Question

A cigarette puts \(1.2 \mathrm{mg}\) of nicotine into the body. Nicotine leaves the body at a continuous rate of \(34.65 \%\) per hour, but more than \(60 \mathrm{mg}\) can be lethal. If a person smokes a cigarette with each of the following frequencies, find the long-run quantity of nicotine in the body right after a cigarette. Does the nicotine reach the lethal level? (a) Every hour (b) Every half hour (c) Every 15 minutes (d) Every 6 minutes (e) Every 3 minutes

Step-by-Step Solution

Answer

None of the schedules reach the lethal level of 60 mg; the highest amount, 57.56 mg, is just below the lethal limit for smoking every 3 minutes.

Step 1: Define Variables and Equations

For continuous decay at rate \( r \) per hour, nicotine in the body decreases by \( 1 - r \) of its current amount in each hour. Here, \( r = 0.3465 \) per hour, thus nicotine keeps \( 1 - 0.3465 = 0.6535 \). Let \( N \) be the nicotine amount right after a cigarette. The recurring nicotine amount equation after smoking a single cigarette is given by: \[ N = \frac{1.2}{1-e^{-0.69315 \cdot t}} \] Where \( t \) is time between cigarettes in hours. The exponential part \( e^{-0.69315 \cdot t} \) arises from the continuous decay rate.

Step 2: Calculate for Each Time Interval

We calculate the steady-state nicotine level \( N \) based on each predefined smoking interval:(a) **Every Hour (\( t = 1 \))** \[ N = \frac{1.2}{1-e^{-0.69315 \cdot 1}} \approx \frac{1.2}{0.3465} \approx 3.46 \text{ mg} \](b) **Every Half Hour (\( t = 0.5 \))** \[ N = \frac{1.2}{1-e^{-0.69315 \cdot 0.5}} \approx \frac{1.2}{0.21634} \approx 5.55 \text{ mg} \](c) **Every 15 Minutes (\( t = 0.25 \))** \[ N = \frac{1.2}{1-e^{-0.69315 \cdot 0.25}} \approx \frac{1.2}{0.08027} \approx 14.95 \text{ mg} \](d) **Every 6 Minutes (\( t = 0.1 \))** \[ N = \frac{1.2}{1-e^{-0.69315 \cdot 0.1}} \approx \frac{1.2}{0.04157} \approx 28.87 \text{ mg} \](e) **Every 3 Minutes (\( t = 0.05 \))** \[ N = \frac{1.2}{1-e^{-0.69315 \cdot 0.05}} \approx \frac{1.2}{0.02085} \approx 57.56 \text{ mg} \]

Step 3: Evaluate Lethal Dose Risk

Compare calculated nicotine levels to the lethal threshold of 60 mg: (a) 3.46 mg is far below 60 mg. (b) 5.55 mg is far below 60 mg. (c) 14.95 mg is below 60 mg. (d) 28.87 mg is below 60 mg. (e) 57.56 mg is below, but close to 60 mg. None of the smoking frequencies reach or exceed the lethal 60 mg level, though smoking every 3 minutes is dangerously close to it.

Key Concepts

Continuous Decay Rate

When discussing the continuous decay rate in the context of nicotine levels, it is important to understand how substances diminish over time in a continuous manner. In this scenario, the amount of nicotine in the body decreases exponentially at a rate of 34.65% per hour. This means that every hour, 34.65% of the nicotine present is eliminated from the body.

The concept of exponential decay is characterized by its smooth and rapid decline of the substance. This can be calculated using the formula:

• Nicotine remaining after one hour = Current nicotine amount × (1 - Decay rate)

By applying exponential decay, we not only model how nicotine clears from the system but also predict long-term consequences under different smoking habits. The formula: \[N = \frac{1.2}{1-e^{-0.69315 \cdot t}}\] applies this decay rate to compute the steady-state nicotine level based on the frequency of smoking, where \( t \) is the time interval between cigarettes. This mathematical representation allows us to understand how quickly the nicotine dissipates relative to its intake.

Nicotine Levels

Nicotine levels in the body depend greatly on the rate and frequency of smoking. Each cigarette introduces an additional 1.2 mg of nicotine. This amount, when combined with continuous smoking, contributes to the overall nicotine concentration in the body over time. The key lies in understanding how often these cigarettes are smoked.

For each proposed interval:

• Every hour, the body effectively retains less nicotine compared to more frequent smoking due to more time for decay. Nicotine levels settle around \(3.46 \text{ mg}\).
• As smoking intervals decrease, say to every half hour or every 15 minutes, the nicotine concentration builds up faster than it can decay, reaching approximately \(5.55 \text{ mg}\) and \(14.95 \text{ mg}\), respectively.

These outcomes show how timing affects the balance between decay and accumulation, with shorter intervals resulting in higher nicotine levels. The formula to calculate nicotine after each interval is crucial in predicting and managing potential health risks.

Lethal Dosage

Understanding the risks associated with lethal dosage is paramount when discussing nicotine consumption. The lethal threshold for nicotine is approximately 60 mg. This value is crucial because exceeding it poses significant health risks and can be fatal.

Through calculations, we comprehend that none of the evaluated smoking intervals reach 60 mg:

• Smoking every hour keeps levels comfortably below lethal limits with only 3.46 mg.
• Frequent smoking, such as every 3 minutes, leads to nearing the danger zone at 57.56 mg.

While none reach the full threshold, smoking every 3 minutes is alarmingly close, highlighting the importance of careful monitoring of intake. It warns us against maintaining high frequency consumption, emphasizing the necessity of moderation. Understanding these nicotine levels is critical in making informed decisions about smoking habits.