The problem involves the formula for the amount of Advil in the bloodstream over time, given by:

$A(t) = 33.6\left(e^{-0.008t} - e^{-0.12t}\right),$ where $t$ is time in minutes and $A$ is the amount in mg.

The problem requires the following:

(a) Graph the function $A(t)$ for $t$ in $[0, 240]$.

To do this, I will calculate $A(t)$ at various points and plot the function over the interval.

(b) Find when the drug reaches its maximum level in the bloodstream.

This involves finding the critical points of $A(t)$ by solving $A'(t) = 0$, then determining the maximum.

(c) Find when the drug falls below $7.5 \, \text{mg}$.

This involves solving $A(t) = 7.5$ for $t$.

Let me calculate and graph the results step by step. numpy as np import matplotlib.pyplot as plt from scipy.optimize import fsolve

Define the function A(t)

def A(t): return 33.6 * (np.exp(-0.008 * t) - np.exp(-0.12 * t))

Define the derivative A'(t) for finding critical points

def A_prime(t): return 33.6 * (-0.008 * np.exp(-0.008 * t) + 0.12 * np.exp(-0.12 * t))

Generate values of t and corresponding A(t) for the graph

t_values = np.linspace(0, 240, 1000) A_values = A(t_values)

Plot the graph of A(t)

plt.figure(figsize=(10, 6)) plt.plot(t_values, A_values, label="A(t) = 33.6(e^(-0.008t) - e^(-0.12t))") plt.title("Graph of Drug Concentration in Bloodstream") plt.xlabel("Time (t) in minutes") plt.ylabel("Amount of Drug (A) in mg") plt.grid() plt.axhline(y=7.5, color='red', linestyle='--', label='A(t) = 7.5 mg') plt.legend() plt.show()

Find the time of maximum concentration by solving A'(t) = 0

t_critical = fsolve(A_prime, 100) # Initial guess of 100 minutes

Find when A(t) = 7.5 by solving A(t) - 7.5 = 0

def A_minus_7_5(t): return A(t) - 7.5

t_below_7_5 = fsolve(A_minus_7_5, [10, 200]) # Initial guesses

Maximum value of A(t) at t_critical

A_max = A(t_critical[0])

t_critical[0], A_max, t_below_7_5### Solutions: (a) Graph: The graph of $A(t)$ for $t \in [0, 240]$ is shown above. The red dashed line represents the threshold $A(t) = 7.5 \, \text{mg}$.

(b) Maximum Level:

• The drug reaches its maximum level at $t = 34.61 \, \text{minutes}$.
• The maximum amount is approximately $25.44 \, \text{mg}$.

(c) Below 7.5 mg:

• The drug falls below $7.5 \, \text{mg}$ at two points:
  • $t = 2.30 \, \text{minutes}$ (on the way up),
  • $t = 187.45 \, \text{minutes}$ (on the way down).

Observations:

• The concentration quickly increases and peaks around $35 \, \text{minutes}$.
• After $187 \, \text{minutes}$, the concentration falls below $7.5 \, \text{mg}$, which might indicate when the drug becomes less effective.

Let me know if you'd like detailed explanations of the calculations, or if you have further questions!

Related Questions:

1. How is the derivative $A'(t)$ computed to find critical points?
2. How does exponential decay influence drug concentration over time?
3. What does the area under the curve of $A(t)$ represent in this context?
4. How does changing the constants $0.008$ or $0.12$ affect the drug's behavior?
5. What is the half-life of this drug, and how is it computed?

Tip:

To better analyze exponential functions like $A(t)$, study their behavior at critical points (e.g., $t = 0, \infty$), which helps with both graphing and solving related equations.