Problem 42 - Applied Calculus

Question

At a time \(t\) hours after it was administered, the concentration of a drug in the body is \(f(t)=27 e^{-0.14 t} \mathrm{ng} / \mathrm{ml}\). What is the concentration 4 hours after it was administered? At what rate is the concentration changing at that time?

Step-by-Step Solution

Answer

At 4 hours, the concentration is approximately 15.43 ng/ml and is decreasing at about 2.16 ng/ml per hour.

Step 1: Understand the Problem

We need to find the concentration of a drug in the body at a specific time, and the rate at which the concentration is changing. The concentration at time \( t \) is given by \( f(t) = 27 e^{-0.14 t} \) ng/ml.

Step 2: Substitute Time into the Equation

Substitute \( t = 4 \) into the equation to find the concentration at 4 hours. So, \( f(4) = 27 e^{-0.14 imes 4} \).

Step 3: Calculate the Exponential Value

First calculate the exponent: \(-0.14 imes 4 = -0.56\). Then, compute \( e^{-0.56} \).

Step 4: Find the Concentration

Multiply the result of \( e^{-0.56} \) by 27 to find the concentration at 4 hours: \( f(4) = 27 imes e^{-0.56} \).

Step 5: Differentiate to Find Rate of Change

Differentiate \( f(t) = 27 e^{-0.14 t} \) with respect to \( t \). The derivative is \( f'(t) = 27 imes (-0.14) e^{-0.14 t} = -3.78 e^{-0.14 t} \).

Step 6: Substitute Time into the Derivative

Substitute \( t = 4 \) into the derivative \( f'(t) = -3.78 e^{-0.14 t} \) to find the rate of change at that time. Calculate \( f'(4) = -3.78 imes e^{-0.56} \).

Step 7: Conclusion

Using \( e^{-0.56} \approx 0.5712 \), the concentration is \( 27 imes 0.5712 \approx 15.43 \text{ ng/ml} \). The rate of change is \( -3.78 imes 0.5712 \approx -2.16 \text{ ng/ml per hour}. \)

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. In simple terms, it allows us to know how quickly or slowly something is happening. When we differentiate a function, we produce what is called the derivative. The derivative tells us the instant rate of change of the original function. Differentiation looks at small changes and helps us understand dynamics in various contexts, like in physics or economics.

To differentiate the function for drug concentration, which is given as \(f(t) = 27 e^{-0.14 t}\), we need to apply the rules of differentiation. Here, \(e^{-0.14 t}\) is an exponential function, where differentiation involves the exponential rule. The result is \(f'(t) = 27 \times (-0.14) e^{-0.14 t}\). This derivative \(f'(t)\) shows how the drug concentration changes over time in the body.

Rate of Change

Rate of change is a critical concept that describes how one quantity changes in relation to another quantity. In the context of the drug concentration in the body, it tells us how quickly the concentration of the drug increases or decreases over time after administration.

The rate of change is derived from differentiation. When we calculated the derivative \(f'(t) = -3.78 e^{-0.14 t}\), it represents the rate at which the drug concentration is changing at any time \(t\). For example, when \(t = 4\), this means 4 hours after the drug is administered, the rate of change calculated as \(f'(4)\) is approximately \(-2.16\) ng/ml per hour.

This negative sign indicates that the concentration is decreasing, which is typical after the maximum concentration is reached, as the body metabolizes and eliminates the drug over time.

Drug Concentration

Drug concentration refers to the amount of drug present in a specified volume of biological fluid, such as blood. It is usually measured in nanograms per milliliter (ng/ml). Understanding how drug concentration changes over time is crucial for determining proper dosage and timing in medical treatments.

In this exercise, the concentration function is defined as \(f(t) = 27 e^{-0.14 t}\). This defines how the drug concentration decreases exponentially over time after being introduced into the body. According to the function, \(t\) represents the time in hours. The exponential term \(e^{-0.14 t}\) suggests a rapid decrease initially, which slows down as time progresses. After substituting \(t = 4\) into the function, we determined that the concentration of the drug in the body is approximately \(15.43\) ng/ml at 4 hours.

Understanding these changes can help medical professionals adjust dosages to achieve optimal therapeutic effects while avoiding overdose or underdosing.