Problem 19 - Precalculus

Question

When a person has a cavity filled, the dentist typically gives a local anesthetic. After leaving the dentist's office, one's mouth often is numb for several more hours. If 100 ml of anesthesia is injected into the local tissue at the time of the procedure \((t=0),\) and the amount of anesthesia still in the local tissue \(t\) hours after the initial injection is given by \(A=100 e^{-0.5 t}\) how much remains in the local tissue 4 hours later?

Step-by-Step Solution

Answer

Approximately 13.53 ml of anesthesia remains after 4 hours.

Step 1: Identify the Given Values

We are given the function that describes the amount of anesthesia remaining, which is \( A = 100 e^{-0.5t} \). We need to determine the remaining amount of anesthesia at \( t = 4 \) hours.

Step 2: Substitute the Time into the Formula

Substitute \( t = 4 \) into the formula: \( A = 100 e^{-0.5 \times 4} \). This will give us the amount of anesthesia remaining after 4 hours.

Step 3: Calculate the Exponent

Calculate the exponent: \(-0.5 \times 4 = -2\). Substitute this into the equation to get \( A = 100 e^{-2} \).

Step 4: Evaluate the Exponential Function

Use a calculator to find the value of \( e^{-2} \). \( e^{-2} \approx 0.1353 \).

Step 5: Compute the Remaining Anesthesia

Substitute the value of \( e^{-2} \) into the equation: \( A = 100 \times 0.1353 \). This simplifies to \( A \approx 13.53 \).

Step 6: Interpret the Result

The calculation shows that approximately \( 13.53 \) ml of anesthesia remains in the local tissue 4 hours after the procedure.

Key Concepts

Anesthesia

Anesthesia is a medical method used to suppress pain during procedures. Anesthetics can be either general, affecting the entire body, or local, affecting just a specific area. Dentists often use local anesthesia to numb the mouth so patients won't feel pain while they work. This is injected into local tissue and then absorbed by the body over time.
- **Local Anesthesia:** Local anesthetics work by blocking nerve signals in a part of the body. This is why your mouth feels numb after a dental procedure. - **Duration:** The effect may vary from person to person, but typically lasts a few hours after injection. Understanding how anesthesia works and its duration is critical in both medical practice and patient planning.

Mathematical Modeling

Mathematical modeling is a process used to represent a real-world situation with mathematical expressions. It helps predict outcomes and understand complex systems.
In the case of anesthesia, mathematical models help predict how long the effects last by quantifying the decay of anesthesia in the body over time. - **Why Use Models:** Provides a safe environment to test outcomes. - **Practical Application:** Allows dentists to estimate when a patient will regain sensation. This knowledge helps healthcare professionals schedule follow-up care and informs patients about what to expect after treatment.

Exponential Function

Exponential functions are a key part of mathematical modeling, especially when analyzing processes involving growth or decay. An exponential function can be written as \( A = a \times e^{kt} \), where \( a \) is the initial amount, \( e \) is the base of the natural logarithm, and \( k \) is the rate of change.In the exercise, \( A = 100 e^{-0.5t} \) describes the exponential decay of anesthesia in the body:- **Initial Value:** At \( t = 0 \), \( A = 100 \) representing the initial 100 ml of anesthesia.- **Rate of Decay:** The value \( -0.5 \) in the exponent shows the anesthesia is decaying over time. A negative sign indicates decay.Understanding exponential functions allows us to calculate remaining anesthesia at any given time, such as finding it to be approximately 13.53 ml after 4 hours.

Calculus

Calculus is a branch of mathematics that deals with rates of change and accumulation of quantities. It involves concepts like derivatives and integrals. - **Differentiation:** In this context, it helps find the rate at which anesthesia concentration decreases over time. - **Exponential Decay:** Calculus helps us understand how the anesthesia dissipates exponentially rather than linearly. For this problem, we calculated exponential decay using calculus, which provides an accurate model for how the anesthesia dissipates after injection. Being able to understand calculus allows one to more deeply predict and analyze how different rates can change over time in the body.