Problem 20 - 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 12 hours later?

Step-by-Step Solution

Answer

Approximately 0.25 ml of anesthesia remains after 12 hours.

Step 1: Understanding the Formula

The formula given is \(A = 100e^{-0.5t}\). This describes the amount of anesthesia remaining at any time \(t\) after the initial injection. The initial amount of anesthesia is 100 ml.

Step 2: Substituting the Time Value

To find how much anesthesia remains after 12 hours, substitute \(t = 12\) into the formula. This gives \(A = 100e^{-0.5 \times 12}\).

Step 3: Calculating the Exponent

First, calculate the exponent: \(-0.5 \times 12 = -6\). Now the formula becomes \(A = 100e^{-6}\).

Step 4: Evaluating the Exponential Component

Use a calculator to find \(e^{-6}\). This is approximately \(0.002478752\).

Step 5: Calculating the Remaining Anesthesia

Now multiply the exponential value by the initial amount: \(100 \times 0.002478752 \approx 0.2478752\).

Step 6: Rounding the Result

Round the result to two decimal places to make it practical: \(A \approx 0.25 \text{ ml}\).

Key Concepts

Local Anesthetic

Local anesthetics are medications used to numb specific areas of the body to prevent pain during procedures such as filling a cavity at the dentist's office. When administered, they temporarily disrupt nerve signal transmission in the targeted area.

Key points about local anesthetics include:

• They provide localized pain relief without affecting consciousness.
• The duration of their effects varies depending on the type, concentration, and size of the dosage.
• In this problem, 100 ml of an anesthetic was used, which gradually reduces in potency over time.

Understanding how local anesthetics dissolve over time helps manage their effects and predict when normal sensation will return. This makes them a crucial tool for medical and dental procedures.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent. It's commonly used to model situations involving exponential growth or decay.

In the context of this exercise, the formula given is an example of exponential decay: \[ A = 100e^{-0.5t} \]This formula signifies how the anesthesia decays or reduces over time. Here,

• 100 is the initial quantity of anesthesia.
• \(e\) is the base of the natural logarithm, approximately equal to 2.718.
• \(t\) represents time in hours since the anesthetic was administered.

To solve such problems, you'll replace \(t\) with the specific time duration you're interested in. This will reveal how much of the substance remains at that time. Exponential functions are vital for understanding phenomena where change happens rapidly and continuously.

Decay Rate

Decay rate is a measure of how quickly a substance decreases over time. It is often expressed with exponential functions in natural decay processes, such as the fading of an anesthetic's effect.

In this case, the decay rate is encoded in the formula as \(-0.5\). This negative value within the exponent signifies a reduction over time:

• A higher absolute value of decay rate means a faster decrease in the anesthetic's potency.
• The decay rate here implies that half of the anesthetic will reduce approximately every hour.
• Understanding the decay rate allows for predicting how long the anesthetic's effects will last, which is crucial for managing pain relief post-procedure.

Managing such decay helps plan effectively for care after a procedure, ensuring comfort and safety for the patient.