Problem 91 - College Algebra

Question

The function \(A(d)\) gives the pain level on a scale of 0 to 10 experienced by a patient with \(d\) milligrams of a pain- reducing drug in her system. The milligrams of the drug in the patient's system after \(t\) minutes is modeled by \(m(t)\). Which of the following would you do in order to determine when the patient will be at a pain level of 4 ? a. Evaluate \(A(m(4))\) b. Evaluate \(m(A(4))\) c. Solve \(A(m(t))=4\). d. Solve \(m(A(d))=4\).

Step-by-Step Solution

Answer

Choose option (c), "Solve \(A(m(t))=4\)".

Step 1: Understanding Function Composition

We are given two functions: \(A(d)\), which represents pain level based on the drug amount \(d\), and \(m(t)\), which represents the drug amount based on time \(t\). The goal is to find the time \(t\) when the pain level is 4.

Step 2: Linking Functions to Pain Level

To find when the pain level is 4, we need \(A(d) = 4\) where \(d\) is the amount of the drug. Since \(m(t)\) gives the drug amount \(d\) based on \(t\), we need to consider \(A(m(t))\).

Step 3: Equation for Pain Level 4

Setting up the equation \(A(m(t)) = 4\) allows us to directly solve for \(t\) as it calculates the pain level based on the drug amount derived from time.

Step 4: Choosing the Correct Option

Option (c), "Solve \(A(m(t))=4\)", is the approach to determine when \(t\) results in a pain level of 4.

Key Concepts

Pain Level Function

The pain level function, often denoted as \(A(d)\), is a mathematical representation of how much pain a patient feels given a certain amount of medication in their body. This function is crucial to understanding how effective a drug is at reducing pain. By inputting different values of \(d\), which represents the milligrams of the drug in the patient's system, we can determine the corresponding pain level on a scale from 0 to 10.

Important aspects of the pain level function include:

• Domain: This is the set of all possible \(d\) values (drug amounts) that you can use in the function.
• Range: This is the set of possible outcomes, in this case, pain levels ranging from 0 (no pain) to 10 (maximum pain).

Understanding this function helps in predicting how much of the drug is necessary to reduce the patient's pain to a desired level. For this exercise, we are particularly interested in when the pain reaches a level of 4.

Drug Concentration Model

The drug concentration model is represented by the function \(m(t)\). This function describes how the concentration of the drug in the patient's body changes over time. It is typically based on pharmacokinetics, which helps in studying how the body absorbs, distributes, metabolizes, and excretes the drug.

Key points to remember include:

• Input: \(t\), which stands for time in minutes since the drug was administered.
• Output: \(d\), the amount of drug in the patient's system.

By understanding \(m(t)\), healthcare providers can determine how long it takes for the drug to reach its optimal levels and how long it remains effective. For example, in the exercise, we aim to find how \(m(t)\) affects \(A(d)\) to reach a pain level of 4 after a specific amount of time \(t\).

Solving Equations

Solving equations in the context of function composition involves finding the unknown variable that satisfies the equation. When we have composed functions, like in our problem \(A(m(t))=4\), we want to find out which value of \(t\) causes the entire expression to equal 4.

Steps typically include:

• Substitute \(m(t)\) into \(A(d)\) to form one equation: \(A(m(t)) = 4\).
• Isolate the variable \(t\) by using algebraic techniques like substitution or inversion.
• Solve for \(t\), which is the time when the specific condition (pain level 4) is met.

This method allows us to determine exactly when the drug has administered the desired pain relief, guiding treatment timing and dosage planning effectively. It's a key process in making medical treatment more precise and tailored to individual needs.