Problem 35 - Algebra Form and Function

Question

The rate \(R\) at which the drug level in the body changes when an intravenous line is used is a function of the amount \(Q\) of the drug in the body. For a certain drug, we have \(R=25-0.08 Q\). The quantity \(Q\) of the drug is a function of time \(t\) with \(Q=\sqrt{t}\) over a fixed time period. Express the rate \(R\) as a function of time \(t\).

Step-by-Step Solution

Answer

Question: Express the rate R as a function of time t, given the expressions R = 25 - 0.08 Q and Q = √t. Answer: R(t) = 25 - 0.08 (√t).

Step 1: Recall the expressions for R(Q) and Q(t)

We are given expressions for the rate of drug level change in the body (\(R\)) as a function of the quantity of the drug (\(Q\)), and the quantity of drug as a function of time (\(t\)): $$ R = 25 - 0.08 Q \quad \text{and} \quad Q = \sqrt{t} $$ Our goal is to find \(R(t)\).

Step 2: Substitute the expression for Q(t) into the equation for R(Q)

Let's substitute the expression for \(Q\) in terms of \(t\) into the equation for \(R\): $$ R(t) = 25 - 0.08 (\sqrt{t}) $$

Step 3: Simplify the resulting expression for R(t)

The expression for \(R(t)\) is already simple, so we have found the answer: $$ R(t) = 25 - 0.08 (\sqrt{t}) $$ Thus, the rate \(R\) is a function of time \(t\) given by \(R(t) = 25 - 0.08 (\sqrt{t})\).

Key Concepts

Rate of Change

The concept of "Rate of Change" is essential in understanding how one quantity varies in relation to another. In this exercise, we focused on how the drug level in the body changes over time. When we refer to the "Rate of Change," we talk about how fast or slow something happens. Mathematically, it's the change in a variable divided by the change in another variable. This helps us understand trends and behaviors of different systems.

• In the given problem, the rate at which the drug level changes is represented by the function \( R = 25 - 0.08Q \).
• This formula shows that the rate decreases as the amount of drug \( Q \) increases.

With a linear decrease in \( R \) as \( Q \) increases, each unit increase in drug amount reduces the rate by 0.08. Recognizing the rate of change helps predict future behaviors and trends in various scenarios.

Function of Time

The notion of a "Function of Time" involves seeing how a quantity changes with time. In our exercise, the drug quantity in the body changes as time passes.Functions of time are particularly useful in real-life applications, like monitoring drug levels. They help in modeling how things evolve.

• The problem provides \( Q = \sqrt{t} \), indicating that the drug amount \( Q \) depends on time \( t \).
• Time is a continuous, independent variable, and \( Q \) is a dependent variable, changing based on \( t \).

Seeing this, we understand how the drug level builds up or depletes over time. Analyzing functions of time is crucial in planning and decision-making in healthcare, physics, and many other fields.

Substitution Method

To solve the exercise, the "Substitution Method" was utilized. This technique involves replacing one variable with another variable's expression.This method is quite common in Algebraic Functions. It simplifies complex problems by making them straightforward step-by-step processes.

• First, we had the equations: \( R = 25 - 0.08Q \) and \( Q = \sqrt{t} \).
• We substituted \( Q = \sqrt{t} \) into \( R = 25 - 0.08Q \) to express \( R \) as a function of \( t \).

Thus, substituting \( Q \), we get \( R(t) = 25 - 0.08\sqrt{t} \). This simplification helps in understanding how the rate of change depends directly on time. The substitution method is a very effective problem-solving technique because it breaks down complex relationships into manageable parts.