Figure 1.19 shows the concentration of a drug in the body, f(t) = a −be^kt , over time, t: What does the constant, a, represent in this context?
(b) What is the sign of the constant a?
(c) What is the relation between the constants a and b?
(d) What is the sign of the constant b?

To understand the context of the given function \( f(t) = a - b e^{kt} \), which represents the concentration of a drug in the body over time \( t \), let's address each part of the question.

### (a) What does the constant, \( a \), represent in this context?

The constant \( a \) represents the **maximum possible concentration of the drug in the body** as \( t \to \infty \). This is because the term \( e^{kt} \) will approach zero over time if \( k < 0 \) (which is typical for decay processes). Therefore, as time increases, the concentration approaches the constant value \( a \), which signifies the maximum steady-state concentration of the drug.

### (b) What is the sign of the constant \( a \)?

The constant \( a \) should be **positive** because drug concentrations are generally non-negative, and \( a \) represents the maximum concentration the drug can reach over time. A negative concentration would not be physically meaningful in this context.

### (c) What is the relation between the constants \( a \) and \( b \)?

The relation between \( a \) and \( b \) is such that:
- \( b \) determines how far the initial concentration is from the maximum concentration \( a \). Specifically, when \( t = 0 \), the function simplifies to \( f(0) = a - b e^{0} = a - b \).
- Therefore, \( b \) represents the difference between the initial concentration and the maximum concentration. In other words, \( f(0) = a - b \), meaning that the initial concentration is \( a - b \), and \( b \) quantifies the initial "distance" from \( a \).

### (d) What is the sign of the constant \( b \)?

The constant \( b \) must be **positive** because it represents a difference in concentrations. Since \( a - b \) gives the initial concentration, \( b \) must be positive to ensure that the initial concentration is less than the maximum concentration \( a \).

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Would you like any more details or have any specific questions?

### Related Questions:
1. What happens to the concentration \( f(t) \) as \( t \to 0 \)?
2. How does the constant \( k \) affect the rate at which the drug concentration approaches \( a \)?
3. If \( b \) were negative, what would the initial concentration \( f(0) \) look like?
4. What does the derivative of \( f(t) \) tell us about the rate of change of drug concentration over time?
5. How can we determine the half-life of the drug concentration in this model?

### Tip:
The value of \( k \) is crucial in determining how fast the drug's concentration decreases (or increases) over time. If \( k < 0 \), it usually models a decaying process (drug elimination).

Figure 1.19 shows the concentration of a drug in the body, f(t) = a − be^kt, over time, t: What does the constant, a, represent in this context? (b) What is the sign of the constant a? (c) What is the relation between the constants a and b? (d) What is the sign of the constant b?

Explore how the constants a and b represent the drug concentration model f(t) = a − be^kt. Learn how a relates to the maximum concentration, and how b determines the initial drug concentration difference. Understand key concepts like exponential decay and limits, ideal for students in Grades 10-12.

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