Problem 18 - Applied Calculus

Question

The functions in Problems \(17-20\) represent exponential growth or decay. What is the initial quantity? What is the growth rate? State if the growth rate is continuous. $$P=7.7(0.92)^{t}$$

Step-by-Step Solution

Answer

Initial quantity is 7.7; decay rate is 8%; rate is discrete.

Step 1: Identify the Initial Quantity

The initial quantity in an exponential function of the form \( P = A (r)^t \) is the coefficient \( A \), which represents the starting value when \( t = 0 \). In the given function \( P = 7.7(0.92)^t \), the initial quantity is 7.7.

Step 2: Determine the Base for Growth or Decay

In the expression \( P = 7.7(0.92)^t \), the base of the exponential function is 0.92. This tells us whether the function represents growth or decay. Since 0.92 is less than 1, this indicates the function represents exponential decay.

Step 3: Calculate the Decay Rate

The decay rate can be found by subtracting the base from 1. The formula for finding the decay rate is \( 1 - r \). Here, \( r = 0.92 \), so the decay rate is \( 1 - 0.92 = 0.08 \). This means the quantity decreases by 8% each time period.

Step 4: Check for Continuous or Discrete Rate

Since the problem uses a specific base \( 0.92 \) for decay, it represents a discrete interval rather than continuous compounding. Continuous growth or decay would require using a base \( e^k \), where \( k \) is a continuous rate.

Key Concepts

Initial Quantity

The concept of "Initial Quantity" is essential when dealing with exponential functions. This specifies the starting point or the beginning amount of the substance before any decay or growth occurs. In mathematical terms, for an exponential function of the form \( P = A(r)^t \), the initial quantity is represented by \( A \). It is the amount present when time \( t = 0 \). Think of it as the foundation or the starting level from where the changes commence. In the equation \( P = 7.7(0.92)^t \), the initial quantity is 7.7. It signifies that at time zero, the quantity starts at 7.7. This initial value is vital for understanding how much you begin with before considering any factors like the growth or decay rates. Recognizing the initial quantity helps in predicting future quantities based on the rate at which they grow or decay. This concept lays the groundwork for further calculations and is central to solving exponential problems.

Decay Rate

The "Decay Rate" is an indicator of how quickly a quantity reduces over time in an exponential decay situation. Unlike growth, where quantities increase, decay involves a consistent percentage reduction. The decay rate is derived from the base \( r \) of the exponential function \( P = A(r)^t \). For exponential decay, this base is a number between 0 and 1. It reflects the stability or instability of the process. To determine the decay rate, you subtract the base \( r \) from 1:

• Decay Rate = \( 1 - r \)

In our problem, with the function \( P = 7.7(0.92)^t \), the base is 0.92. Thus, the decay rate is \( 1 - 0.92 = 0.08 \). This means an 8% reduction occurs in each time period. Points to consider:

• A higher decay rate implies a faster reduction.
• A lower decay rate indicates a more gradual decrease.

Understanding the decay rate helps us predict the future number of items or quantities after certain time intervals.

Continuous vs. Discrete Growth

Understanding whether growth or decay is continuous or discrete is valuable in interpreting exponential functions. Here, we are focusing on how continuous and discrete changes differ. **Discrete Growth or Decay:** This occurs when changes happen in specific increments or intervals. In the provided function \( P = 7.7(0.92)^t \), the decay uses a constant base of 0.92. This base implies that decay happens at separate, defined intervals. **Continuous Growth or Decay:** Continuous change, on the other hand, is smooth and ongoing at every moment. It utilizes the natural base \( e \) and is expressed as \( e^{kt} \), where \( k \) indicates a continuous rate. In practical terms:

• Discrete processes are like a bucket being filled with water by constant drips every minute.
• Continuous processes can be imagined as water flowing from a tap steadily.

In our context, since decay is not governed by the base \( e \), it is discrete. This simply means quantities reduce at distinct regular intervals rather than continuously. Understanding these distinctions helps one decide the appropriate model depending on the situation and accurately predict future scenarios.