Problem 17 - 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. $$P=5(1.07)^{t}$$

Step-by-Step Solution

Answer

Initial quantity is 5. Growth rate is 7%. Not continuous.

Step 1: Identifying the Initial Quantity

The function given is \( P = 5(1.07)^t \). The initial quantity in an exponential function of the form \( P = a(b)^t \) is represented by \( a \). So, the initial quantity here is 5.

Step 2: Identifying the Growth Rate

In the exponential function \( P = a(b)^t \), the base \( b \) (where \( b > 1 \)) represents the growth factor. Here, \( b = 1.07 \), which means that the growth rate is 7%.

Step 3: Checking for Continuous Growth Rate

Since \( P = 5(1.07)^t \) represents discrete growth with a growth factor of 1.07 per unit time, it does not represent a continuous growth rate. Continuous growth would typically be modeled by an exponential function of the form \( P = ae^{kt} \).

Key Concepts

Initial Quantity

In any exponential function written in the form \( P = a(b)^t \), the initial quantity is a critical component to understand. The initial quantity, denoted by \( a \), is the starting amount or size of the population or item being measured at the time \( t = 0 \). It essentially tells you how much you have before any growth or decay begins to occur.

• Consider the function \( P = 5(1.07)^t \). Here, \( a = 5 \), making the initial quantity 5. This would be the quantity at the beginning or at time zero.
• The initial quantity sets the baseline for calculations. As time progresses, the function calculates the new quantity by adding the effects of growth (or decay) to this starting point.

Understanding initial quantity helps anchor your calculations, giving you a clear point of reference from which changes due to growth or decay can be compared.

Growth Rate

The growth rate is a fundamental aspect of exponential functions that helps to determine how quickly something is expanding or contracting over time. In an exponential function of the form \( P = a(b)^t \), the growth rate is derived from the base \( b \).

• If \( b > 1 \), the function describes growth. The percentage increase is calculated by subtracting 1 from \( b \) and then converting to a percentage. For example, if \( b = 1.07 \), as it is in our function \( P = 5(1.07)^t \), the growth rate is 7% because \( 1.07 - 1 = 0.07 \) or 7%.
• It's essential to differentiate between the growth factor \( b \) and the growth rate. The growth factor \( b \) is the multiplicative factor applied for each period, while the growth rate is the nominal percentage change expressed.
• Growth rates help in predicting future values of the quantity as they tell us year over year (or whatever the time period may be) how much the quantity will expand.

Continuous Growth Rate

Continuous growth rate is a concept where growth or decay happens seamlessly over time rather than at discrete intervals. In real-world applications, this is modelled with the formula \( P = ae^{kt} \), where \( e \) is the base of the natural logarithm, and \( k \) represents the continuous growth rate.

• The function \( P = 5(1.07)^t \) presented in this exercise actually represents discrete growth because it involves a growth factor applied at certain intervals of time indicated by \( t \). Since it uses a base other than \( e \), it is not classified as continuous growth.
• Continuous growth is more commonly used when a process is ongoing and constant, which means it’s better suited for phenomena like population growth in a natural habitat or certain reaction rates in chemistry.
• Understanding whether a process is modeled by continuous or discrete growth helps determine which mathematical tools and formulas are best used for accurate modeling and prediction in various scenarios.