Problem 17 - Applied Calculus

Question

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

The initial quantity is 5, the growth rate is 7%, and it is not continuous.

Step 1: Identify the formula components

The given formula is \( P = 5(1.07)^t \), which represents exponential growth or decay. Here, 5 is the initial quantity, and 1.07 is the base of the exponential expression.

Step 2: Examine the initial quantity

The initial quantity in the formula \( P = 5(1.07)^t \) is 5. This is the value of \( P \) when \( t = 0 \).

Step 3: Determine growth or decay

Since the base of the exponent, 1.07, is greater than 1, this suggests exponential growth.

Step 4: Calculate the growth rate

To find the growth rate, subtract 1 from the base: 1.07 - 1 = 0.07, or 7%. This is the growth rate per time period.

Step 5: Assess if growth rate is continuous

The given equation uses a discrete growth rate formula, so the growth rate in this context is not continuous, but rather discrete per time unit.

Key Concepts

Understanding Initial Quantity

In the context of exponential growth, the initial quantity acts as the starting point of our calculations. In our example with the formula \( P = 5(1.07)^t \), the initial quantity is 5. This represents the original amount before any growth happens. Think of it as the base value:

• The initial quantity is crucial because it sets the stage for future growth.
• It is often the value of \( P \) when \( t = 0 \). For any exponential function, this time zero value gives us a snapshot of where we begin.

It's important to identify this correctly because it lays the foundation for how the growth unfolds over time.

Calculating the Growth Rate

The growth rate is a key element in understanding how quickly or slowly an exponential process unfolds over time. In our formula \( P = 5(1.07)^t \), the expression \( 1.07 \) is the base of the exponent:

• To uncover the growth rate, we subtract 1 from the base. Here, \( 1.07 - 1 = 0.07 \), which translates into a 7% growth rate.
• This number is a percentage that tells us how much the quantity increases in each time unit, reflecting exponential augmentation or decay.

Understanding this rate is important for predicting how fast a process will grow, enabling more accurate forecasting and planning.

Exploring Discrete Growth

The term 'discrete growth' refers to situations where growth happens at specific intervals. Unlike continuous growth, which happens smoothly and constantly over time, discrete growth takes place in separate steps. In our equation \( P = 5(1.07)^t \), growth occurs per discrete time unit:

• Each time period represents a distinct growth step, often seen in scenarios like yearly population growth or financial interest compounding yearly.
• This method allows for clear measurement and assessment at regular intervals.

Understanding discrete growth helps in analyzing situations where changes occur at intervals, allowing for strategic decision-making based on periodic evaluations.