Problem 19 - 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. $$P=15 e^{-0.06 t}$$

Step-by-Step Solution

Answer

Initial quantity: 15; decay rate: 0.06 (continuous).

Step 1: Identify the Initial Quantity

The function given is \( P = 15 e^{-0.06 t} \). The initial quantity in an exponential function can be identified as the coefficient of the exponential term. Here, the initial quantity is \( 15 \) since it is the constant multiplier in front of the exponential function.

Step 2: Determine the Exponential Decay Rate

In the function \( P = 15 e^{-0.06 t} \), the exponent has a negative sign, indicating decay. The decay rate is \( 0.06 \), which is the absolute value of the coefficient accompanying \( t \) in the exponential function, \( -0.06 \).

Step 3: Determine if the Decay Rate is Continuous

The function \( P = 15 e^{-0.06 t} \) is in the form \( P = P_0 e^{rt} \), where \( P_0 \) is the initial quantity and \( r \) is the continuous decay rate. Since \( r = -0.06 \) appears directly as the exponent of \( e \), the decay rate, 0.06, is already in continuous form.

Key Concepts

Initial Quantity

Every exponential function can be easily identified by its form, which follows the pattern \( P(t) = P_0 e^{rt} \), where \( P_0 \) represents the initial quantity. This initial quantity is simply the starting value of the function when \( t=0 \). It acts as the baseline from which growth or decay processes are measured over time.

In our function, \( P = 15 e^{-0.06 t} \), the initial quantity is clearly visible as the coefficient, \( 15 \). This number signifies the amount present at the beginning before any changes due to time are factored in.

Understanding this initial quantity is crucial because it lays the foundation for all future calculations and gives context to the changes described by the exponential function. It is essentially the platform from which all growth or decay activity begins.

Growth Rate

The growth rate in an exponential function indicates how quickly the quantity is increasing or decreasing over time. When looking at the function \( P = 15 e^{-0.06 t} \), the exponential part \( e^{-0.06 t} \) reveals our growth or decay rate. This portion of the formula determines the speed of change, whether increasing or decreasing.

The value coefficient in the exponent, in this case \(-0.06\), is critical to understanding whether the function is growing or decaying. If the rate is negative, as here, it signifies **exponential decay**. Conversely, a positive rate would indicate growth. In simpler terms, this coefficient tells us by what percentage the initial quantity changes over each time unit.

Additionally, the rate in our function is continuous, meaning the changes occur without sudden jumps, offering a smooth transition over time.

Exponential Decay

Exponential decay is the process where quantities reduce rapidly at first, and then slowly over time. This type of function is common in real-life scenarios such as cooling of substances, depreciation of values, and radioactive decay. Each of these processes showcase how exponential decay functions are useful in predicting future states based on current data.

In the function \( P = 15 e^{-0.06 t} \), the decay rate is \( 0.06 \), indicating a consistent and continuous decline over time by that fractional amount. The negative sign in front of the rate confirms it is a decay. Such decay implies each unit of time reduces the quantity more when it's higher, but less and less significantly as the quantity diminishes.

Grasping the concept of exponential decay helps in understanding the long-term behavior of phenomena represented by these functions and reflects natural processes of diminishing.