Problem 27 - Applied Calculus

Question

In Problems \(25-28,\) put the functions in the form \(P=P_{0} e^{k t}\). $$P=174(0.9)^{t}$$

Step-by-Step Solution

Answer

The function in the form \( P = P_0 e^{kt} \) is \( P = 174 e^{t \ln(0.9)} \).

Step 1: Identify the Given Function

The given function is \( P = 174(0.9)^t \). This function needs to be converted into the form \( P = P_0 e^{kt} \).

Step 2: Compare with the Exponential Form

In the form \( P = P_0 e^{kt} \), \( P_0 \) is the initial value, and \( e^{kt} \) represents the exponential growth or decay. Here, \( 174 \) corresponds to \( P_0 \), so \( P_0 = 174 \).

Step 3: Convert the Base to Euler's Number

To convert the function into the desired form, replace the base of the exponent from \( 0.9^t \) to an exponential with base \( e \). Recall that any number can be written as a power of \( e \):\[0.9 = e^{\ln(0.9)}\]

Step 4: Substitute and Simplify

Substitute \( 0.9 \) with \( e^{\ln(0.9)} \) in the original equation:\[P = 174(e^{\ln(0.9)})^t = 174 e^{t \ln(0.9)}\]Here, \( kt = t \ln(0.9) \), so \( k = \ln(0.9) \).

Step 5: Write the Final Expression

Now the function is in the form \( P = P_0 e^{kt} \), where \( P_0 = 174 \) and \( k = \ln(0.9) \). The final expression is:\[P = 174 e^{t \ln(0.9)}\]

Key Concepts

Exponential Growth and Decay

Exponential functions are used to model situations where quantities grow or decay at a rate proportional to their size. This is seen in continuous growth like populations or continuous decay like radioactive substances. For the expression \( P = P_0 e^{kt} \), the term \( k \) measures the rate. If \( k > 0 \), it's exponential growth; if \( k < 0 \), it's exponential decay.
For example, in the function \( P = 174 e^{t \, \ln(0.9)} \), \( k = \ln(0.9) \). Since \( \ln(0.9) \) is negative, this indicates decay. The value \( 0.9 \) represents a reduction, meaning the original quantity shrinks by 10% every unit of time.
It's critical to understand that the base transition from \( 0.9^t \) to \( e^{t \ln(0.9)} \) maintains the decay nature but changes the base for easier mathematical manipulation.

Base Conversion

Base conversion involves changing the base of an exponential expression while retaining the same value. In exponential equations, this allows us to express functions in a standard form that uses Euler's number, \( e \).
For any number, say \( a \), we can convert it to the form \( e^{\ln(a)} \). For instance, in \( 0.9^t \), this becomes \( (e^{\ln(0.9)})^t \), resulting in \( e^{t \ln(0.9)} \).
The conversion makes handling exponential equations more manageable since many mathematical models and solutions rely on the natural exponential function with base \( e \). This simplification improves calculation and interpretation while maintaining equivalence with the original expression.

Euler's Number

Euler's number, commonly known as \( e \), is approximately 2.71828. It is a mathematical constant used as the base of natural logarithms. Euler's number is significant in mathematics due to its natural growth and decay properties, which appear in various fields like calculus, complex numbers, and natural sciences.
By converting an exponential function to have a base \( e \), such as moving from \( 0.9^t \) to \( e^{t \ln(0.9)} \), we create a standard form widely used for differential equations and integrals.
The unique properties of \( e \) make it the preferred base in continuous growth or decay models, simplifying calculus operations like differentiation and integration related to growth processes.