Problem 21 - Applied Calculus

Question

A quantity \(P\) is an exponential function of time \(t .\) Use the given information about the function \(P=P_{b} a^{t}\) to: (a) Find values for the parameters \(a\) and \(P_{0}\). (b) State the initial quantity and the percent rate of growth or decay. \(P_{0} a^{4}=18 \quad\) and \(\quad P_{0} a^{3}=20\)

Step-by-Step Solution

Answer

The initial quantity is approximately 27.42, and there is a 10% rate of decay.

Step 1: Set up the equations

We are given two equations based on different times:1. \( P_{0} a^{3} = 20 \)2. \( P_{0} a^{4} = 18 \)These equations will help us find the values of \( a \) and \( P_{0} \).

Step 2: Divide the equations

To find \( a \), we can divide the second equation by the first equation:\[\frac{P_{0} a^{4}}{P_{0} a^{3}} = \frac{18}{20}\]This simplifies to:\[a = \frac{18}{20} = 0.9\]

Step 3: Solve for \( P_{0} \)

Substitute \( a = 0.9 \) back into one of the original equations to solve for \( P_{0} \), using the first equation:\[P_{0} (0.9)^{3} = 20\]\[P_{0} \times 0.729 = 20\]\[P_{0} = \frac{20}{0.729} \approx 27.42\]

Step 4: Interpret the results

The initial quantity, \( P_{0} \), is approximately 27.42. The base \( a = 0.9 \) implies a decay since it is less than 1. To find the percent rate of decay, subtract \( a \) from 1 and multiply by 100:\[1 - a = 1 - 0.9 = 0.1\]\[0.1 \times 100 = 10\%\]

Step 5: State the findings

The initial quantity is approximately 27.42, and the percent rate of decay is 10%.

Key Concepts

parameters of exponential functions

Exponential functions are essential in modeling scenarios where a quantity grows or decays at a consistent rate over time. The parameters of an exponential function are crucial in determining its behavior. For an exponential function expressed in the form \( P = P_{0} a^{t} \), two primary parameters stand out:

• \( P_{0} \) (Initial Quantity): This represents the starting amount or initial value of the quantity when time \( t = 0 \). It acts as the scale factor for the entire function.


• \( a \) (Base of the Exponential Function): This parameter determines the rate at which the quantity grows or decays over each time unit. When \( a > 1 \), the function represents growth; when \( a < 1 \), it indicates decay.

Understanding these parameters allows us to predict how the quantity changes over time, giving us insights into various natural and economic processes modeled by exponential functions.

initial quantity

The initial quantity, denoted as \( P_{0} \), is fundamental to understanding any exponential function. It signifies the quantity's value at the start—meaning at time \( t = 0 \). Think of it as the launching point from which growth or decay begins.

In our exercise, \( P_{0} \) was calculated to be approximately 27.42. This tells us that initially, the quantity was about 27.42 units. It serves as a baseline and becomes a foundational figure in analyzing the function's future behavior.

Without knowing \( P_{0} \), predicting the exact path or trajectory of the exponential function would be challenging, since it sets the framework for what follows as time progresses.

percent rate of growth or decay

In exponential functions, the percent rate of growth or decay is derived from the base \( a \). This percentage reveals how quickly or slowly a quantity changes over each time unit.

If \( a < 1 \), like in our example where \( a = 0.9 \), we experience a decay. The percent rate of decay is calculated by subtracting \( a \) from 1 and multiplying the result by 100. Here, it was found to be 10%, indicating a 10% reduction each time period.

• Growth: When \( a > 1 \), the quantity grows. The percent increase is \((a - 1) \times 100\)%.


• Decay: When \( a < 1 \), the quantity declines, calculated as \((1 - a) \times 100\)%.

This rate is crucial for understanding the dynamics of the function, helping to anticipate future changes or make informed decisions based on projected trends.