Problem 23 - 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. \(P=1600\) when \(t=3\) and \(P=1000\) when \(t=1\)

Step-by-Step Solution

Answer

Parameters are \(P_0 \approx 790.57\) and \(a \approx 1.2649\). Initial quantity is approximately 790.57; growth rate is about 26.49%.

Step 1: Set Up the System of Equations

We know the form of the exponential function is \(P = P_0 a^t\). We have two data points: \(P = 1600\) when \(t = 3\) and \(P = 1000\) when \(t = 1\). Substituting these into the equation gives us two equations: \(1600 = P_0 a^3\) and \(1000 = P_0 a^1\).

Step 2: Eliminate \(P_0\) by Division

To eliminate \(P_0\), divide the first equation by the second: \(\frac{1600}{1000} = \frac{P_0 a^3}{P_0 a^1}\), which simplifies to \(1.6 = a^2\).

Step 3: Solve for \(a\)

Taking the square root of both sides of \(1.6 = a^2\) gives us \(a = \sqrt{1.6}\). Calculate \(\sqrt{1.6}\) to find \(a \approx 1.2649\).

Step 4: Solve for \(P_0\)

Use the second equation \(1000 = P_0 a\). Substitute \(a \approx 1.2649\) into this equation to find \(P_0 = \frac{1000}{1.2649}\). This calculation gives \(P_0 \approx 790.57\).

Step 5: Determine the Initial Quantity

The initial quantity \(P_0\) is the value of \(P\) when \(t = 0\). From our calculation \(P_0 \approx 790.57\).

Step 6: Determine the Percent Rate of Growth

The growth factor \(a \approx 1.2649\) represents a \((1.2649 - 1) \times 100\%\) growth per unit of time. Thus, the percent rate of growth is approximately \(26.49\%\).

Key Concepts

Initial Quantity

In any exponential growth scenario, the **initial quantity** is a crucial component. It represents the starting value before any growth has occurred. In the context of our exponential function \(P = P_0 a^t\), the initial quantity is denoted by \(P_0\). This value tells us the amount present at time \(t = 0\), serving as a baseline for future growth or decay.

Here, we found this initial quantity by solving for \(P_0\) using the information that \(P = 1000\) when \(t = 1\). Once we computed the growth factor, we calculated \(P_0 = \frac{1000}{1.2649} \approx 790.57\). This means that at the start, the quantity was approximately 790.57.

Growth Factor

The **growth factor** is key to determining how much the quantity grows over time. It is represented by the variable \(a\) in the exponential function \(P = P_0 a^t\). This factor shows how the quantity changes per unit of time. If \(a > 1\), it indicates growth, while \(a < 1\) indicates decay.

In our equation, we found that \(a\) is approximately 1.2649 by setting up and solving a system of equations. Specifically, we divided two equations derived from the given data points to isolate \(a\) and solved \(1.6 = a^2\), leading to \(a = \sqrt{1.6} \approx 1.2649\). The growth factor of 1.2649 indicates that our quantity grows by a factor of 1.2649 each time period.

Percent Rate of Growth

Understanding the **percent rate of growth** is essential in interpreting how quickly a quantity is increasing over time. It allows us to express the growth factor in percentage terms, which tend to be more intuitive. The percent rate is calculated as \((a - 1) \times 100\%\), where \(a\) is the growth factor.

For our problem, with a growth factor of approximately 1.2649, the percent rate of growth is \((1.2649 - 1) \times 100\% \approx 26.49\%\). This means the quantity grows by 26.49% per time unit.

Exponential Function

An **exponential function** models situations where growth is proportional to the current value. It is expressed in the form of \(P = P_0 a^t\) and is characterized by a consistent percentage change over regular intervals. These functions are widespread in modeling natural phenomena and financial growth.

The core components of an exponential function include:

• **Initial Quantity \(P_0\)**: The starting point of the function when \(t = 0\).
• **Growth Factor \(a\)**: Indicates how much the quantity multiplies per time period.
• **Time \(t\)**: The independent variable representing the passage of time.

By analyzing these components, one can understand how the quantity changes over time, helping to predict future outcomes or reconstruct past events.