Problem 123 - Elementary and Intermediate Algebra

Question

Use the formula \(P=P_{0} e^{r t}\) to verify that \(P\) will be twice \(P_{0}\) when \(t=\frac{\ln 2}{r}\)

Step-by-Step Solution

Answer

When \(t = \frac{ln(2)}{r}\), \(P\) is twice \(P_{0}\).

Step 1: Understanding the given equation

The formula given is \(P = P_{0} e^{r t}\), which describes an exponential growth process where \(P_{0}\) is the initial amount, \(r\) is the growth rate, and \(t\) is the time. Our goal is to find \(t\) such that \(P\) becomes twice \(P_{0}\), or \(P = 2P_{0}\).

Step 2: Setting up the equation to solve for t

Substitute \(P = 2P_{0}\) into the equation \(P = P_{0} e^{r t}\). This gives us the equation \(2P_{0} = P_{0} e^{r t}\).

Step 3: Cancel out common terms

Since \(P_{0}\) appears on both sides of the equation and is non-zero, we can divide both sides by \(P_{0}\) to simplify the equation to \(2 = e^{r t}\).

Step 4: Solving for the exponent rt

Take the natural logarithm of both sides to solve for \(rt\). This gives \(ln(2) = ln(e^{r t}) = r t\).

Step 5: Isolating t

To solve for \(t\), divide both sides by \(r\), resulting in \(t = \frac{ln(2)}{r}\).

Step 6: Verification

Substitute \(t = \frac{ln(2)}{r}\) back into the original equation to check if \(P = 2P_{0}\). We have \(P = P_{0} e^{r \left(\frac{ln(2)}{r}\right)} = P_{0} e^{ln(2)} = P_{0} \cdot 2 = 2P_{0}\), confirming that the equation holds.

Key Concepts

Understanding the Growth Rate

In any process of change over time, such as population increase or financial investment, we rely on the concept of growth rate. Growth rate signifies the multiplier applied to a quantity per unit of time.

In exponential growth, which is a specific type of growth, the change rate becomes constant relative to the existing quantity. The formula used for exponential growth is denoted as \( P = P_{0} e^{r t} \), where \( r \) represents the growth rate. This rate remains consistent and proportional through time.

Key points to consider about growth rate:

• It's usually a percentage, like 5% annual growth, translating to \( r = 0.05 \).
• It dictates how quickly the quantity increases — a higher \( r \) means faster growth.
• It helps predict future values based on current data.

Understanding growth rates helps with more precise forecasts and strategies, allowing better decision-making in various fields.

Decoding the Natural Logarithm

The natural logarithm is an essential mathematical concept, denoted as \( \ln \). It plays a crucial role in solving equations involving exponential growth. A logarithm essentially gives the power to which a number must be raised to obtain another number.

The natural logarithm specifically uses the base 'e', which is approximately 2.718. When you see \( \ln(x) \), it asks "to what power must \( e \) be raised in order to yield \( x \)?"

Significance of the natural logarithm:

• It helps transition from multiplicative to additive processes, simplifying calculations.
• In exponential growth problems, it enables solving for time or rate by isolating variable terms.
• It is widely applied in various scientific fields because of its natural connection to growth and decay processes.

Natural logarithms are an invaluable tool for simplifying and solving exponential equations.

Exploring the Exponential Function

The exponential function is a mathematical function that represents rapid growth or decay over time. In the formula \( P = P_{0} e^{r t} \), \( e^{r t} \) is the exponential function where every unit of time increases the quantity by a consistent percentage.

Properties of the exponential function include:

• The base \( e \) is an irrational constant approximately equal to 2.718.
• Exponential functions can model real-world processes like population growth, radioactive decay, and interest compounding.
• They exhibit constant relative growth, meaning the rate of growth is proportional to the current amount.

Understanding exponential functions helps analyze phenomena that change, not just linearly, but by percentage increases. They're crucial in sciences, economics, and even computer science to model natural and man-made systems over time.