Problem 124 - Elementary and Intermediate Algebra

Question

Use the formula \(P=P_{0} e^{r t}\) to verify that \(P\) will be three times as large as \(P_{0}\) when \(t=\frac{\ln 3}{r}\)

Step-by-Step Solution

Answer

Yes, when \( t = \frac{\ln 3}{r} \), \( P \) is three times \( P_0 \).

Step 1: Understand the Given Formula

We have the exponential growth function given by \( P = P_0 e^{r t} \), where \( P \) is the future value, \( P_0 \) is the initial value, \( r \) is the rate of growth, and \( t \) is time. We need to verify if \( P \) is three times \( P_0 \) for \( t = \frac{\ln 3}{r} \).

Step 2: Substitute the Expression for t

Substitute \( t = \frac{\ln 3}{r} \) into the formula. This gives us:\[ P = P_0 e^{r \left( \frac{\ln 3}{r} \right)} = P_0 e^{\ln 3} \]

Step 3: Simplify the Exponential Expression

Using the property of exponents \( e^{\ln a} = a \), we simplify:\[ P = P_0 \cdot 3 \] This means \( P \) becomes three times \( P_0 \).

Step 4: Verify the Relationship

Since \( P = 3P_0 \) holds true with the given substitution, we have verified that the formula confirms the condition.

Key Concepts

Exponential Function

An exponential function is a mathematical expression often used to model situations where something grows or decays at a constant percentage rate. At its core, the function is expressed as \( P = P_0 e^{r t} \). Here, \( P \) is the quantity of interest over time, \( P_0 \) is the initial amount or starting value, \( r \) represents the rate, and \( t \) is time. The base of the exponential function, \( e \) (approximately 2.718), is a special irrational number known as Euler's number. It is used because it naturally describes growth processes.Exponential functions are crucial in fields such as finance, biology, and physics because they capture real-world phenomena like population growth, interest accrual, and radioactive decay. For example, if you deposit money into a bank account with continuous compounding interest, the exponential function helps you calculate how much your money will grow over time. Similarly, it's used to predict how a population might increase when living conditions are ideal and resources are unlimited.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the inverse of the exponential function with base \( e \). It helps solve equations where the unknown is in the exponent, bringing the variable down to a linear form that is easier to handle. For instance, when dealing with the formula \( P = P_0 e^{r t} \), the logarithm can be handy if we need to find the duration \( t \) required for a specific growth, where \( t \) is isolated as \( t = \frac{\ln(P/P_0)}{r} \).The natural logarithm tells us the power to which \( e \) must be raised to achieve the same number. This property is utilized to simplify expressions involving exponentials. In practice, natural logarithms are frequently used in contexts involving growth processes. They encapsulate many complex multiplicative relationships found in the real world into sums, which are simpler to analyze and understand.

Rate of Growth

The rate of growth \( r \) in an exponential function determines how quickly the variable of interest increases over time. This rate is expressed as a positive decimal or percentage, indicating the constant proportion by which a value increases per unit of time. For example, if \( r = 0.05 \), this would imply a 5% growth rate per time period. In the context of the given formula \( P = P_0 e^{r t} \), \( r \) directly impacts the speed and scale of growth. A higher \( r \) means faster growth and vice versa.Understanding the rate of growth is essential for predicting future values in various contexts. From biological systems like population increases to financial systems such as investment growth, the rate of growth provides insights into how dynamic or static a system might be over time. In solving problems, knowing \( r \) allows us to forecast and strategize based on expected future outcomes.