Problem 90 - College Algebra with Modeling and Visualization

Question

Solve \(P=P_{0} e^{r\left(t-t_{0}\right)}+5\) for \(t\)

Step-by-Step Solution

Answer

\( t = \frac{\ln \left( \frac{P-5}{P_0} \right)}{r} + t_0 \)

Step 1: Isolate the Exponential Term

Start by subtracting 5 from both sides to isolate the exponential term. This gives:\[ P - 5 = P_0 e^{r(t-t_0)} \]

Step 2: Divide by Initial Condition

Divide both sides by \( P_0 \) to further isolate the exponential term:\[ \frac{P-5}{P_0} = e^{r(t-t_0)} \]

Step 3: Take the Natural Logarithm

Apply the natural logarithm (\(\ln\)) to both sides to remove the exponential:\[ \ln \left( \frac{P-5}{P_0} \right) = r(t-t_0) \]

Step 4: Solve for Time Variable

Finally, solve for \( t \) by dividing both sides by \( r \) and adding \( t_0 \):\[ t = \frac{\ln \left( \frac{P-5}{P_0} \right)}{r} + t_0 \]

Key Concepts

Natural Logarithms

Natural logarithms are a special type of logarithm, where the base is the constant \( e \), approximately equal to 2.71828. They are denoted as \( \ln \). When dealing with exponential equations, natural logarithms play a critical role. Why? Because they can effectively "undo" an exponentiation of the base \( e \).

This property is extremely useful in solving equations where the unknown variable is in an exponent. For instance, if you have an equation \( e^x = y \), taking the natural logarithm of both sides gives \( x = \ln(y) \). This is because \( \ln(e^x) = x \). The ability of natural logarithms to simplify such equations makes them indispensable in mathematics.

• They help in linearizing exponential relationships.
• They are used extensively in calculus for integration and differentiation.

In summary, natural logarithms are a key tool for handling and solving exponential equations.

Isolating Variables

Isolating variables is a fundamental technique in solving equations. It involves manipulating the equation to get the variable of interest on one side by itself. This is a crucial step because it allows you to clearly understand the relationship and find the value of the variable you're looking to solve.

When dealing with exponential equations, isolating the variable often requires:

• Subtracting constants from both sides.
• Dividing all terms by the coefficient of the variable or any multiplicative value attached to it.

In the context of the provided exercise, isolating the variable \( t \), involved steps like subtracting 5 and dividing by \( P_0 \). By breaking the problem into smaller, more manageable pieces, each step brought us closer to isolating \( t \) perfectly.

This methodical approach can apply not only to exponential equations but to any mathematical problem, serving as a universal problem-solving strategy.

Growth Models

Growth models are mathematical representations used to describe how a quantity increases over time. They are often expressed through exponential functions due to their natural fit to growth phenomena, such as population growth and interest accumulation.

In the context of this exercise, an exponential growth model helps predict the behavior of the variable \( P \). Growth models typically depend on parameters like:

• Initial Condition (e.g., \( P_0 \) in the formula).
• Growth Rate (e.g., \( r \) in the formula).
• Time (e.g., \( t \) in the formula).

Exponential models assume the rate of growth is proportional to the current size. This causes the quantity to grow at an accelerating rate, characteristic of exponential functions. Understanding growth models allows for predicting how changes to the initial conditions or growth rate affects the outcome over time.
By learning how to solve equations related to growth models, you can gain insights into diverse fields such as finance, biology, and environmental science.