Problem 439 - Precalculus

Question

A logistic moden by the equation \(P(t)=\frac{90}{1+5 e^{-0.42 t}}\) To the nearest hundredth, for what value of \(t\) does \(P(t)=45 ?\)

Step-by-Step Solution

Answer

The value of \( t \) is approximately 3.76.

Step 1: Set Up the Equation

The given logistic model is \( P(t) = \frac{90}{1+5e^{-0.42t}} \). We are asked to find the value of \( t \) when \( P(t) = 45 \). Therefore, we need to solve the equation \( \frac{90}{1+5e^{-0.42t}} = 45 \).

Step 2: Rearrange the Equation

Start by simplifying the equation: \( 45 = \frac{90}{1+5e^{-0.42t}} \). Multiply both sides by \( 1+5e^{-0.42t} \) to eliminate the fraction: \( 45(1+5e^{-0.42t}) = 90 \). This expands to \( 45 + 225e^{-0.42t} = 90 \).

Step 3: Simplify to Isolate the Exponential Term

Subtract 45 from both sides to isolate the exponential term: \( 225e^{-0.42t} = 90 - 45 \). This simplifies to \( 225e^{-0.42t} = 45 \).

Step 4: Solve for the Exponential Expression

Divide both sides by 225: \( e^{-0.42t} = \frac{1}{5} \).

Step 5: Apply the Natural Logarithm

To solve for \( t \), take the natural logarithm of both sides: \( \ln(e^{-0.42t}) = \ln\left(\frac{1}{5}\right) \). Using the property \( \ln(e^x) = x \), this simplifies to \( -0.42t = \ln\left(\frac{1}{5}\right) \).

Step 6: Solve for t

Divide both sides by -0.42 to find \( t \): \( t = \frac{\ln\left(\frac{1}{5}\right)}{-0.42} \). Calculate this expression using a calculator to find \( t \approx 3.76 \).

Key Concepts

Solving Equations

Solving equations is the core process in finding unknown values within mathematical expressions. In the exercise above, our task was to find the time, \( t \), when the population reaches 45 using the given logistic model equation: \( P(t) = \frac{90}{1 + 5e^{-0.42t}} \). To solve this:

• First, we equate the function to 45: \( \frac{90}{1+5e^{-0.42t}} = 45 \).
• Next, eliminate the fraction by multiplying both sides by the denominator: \( 45(1+5e^{-0.42t}) = 90 \).
• Expand and simplify to isolate terms with \( t \): \( 225e^{-0.42t} = 45 \).
• Finally, divide both sides by 225 to isolate the exponential : \( e^{-0.42t} = \frac{1}{5} \).

This process of rearranging and simplifying is essential in solving many types of equations. It requires careful manipulation to isolate the unknown and eventually find its value.

Natural Logarithm

The natural logarithm (usually written as \( \ln(x) \)) is a powerful tool that helps us solve equations involving exponential expressions. Natural logarithms have a base \( e \), where \( e \approx 2.718 \). They possess properties that simplify the process of solving for unknowns:

• One key property is \( \ln(e^x) = x \), which allows us to simplify expressions.

In our logistic model example:

• After isolating the exponential term \( e^{-0.42t} = \frac{1}{5} \), we took the natural logarithm of both sides to solve for the unknown \( t \).
• This gave us \( \ln(e^{-0.42t}) = \ln\left(\frac{1}{5}\right) \).
• Simplifying, we used the property \( \ln(e^x) = x \) to rewrite the left side as \( -0.42t \).
• By doing so, we transformed an exponential equation into a linear one, making it much easier to solve for \( t \).

Exponential Functions

Exponential functions are a type of function where the variable appears in the exponent. These functions often describe growth and decay processes in real-life scenarios. In our problem, the exponential function is a key part of the logistic model:

• The model \( P(t) = \frac{90}{1 + 5e^{-0.42t}} \) involves an exponential expression \( e^{-0.42t} \).
• This expression illustrates how the population grows over time, gradually approaching a maximum limit as \( t \) increases.

Exponential functions have distinctive properties:

• Their base, \( e \), is unique due to its constant rate of growth, making it ideal for modeling continuous natural processes.
• They are characterized by rapid growth or decay, with the base number raised to the power of the variable.

Understanding the nature of exponential functions helps interpret models like the logistic equation, predicting behaviors accurately and providing insights into dynamics such as population growth, radioactive decay, or interest accumulation.