Problem 85 - Intermediate Algebra

Question

Ants. Shortly after an explorer ant discovers a food source, a recruitment process begins in which numerous additional ants travel to the source. The number of ants at the source grows exponentially according to the function \(a(t)=1.36\left(\frac{e}{2.5}\right)^{t}\) where \(t\) is the number of minutes since the explorer discovered the food. How many ants will be at the source in 40 minutes?

Step-by-Step Solution

Answer

Approximately 9.11 ants will be at the source in 40 minutes.

Step 1: Identify the Formula

We are given the exponential growth function \( a(t) = 1.36 \left( \frac{e}{2.5} \right)^{t} \). This formula defines how the number of ants at the source grows over time, \(t\), in minutes.

Step 2: Substitute the Given Time

To find out how many ants will be at the source in 40 minutes, substitute \( t = 40 \) into the function: \( a(40) = 1.36 \left( \frac{e}{2.5} \right)^{40} \).

Step 3: Evaluate the Exponential Term

'Evaluate the exponential term by computing \( \left( \frac{e}{2.5} \right)^{40} \). Since \( e \approx 2.71828 \), calculate \( \frac{2.71828}{2.5} \approx 1.087312 \) and then raise it to the power of 40.

Step 4: Multiply by the Constant

Multiply the result obtained from the exponential calculation by 1.36 to find the number of ants: \( a(40) = 1.36 \times (\text{result from Step 3}) \).

Step 5: Calculate the Final Answer

Complete the calculation to find the number of ants: \( a(40) \approx 1.36 \times (6.697296) \). After performing the multiplication, you will have the final answer.

Key Concepts

Understanding Exponential Functions

Exponential functions are a fundamental concept when dealing with growth and decay processes. An exponential function typically has the form \( f(x) = a \, b^{x} \), where \( a \) is a constant and \( b \) is the base. The variable \( x \) often represents time. What's unique about these functions is how rapidly they grow or shrink.
For instance, in the context of our ant problem, the base of our exponential function is \( \frac{e}{2.5} \), showing a continuous rate of growth over time \( t \). Key characteristics of exponential functions include:

• Constant Ratio: Each unit increase in \( x \) results in multiplication of the previous function value by the base \( b \).
• Initial Value: The constant \( a \) represents the initial quantity or amount present when \( x = 0 \).
• Continuous Growth: Exponential functions can model scenarios where growth conditions are consistent over time.

By understanding these elements, anyone can better grasp situations involving exponential changes, such as the rapid recruitment of ants to a food source.

Solving Exponential Equations

Exponential equations are solved by manipulating them algebraically to isolate the variable of interest, often through substitutions or logarithms. Our ant example involves solving \( a(t) = 1.36 \left( \frac{e}{2.5} \right)^{t} \) for a specific \( t \), which is when \( t = 40 \).
Here's how to approach solving an exponential equation:

• Substitution: Start by substituting the given value of \( t \) into the equation.
• Compute the Exponential: Determine the value of the exponential expression \( \left( \frac{e}{2.5} \right)^{t} \) using approximation for \( e \).
• Evaluate the Expression: Multiply the resulting value by the initial constant to find the desired result.

This method allows you to pinpoint specific points along the curve described by the exponential function, such as determining how many ants gather at the food source after a set period.

Applications of Mathematical Modeling in Growth Processes

Mathematical modeling is an essential tool used to simulate real-world situations, making it easier to predict and understand various processes. When it comes to exponential growth, such as in our ant recruitment scenario, mathematical models can provide precise predictions and insights.
Here are some applications and benefits:

• Predicting Future Outcomes: Models can forecast future events based on current data, like predicting how many ants will gather after a certain time.
• Understanding Dynamics: By modeling with exponential functions, the dynamics of population growth, spread of diseases, and financial interest can be accurately captured.
• Supporting Decision Making: Decision-makers can use these models to prepare for expected changes, optimizing processes in various fields.

Ultimately, understanding and utilizing mathematical modeling allows us to simplify and solve complex problems in a structured and scientifically valid manner, enhancing our ability to plan, analyze, and interpret real-world phenomena.