Problem 85 - Elementary and Intermediate Algebra

Question

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 63,627 ants will be at the source in 40 minutes.

Step 1: Understand the Problem

We need to find the number of ants at the food source after 40 minutes. The number of ants is given by the function \(a(t)=1.36\left(\frac{e}{2.5}\right)^{t}\) where \(t\) is time in minutes.

Step 2: Substitute the Time Variable

Substitute \(t = 40\) into the function to find \(a(40)\). This gives us the expression: \[a(40) = 1.36 \left(\frac{e}{2.5}\right)^{40}\]

Step 3: Calculate the Base of the Exponent

Calculate the value of \(\frac{e}{2.5}\). Given that \( e \approx 2.718\), \[ \frac{2.718}{2.5} \approx 1.0872 \]

Step 4: Raise the Base to the Power of 40

Raise \(1.0872\) to the power of 40. Using a calculator, this results in approximately 46,783.

Step 5: Multiply by Initial Value

Multiply the result from the previous step by 1.36 to get the total number of ants. \[ 1.36 \times 46783 \approx 63,627 \]

Key Concepts

Algebra and Functions

Algebra is a fundamental part of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations or describe relationships. In the context of our problem, algebra helps us understand how the function describing the number of ants operates over time. When we use a function like \(a(t)=1.36\left(\frac{e}{2.5}\right)^{t}\), we are using algebra to express how the number of ants changes with time.

• This equation is a function of \(t\), representing time in minutes.
• The constant \(1.36\) is the initial multiplier, indicating that the process starts with a small, fixed number of ants arriving immediately after discovery.

By substituting different values for \(t\), we can predict how many ants will be at the food source at any given minute past the initial discovery. Algebra allows us to manipulate and understand these equations, providing a clear method to solve real-world problems like this one.

Understanding Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to the power of a variable, often denoted as \(a(t) = b^{t}\). In our problem, such a function reflects rapid growth over time. The base of our exponential function is \(\frac{e}{2.5}\), where \(e\) is Euler's number, approximately 2.718, a fundamental constant in mathematics.

• Because the base is greater than 1 (i.e., about 1.0872), the function describes growth.
• This means that as time increases, the number of ants grows significantly.

Exponential growth is often seen in processes that increase in size rapidly, such as populations or compounds interest. This growth type is key in understanding how a small beginning can swell into a much larger result in relatively short periods. Calculating with exponential functions requires understanding of powers and how quickly they can skyrocket a result.

Problem Solving Techniques

When tackling a math problem, especially with exponential growth, a structured approach can significantly ease the process. In this particular ant problem, systematic steps lead us to the solution.

• Identify What is Asked: Start by understanding the question; how many ants after 40 minutes?
• Substitute Known Variables: Replace \(t\) with 40 in the function \(a(t)\) to get \(a(40)\).
• Simplify Incrementally: Calculate intermediate values such as the base \(\frac{e}{2.5}\) and raise it to the power needed.
• Final Calculation: Multiply results together, keeping track of operations.

Each step logically follows from the previous one and should be done carefully, ensuring accuracy. This disciplined approach is instrumental in executing solutions correctly and can be applied to numerous other problems involving exponential functions or algebraic equations.