Problem 55 - Applied Calculus

Question

BEHAVIORAL SCIENCE: Mazes In a psychology experiment, rats were placed in a T-maze, and the proportion of rats who required more than \(t\) seconds to reach the end was \(\int_{t}^{\infty} 0.05 e^{-0.05 s} d s .\) Use this formula to find the proportion of rats who required more than 10 seconds.

Step-by-Step Solution

Answer

The proportion of rats requiring more than 10 seconds is approximately 0.6065.

Step 1: Identify the Given Information

We need to find the proportion of rats requiring more than 10 seconds to reach the end of the T-maze. The formula provided is \( \int_{t}^{\infty} 0.05 e^{-0.05 s} ds \), where \( t \) is the time in seconds.

Step 2: Substitute the Value of t

Substitute \( t = 10 \) into the integral because we want to find the proportion for rats taking more than 10 seconds. Our integral becomes \( \int_{10}^{\infty} 0.05 e^{-0.05 s} ds \).

Step 3: Integrate the Function

The integral we need to solve is \( \int 0.05 e^{-0.05 s} ds \). The integral of \( e^{-0.05 s} \) is \( \frac{-1}{0.05} e^{-0.05 s} = -20 e^{-0.05 s} \). Therefore, the definite integral \( \int_{10}^{\infty} 0.05 e^{-0.05 s} ds = \left[ -e^{-0.05 s} \right]_{10}^{\infty} \).

Step 4: Evaluate the Integral Limits

Evaluate the expression \( \left[ -e^{-0.05 s} \right]_{10}^{\infty} = \lim_{b \to \infty} -e^{-0.05 b} + e^{-0.5} \). Since \( e^{-0.05 b} \) approaches 0 as \( b \to \infty \), we have \( 0 + e^{-0.5} \).

Step 5: Calculate the Final Value

Now calculate \( e^{-0.5} \). Approximating using a calculator, \( e^{-0.5} \approx 0.60653 \). Therefore, the proportion of rats requiring more than 10 seconds is approximately 0.6065.

Key Concepts

T-maze

The T-maze is a fascinating tool used in behavioral science, particularly in studies involving rodents. It is called the T-maze because of its shape, resembling the letter "T." The long base of the T serves as the start position, while the top arms of the T create a choice point where the subject, typically a rat, decides which path to follow.

The T-maze helps researchers understand learning, memory, and decision-making processes in animals. Its simplicity allows for controlled experiments where variables such as rewards or cues can be manipulated.

• Rats are often used in T-maze studies because they have a natural curiosity and an ability to learn mazes quickly.
• In these experiments, researchers might observe how long it takes a rat to choose a path or how repeatedly choosing a particular arm can indicate learning or memory.

Understanding the behavior in a T-maze helps infer general behaviors or neurological patterns applicable to broader contexts, including human psychology.

Exponential function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. It is expressed in the form \(f(x) = a \cdot e^{bx}\), where \(a\) and \(b\) are constants, \(e\) is Euler's number (approximately 2.718), and \(x\) is the variable.

Exponential functions are crucial in modeling growth and decay processes. In our exercise, the function \(0.05 e^{-0.05s}\) is an exponential decay function. Here:

• The base \(e\) is a natural choice for continuous growth or decay.
• The negative sign in the exponent \(-0.05s\) indicates a decay process.

Such functions are helpful to describe processes where the rate of change is proportional to the current value, like radioactive decay, population growth, or even certain financial applications. They are often used in probability and statistics to model time-related events, such as waiting times or survival analysis.

Definite integral

The concept of a definite integral can be thought of as the "area under the curve" for a given function between two boundaries. In calculus, the definite integral of a function \(f(x)\) between two limits \(a\) and \(b\) is expressed as \(\int_{a}^{b} f(x) \) dx. This integral helps in determining quantities like the total accumulation of a quantity, such as distance, probability, or in our case, the proportion of rats.

In our specific problem, we use the definite integral \(\int_{10}^{\infty} 0.05 e^{-0.05s} ds\) to find the proportion of rats needing more than 10 seconds in a T-maze task:

• It represents the probability that a randomly selected rat takes more than 10 seconds to complete the T-maze.
• Evaluating a definite integral sometimes involves considering limits at infinity, as seen here, where the integration results in an exponential function reaching its limit.

The usage of definite integrals in scenarios like these helps mathematicians and scientists not only understand behaviors but also make predictions based on empirical data.