Problem 72 - Algebra and Trigonometry with Analytic Geometry

Question

It has been estimated that 1000 curies of a radioactive substance introduced at a point on the surface of the open sea would spread over an area of 40,000 \(\mathrm{km}^{2}\) in 40 days. Assuming that the area covered by the radioactive substance is a linear function of time \(t\) and is always circular in shape, express the radius \(r\) of the contamination as a function of \(t\).

Step-by-Step Solution

Answer

The radius of contamination as a function of time is \( r(t) = \sqrt{\frac{1000t}{\pi}} \).

Step 1: Understand the Problem

We are given that a radioactive substance spreads over an area that increases linearly with time. We need to express the radius of the contaminated area as a function of time.

Step 2: Linear Function of Area

Since the area is a linear function of time, we represent it as: \( A(t) = kt \), where \( k \) is a constant. We know from the problem description that \( A(40) = 40000 \, \text{km}^2 \). Hence, \( 40000 = 40k \).

Step 3: Solve for the Constant k

From the equation \( 40000 = 40k \), solve for \( k \): \( k = \frac{40000}{40} = 1000 \). So, \( A(t) = 1000t \).

Step 4: Area of a Circle Formula

The area of a circle is given by: \( A = \pi r^2 \). Set this equal to the linear area function: \( \pi r^2 = 1000t \).

Step 5: Solve for the Radius r

Solve the equation \( \pi r^2 = 1000t \) for \( r \) by isolating \( r \): \( r^2 = \frac{1000t}{\pi} \). Thus, \( r = \sqrt{\frac{1000t}{\pi}} \).

Step 6: Final Expression

The radius \( r \) as a function of time \( t \) is: \( r(t) = \sqrt{\frac{1000t}{\pi}} \).

Key Concepts

Linear Functions

In mathematics, a linear function is a function that creates a straight line when graphed. Its general form is given as \( f(x) = mx + b \), where \( m \) represents the slope, and \( b \) the y-intercept. Linear functions are crucial for modeling relationships where one quantity changes at a constant rate with respect to another.

• **Constant Rate**: The key characteristic of linear functions is the consistent rate of change, signified by the slope \( m \).
• **Real-World Applications**: These functions are often used to describe scenarios in economics, physics, and various scientific fields where something grows or decreases uniformly over time.
• **In Our Scenario**: The area over which the radioactive substance spreads is described as increasing linearly with time, making it a perfect example of this concept. The linear relationship \( A(t) = kt \) implies that the constant \( k \) determines how fast the area grows every unit of time.

By understanding how linear functions operate, we can predict and model scenarios like the spread of a substance over time more accurately.

Area of Circle

The area of a circle is an essential concept in geometry, given by the formula \( A = \pi r^2 \). Here, \( r \) is the radius of the circle and \( \pi \approx 3.14159 \) is a mathematical constant.

• **Formula Components**: The formula combines the radius squared with the constant \( \pi \), which inherently ties the circular shape's area to its radius.
• **Real-World Uses**: The formula is crucial when determining the space occupied by circular objects and understanding phenomena in natural processes, astronomy, and engineering.
• **In Application**: In our radioactive decay model, the spreading area of the contamination is assumed to be circular due to the natural diffusion properties of the substance in a fluid body like the ocean.

By setting the formula for the area of a circle equal to the linear area function \( \pi r^2 = 1000t \), the calculation progresses to find the radius \( r \) as a function of time \( t \), which is a critical step in the solution.

Mathematical Modeling

Mathematical modeling involves using mathematical structures to represent real-world systems. It allows us to simplify, interpret, and predict complex, dynamic situations through mathematics.

• **Model Structure**: A model usually defines variables and relationships, often through equations or inequalities, to create a framework that mirrors the phenomenon being studied.
• **Application in Sciences**: It's extensively used in scientific research, engineering, economics, and environmental studies to simulate scenarios and predict future trends or results.
• **Current Context**: In the problem of radioactive diffusion over the sea, modeling is used to predict how the contaminated area grows over time and provides a function \( r(t) \) representing the radius dependent on time \( t \).

This problem illustrates how mathematical models are powerful tools in making sense of complex environmental processes and serves the purpose of planning, management, and safety measures.

Function of Time

A function of time often represents how a system or situation evolves over a given period. It's expressed typically as \( f(t) \), where \( t \) symbolizes the time variable.

• **Dynamic Scenarios**: Such functions are employed in scenarios where growth, decay, or change is being tracked, incorporating time as a pivotal component.
• **Understanding Time Dependence**: By analyzing \( f(t) \), one can understand how a situation progresses - either grows, shrinks, or fluctuates - which is fundamental in both natural and social sciences.
• **Specific Use**: In our solved problem, the function \( r(t) = \sqrt{\frac{1000t}{\pi}} \) directly establishes the relationship between the radius \( r \) of the contamination area and time \( t \). It shows how the area of contamination spreads circularly over time.

Functions of time are essential in scenarios that exhibit temporal evolution, and they help in predicting the future state of many dynamic systems.