Problem 94 - College Algebra

Question

A forest fire leaves behind an area of grass burned in an expanding circular pattern. If the radius of the circle of burning grass is increasing with time according to the formula \(r(t)=2 t+1,\) express the area burned as a function of time, \(t\) (minutes).

Step-by-Step Solution

Answer

The area burned as a function of time is \( A(t) = 4\pi t^2 + 4\pi t + \pi \).

Step 1: Understand the Formula for Area of a Circle

The area of a circle is given by the formula \( A = \pi r^2 \), where \( A \) is the area and \( r \) is the radius. We need this formula as the basis to express the area as a function of time.

Step 2: Substitute the Given Radius Function

Given that the radius of the circle is \( r(t) = 2t + 1 \), we substitute this into the area formula. This gives us \( A(t) = \pi(2t + 1)^2 \).

Step 3: Expand the Square

To simplify \( A(t) = \pi(2t + 1)^2 \), we first expand the square: \((2t + 1)^2 = (2t)^2 + 2(2t)(1) + 1^2 = 4t^2 + 4t + 1\).

Step 4: Express the Area Function

After expanding the square, we can now write the area function as \( A(t) = \pi(4t^2 + 4t + 1) \). Thus, the area burned as a function of time is \( A(t) = 4\pi t^2 + 4\pi t + \pi \).

Key Concepts

Radius Function

The concept of a radius function is foundational in understanding problems involving circles. Here, the radius isn't just a constant number; it changes as time passes. This is what we call a "radius function." In our problem, the radius of the circle, where the fire burns, evolves according to time with the formula \(r(t) = 2t + 1\). This formula indicates that for every unit increase in time \(t\) (in minutes), the radius increases by 2 units plus an initial 1 unit increase right from the start.

• The function \(r\) indicates dependence on time \(t\).
• It shows linear growth, meaning the radius increases at a constant rate.

Understanding this relationship helps in finding out how fast the area of the circle is expanding, which is vital for our exercise.

Expanding Circle

In real-world scenarios, like a forest fire, the affected area often expands over time. An "expanding circle" refers to how the boundary of the burned area grows. The rate at which it grows is determined by the radius. Given our radius function \(r(t) = 2t + 1\), the circle doesn't just grow in size; it grows in a predictable pattern and the circle's boundary expands outwards consistently over time.

• This growth is uniform, meaning the shape remains a circle as time progresses.
• The radius's increase directly impacts how drastically the area (circle) grows.

The bigger the circle grows, the more area is covered, hence the importance of understanding the expanding nature of this phenomenon.

Function of Time

In mathematical contexts, especially in dynamic systems like expanding circles, understanding variables as a "function of time" is crucial. In our problem, both the radius and the area of the circle are expressed in terms of time \(t\). The area burned over time is given by the function \(A(t) = \pi(2t+1)^2\).

• This expression explicitly defines how the area changes as time changes.
• The notation \(A(t)\) represents the area as a variable dependent on time.

By investigating \(A(t)\), we grasp how dynamic the situation is, capturing the progressive influence of time on the area's growth. It teaches us how one variable (area) can rely on another evolving factor (time).

Expansion of Squares

The "expansion of squares" is a mathematical technique crucial for simplifying quadratic expressions. In the context of our problem, we expand \((2t + 1)^2\) to turn it into a more manageable expression. This expansion follows the formula \((a + b)^2 = a^2 + 2ab + b^2\). Applying this to our function, we expanded:

• \((2t)^2 = 4t^2\)
• \(2 \cdot (2t) \cdot 1 = 4t\)
• \(1^2 = 1\)

These terms are then combined to form \(4t^2 + 4t + 1\). This process simplifies the function of time for the area, making calculations easier and more comprehensible. Keeping an eye on such techniques not only aids in calculations but also unveils patterns within expressions.