Problem 94 - College Algebra with Corequisite Support

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 Given Formula

We know that the radius of a circle is increasing over time given by the formula \( r(t) = 2t + 1 \). Here, \( t \) is the time in minutes and \( r(t) \) gives the radius of the circle in the area burned by the fire.

Step 2: Recall the Formula for the Area of a Circle

The area \( A \) of a circle as a function of radius \( r \) is given by \( A = \pi r^2 \). We need to use this formula to find the area of the burned region at any time \( t \).

Step 3: Substitute the Expression for Radius into the Area Formula

Substitute \( r(t) = 2t + 1 \) into the area formula \( A = \pi r^2 \). This gives us:\[ A(t) = \pi (2t + 1)^2 \]

Step 4: Expand and Simplify the Expression

Expand the expression \( (2t + 1)^2 \):\((2t + 1)^2 = (2t + 1)(2t + 1) = 4t^2 + 4t + 1\).Substitute back into the area function:\[ A(t) = \pi (4t^2 + 4t + 1) \].

Step 5: Finalize the Expression for the Area

Distribute \( \pi \) across the terms:\[ A(t) = 4\pi t^2 + 4\pi t + \pi \].This is the final simplified expression for the area burned as a function of time.

Key Concepts

circle area formula

To find the area of a circle, we rely on a fundamental formula:

• Area, \( A = \pi r^2 \)
• \( r \) represents the radius
• \( \pi \) is a constant approximately equal to 3.14159

The formula tells us how much space is inside the circle based on the length of the radius. It's crucial when dealing with problems involving circular shapes, as it connects the dimensions of the circle to its area.

Breaking it down, this formula states that the area is proportional to the square of the radius, multiplied by \( \pi \). This means that if you double the radius, the area grows four times larger (since \( (2r)^2 = 4r^2 \)). Understanding this relationship helps in numerous mathematical contexts, especially when the size of the circle changes over time, like in the expanding circle problem.

expanding circle problem

Let's dive into the intriguing problem of an expanding circle, often encountered in real-world scenarios like spreading fires or ripples on water.

The characteristic feature of this problem is a dynamic radius. In our exercise, this radius changes according to the formula \( r(t) = 2t + 1 \), signifying that as time \( t \) increases, so does the radius.

• The term \( 2t \) indicates a linear growth of the radius over time
• The constant \( +1 \) adds an initial radius

With the radius dependent on time, calculating the area becomes more complex because we now have to account for a constantly changing dimension.

We solve this by substituting the time-dependent expression of the radius, \( 2t + 1 \), back into the circle area formula. This substitution allows us to see how the area changes as the radius changes, resulting in a time-dependent area function that describes the progress of the fire.

algebraic expression

An algebraic expression is a mathematical phrase involving numbers, variables, and operations. It is used to translate a word problem into a mathematical model that can be manipulated.

In this exercise, creating the function for the burned area requires expanding the expression \( (2t + 1)^2 \). Here's a step-by-step look:

• Begin with \( (2t + 1)\times (2t + 1) \)
• Use distributive property to expand it: \( 4t^2 + 4t + 1 \)

Once expanded, we substitute this expanded form into the area equation to give us \( A(t) = \pi(4t^2 + 4t + 1) \). This allows us to further simplify and factor in the constant \( \pi \), resulting in \( A(t) = 4\pi t^2 + 4\pi t + \pi \)

Mastering the manipulation of algebraic expressions is vital as it reveals the underlying mathematical relationships in dynamic systems, such as the spreading of a forest fire.