Problem 229 - Calculus Volume 1

Question

The area \(A(t)\) of a circular shape is growing at a constant rate. If the area increases from 4\(\pi\) units to 9\(\pi\) units between times \(t=2\) and \(t=3,\) find the net change in the radius during that time.

Step-by-Step Solution

Answer

The net change in the radius is 1 unit.

Step 1: Understand the problem statement

We are given that the area of a circular shape increases at a constant rate from \(4\pi\) to \(9\pi\) between times \(t=2\) and \(t=3\). We need to find the net change in the radius during this time period.

Step 2: Write the Area Formula

The area of a circle is given by the formula \( A = \pi r^2 \). We can use this formula to understand how the radius changes as the area changes.

Step 3: Calculate Initial and Final Radius

First, find the initial radius when \( A = 4\pi \): \( \pi r_1^2 = 4\pi \), which simplifies to \( r_1^2 = 4 \). Thus, \( r_1 = 2 \).Next, find the final radius when \( A = 9\pi \): \( \pi r_2^2 = 9\pi \), which simplifies to \( r_2^2 = 9 \). Thus, \( r_2 = 3 \).

Step 4: Determine Net Change in Radius

The net change in radius is the difference between the final radius \( r_2 \) and the initial radius \( r_1 \). Hence, \( \Delta r = r_2 - r_1 = 3 - 2 = 1 \).

Key Concepts

Area of a Circle

The area of a circle can be understood through a very simple yet powerful formula. It is given by \( A = \pi r^2 \). Here, \( A \) represents the area, \( r \) stands for the radius, and \( \pi \) is a constant approximately equal to 3.14159. This formula is essential to determine how much space is encompassed within the boundary of a circle.

• The area increases quadratically with the radius. This means if the radius doubles, the area becomes four times larger.
• This relationship plays a key role in solving problems that deal with changes in circle dimensions.

So whenever dealing with issues related to circles, this formula is your go-to tool. It establishes a direct connection between the radius and the area, which helps in calculating changes efficiently.

Rate of Change

The concept of the rate of change is central to calculus and can be simply described as how a quantity changes over time. In this problem, the area of the circle is changing at a constant rate.

• A constant rate implies that the same amount of change happens over equal intervals.
• To find the rate of change of the area, one can look at how much the area increases over a given time period.

For example, if the area increases by \( 5\pi \) square units in one time unit, that describes the rate of change in the area. Understanding this rate helps us to further analyze the changes in the radius.

Radius of a Circle

The radius is a vital component of any circular shape, and is the distance from the center to any point on the edge of the circle. Because of its squared presence in the area formula, changes in radius have a significant impact. Let's examine further:

• Small changes in the radius lead to larger changes in the area due to the \( r^2 \) term.
• Understanding the radius helps in predicting how the circle's size will change when adjusting its dimensions.

Moreover, in problems involving area change, altering the radius is commonly the underlying factor. Finding the initial and final radius can tell us how the circle's size evolves over time.

Net Change

Net change is a term used to define the total change in a quantity over a period of time. In this case, it concerns the radius of the circle.

• It is calculated by finding the difference between the initial and final values.
• For the radius, net change will show how much the radius has increased or decreased over the specific period.

The calculation in the example given involves finding \( r_2 - r_1 \), which signifies the net change from the initial radius to the final one. In this instance, the radius increased by 1 unit.

Mathematical Analysis

Mathematical analysis involves using logical reasoning and calculations to solve a problem. In the context of this problem, several analytical steps are applied.

• Firstly, apply the area formula to find changes in the dimensions.
• Use consistent units and precise calculations to find net changes.

Analysis often involves breaking a problem into smaller parts. In this case, it could mean examining each rate of change, the specific differences in area and radius, and systematically solving for the net change. These analytical practices combine mathematical formulas and logical thinking for effective problem-solving.