Problem 96 - College Algebra Essentials

Question

determine whether each statement makes sense or does not make sense, and explain your reasoning. Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25 \quad\) and \((x-2)^{2}+(y+3)^{2}=36\)

Step-by-Step Solution

Answer

The area of the donut-shaped region bounded by the two given circles is \(11\pi\).

Step 1: Area formula

Recall the formula for the area of a circle: \(\pi r^{2}\), where \(r\) is the radius of the circle. This will be essential for finding the areas of the two given circles.

Step 2: Find the radii

The equations of the circles are given in the format \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \((h,k)\) is the center of the circle and \(r\) is the radius. From the equations, it can be inferred that the radii of the two circles are \(r_{1}=\sqrt{25}=5\) and \(r_{2}=\sqrt{36}=6\).

Step 3: Find the areas of the circles

Using the area formula, find the area of both circles. The area of the first circle is \(\pi\cdot5^{2}=25\pi\) and the area of the second circle is \(\pi\cdot6^{2}=36\pi\).

Step 4: Find the area of the donut-shaped region

Subtract the area of the smaller circle from the area of the larger circle to find the area of the donut-shaped region. Hence, the area is \(36\pi - 25\pi = 11\pi\).

Key Concepts

Circle equations

In mathematics, circles are defined by equations that describe every single point along the curve. A general equation for a circle is

• \((x-h)^2 + (y-k)^2 = r^2\)

where:

• \(h\) and \(k\) represent the center coordinates of the circle.
• \(r\) is the radius.

Circle equations are useful because they let us know exactly where the circle is located on a coordinate plane.
In the given exercise, both circles have equations centered at

• \((x-2)^2 + (y+3)^2 = 25\) and
• \((x-2)^2 + (y+3)^2 = 36\).

This tells us both circles share the same center at the point (2, -3).
The circle's radius affects its size, resulting in a larger or smaller circle. Circle equations are foundational in understanding the shape and layout of circles in geometry.

Radius of a circle

The radius is a key component in understanding circles. It is the distance from the center of the circle to any point on its circumference.
To find the radius in a circle equation, you can follow this simple rule: take the square root of the number on the right side of the equation.
For example, if we have the equation

• \((x-2)^2 + (y+3)^2 = 25\)

the radius \(r\) is:

• \(\sqrt{25} = 5\)

Similarly, for the equation

• \((x-2)^2 + (y+3)^2 = 36\)

the radius becomes:

• \(\sqrt{36} = 6\)

The radii are crucial in solving problems related to the size and area of circles.
Knowing the radius helps us calculate the area, which leads directly to understanding any donut-shaped region between two circles.

Area of a circle

The area of a circle shows how much space it occupies. The formula to find this area is straightforward:

• \(\pi r^2\)

where \(r\) is the radius.
So, if you have a radius, you can find the area by squaring it and multiplying the result by \(\pi\).
For instance, in our exercise, given the first circle:

• Radius = 5, Area = \(\pi \times 5^2 = 25\pi\)

And for the second circle:

• Radius = 6, Area = \(\pi \times 6^2 = 36\pi\)

The problem specifically asks for the area of the donut-shaped region between these two circles.
This is found by subtracting the area of the smaller circle from the larger one:

• \(36\pi - 25\pi = 11\pi\)

Knowing the area of a circle allows us to understand how much of a plane it covers which is particularly important in applications related to geometry and physics.