Problem 96 - Precalculus Essentials

Question

Find the area of the donut-shaped region bounded by the graphs of \((x-2)^{2}+(y+3)^{2}=25\) and \((x-2)^{2}+(y+3)^{2}=36\).

Step-by-Step Solution

Answer

The area of the 'donut' region is \( 11\pi \) square units.

Step 1: Find the Area of the Larger Circle

Use the formula for the area of a circle, \( \pi r^{2} \), with \( r = 6 \) (radius of the larger circle). This gives us \( \pi(6)^{2} = 36\pi \).

Step 2: Find the Area of the Smaller Circle

Again use the formula for the area of a circle, but now with \( r = 5 \) (radius of the smaller circle). This gives us \( \pi(5)^{2} = 25\pi \).

Step 3: Get the Area of the 'Donut'

The area of the 'donut' is the area of the larger circle minus the area of the smaller circle. This can be obtained by subtracting the smaller area from the larger one: \( 36\pi - 25\pi = 11\pi \).

Key Concepts

Area of Circles

The area of a circle is a key concept in geometry and is represented mathematically by the formula \( \pi r^2 \). Here, \( \pi \) is approximately 3.14159, a constant known as Pi. The symbol \( r \) stands for the radius of the circle, which is the distance from the center of the circle to any point on its circumference.
Consider a circle like a big pizza; the larger the radius, the bigger the pizza. The area is a measure of how much space is inside that pizza.
You only need one piece of information, the radius, to find the area of a circle. With this simple formula, you can calculate the space enclosed within any circle.
Understanding the area of circles is foundational to solving more complex problems, such as finding the area of a donut-shaped region, where the area of two circles plays a crucial role.

Donut-Shaped Region

Imagine a donut; it looks like a circle with a hole in the middle. In mathematics, we call this a donut-shaped region, the area between two concentric circles (circles with the same center).
In the problem, we have two circles: a larger circle, which forms the outer boundary, and a smaller one that makes the hole. The larger circle's equation is \((x-2)^2 + (y+3)^2 = 36\) and the smaller circle's equation is \((x-2)^2 + (y+3)^2 = 25\).
To find the area of the donut-shaped region, subtract the area of the smaller circle from the area of the larger circle. This leaves us with the area that's the space between the two circles. Just picture cutting out a smaller pizza from a larger one. The remaining part is what we're interested in. This concept is useful in real-life situations, like designing rings or planning spaces within other shapes.

Radius Calculation

The radius of a circle is a vital measurement when calculating the area. It is essential to determine the radius accurately from given circle equations. In standard form, the equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
In our exercise, both circles have the same center \((2, -3)\) derived from the equations \((x-2)^2 + (y+3)^2 = 25\) and \((x-2)^2 + (y+3)^2 = 36\).
To find the radius, solve for \(r\) by taking the square root of the number on the right side of the equation.

• For the larger circle, \((x-2)^2 + (y+3)^2 = 36\), the radius \(r\) is \(\sqrt{36} = 6\).
• For the smaller circle, \((x-2)^2 + (y+3)^2 = 25\), \(r\) is \(\sqrt{25} = 5\).

Learning how to accurately calculate the radius is essential for determining areas and handling more complex geometric problems.