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Chapter 10: Problem 17

Find the area of the region between the two concentric circles \(r=7\) and \(r=10\)

Short Answer

Expert verified
The area of the region is \(51\pi\) square units.

Step by step solution

01

Understand the Problem

We want to find the area of the region between two concentric circles. These circles have radii 7 and 10 units, respectively. The target area is shaped like a ring or annulus.
02

Calculate the Area of the Larger Circle

The formula for the area of a circle is \( A = \pi r^2 \). For the larger circle with radius 10, the area is \( A_{\text{large}} = \pi (10)^2 = 100\pi \) square units.
03

Calculate the Area of the Smaller Circle

Using the same formula, for the smaller circle with radius 7, the area is \( A_{\text{small}} = \pi (7)^2 = 49\pi \) square units.
04

Determine the Area of the Annulus

The area of the annulus (the region between the two circles) is the difference between the area of the larger circle and the smaller circle. So, it's \( A_{\text{annulus}} = A_{\text{large}} - A_{\text{small}} = 100\pi - 49\pi = 51\pi \) square units.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Area of a Circle
To begin with, knowing how to find the area of a circle is fundamental in geometry. The area measures the space enclosed within the circle's boundary. The crucial formula you need is:
  • Formula: \( A = \pi r^2 \)
  • "\( A \)" stands for the area.
  • "\( r \)" represents the radius, the distance from the circle鈥檚 center to its edge.
The formula hinges on "pi" (\( \pi \)), approximately 3.14159. Multiplying \( \pi \) by the square of the radius gives the area in square units.
Knowing this formula allows you to tackle more complex shapes like an annulus, a ring-shaped space between two concentric circles.
Exploring the Annulus
An annulus is a fascinating geometric shape that forms a ring. It's essentially the leftover space when a smaller circle is removed from a larger circle, both having the same center.
Think of a donut or a washer; these are practical examples of an annulus. To find its area, you:
  • Calculate the area of the larger circle using its radius with \( A_{ ext{large}} = \pi R^2 \).
  • Calculate the area of the smaller circle using its radius \( A_{ ext{small}} = \pi r^2 \).
  • Subtract the area of the smaller circle from the larger circle: \( A_{ ext{annulus}} = A_{ ext{large}} - A_{ ext{small}} \).
This subtraction gives you the area of the annulus, capturing just the ring's section.
Effective Geometry Problem-Solving
Geometry problem-solving often requires a strategy that simplifies the problem and makes it manageable. Here's how you might approach solving an area problem involving concentric circles:
  • First, clearly gather all necessary information. For instance, the radii of both circles must be known.
  • Second, utilize a step-by-step plan. For the area between two concentric circles, calculate the area of each circle separately before subtracting to find the annulus.
  • Double-check each computation to avoid small errors. Confirm calculations with the use of a calculator or verify using another method.
This orderly method ensures accuracy and clarity, which is vital when tackling geometry problems. Employ these steps and strategies to effectively solve various geometric challenges.

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Most popular questions from this chapter

The slope of the tangent line to the parabola \(y^{2}=5 x\) at a certain point on the parabola is \(\sqrt{5} / 4\). Find the coordinates of that point. Make a sketch.

Find the equation of the given central conic. Vertical hyperbola with eccentricity \(\sqrt{6} / 2\) that passes through (2,4)

In Problems \(1-8\), find the coordinates of the focus and the equation of the directrix for each parabola. Make a sketch showing the parabola, its focus, and its directrix. \(y^{2}=4 x\)

Using the same axes, draw the conics \(y=\) \(\pm\left(a x^{2}+1\right)^{1 / 2}\) for \(-2 \leq x \leq 2\) and \(-2 \leq y \leq 2\) using \(a=\) \(-2,-1,-0.5,-0.1,0,0.1,0.6,1 .\) Make a conjecture about how the shape of the figure depends on \(a\).

Sketch the graph of the given equation, indicating vertices, foci, and asymptotes (if it is a hyperbola). \(16 x^{2}+4 y^{2}=32\)
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