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Chapter 3: Problem 6

In Exercises 1-12, find the exact length of each arc made by the indicated central angle and radius of each circle. $$ \theta=\frac{\pi}{4}, r=10 \mathrm{in} $$

Short Answer

Expert verified
The arc length is \( \frac{5\pi}{2} \) inches.

Step by step solution

01

Understand the Arc Length Formula

The formula to find the arc length (s) of a circle is given by \( s = r\theta \), where \( r \) is the radius of the circle and \( \theta \) is the central angle in radians.
02

Identify the Given Values

From the problem, we have: \( \theta = \frac{\pi}{4} \) radians and \( r = 10 \) inches. These are the values we'll substitute into the formula.
03

Substitute the Values into the Formula

Substitute the given values \( r = 10 \) and \( \theta = \frac{\pi}{4} \) into the arc length formula: \( s = 10 \times \frac{\pi}{4} \).
04

Perform the Calculation

Simplify the expression: \( s = 10 \times \frac{\pi}{4} = \frac{10\pi}{4} = \frac{5\pi}{2} \).
05

State the Final Answer

The exact length of the arc is \( \frac{5\pi}{2} \) inches.

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Key Concepts

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

circle
A circle is one of the most fundamental shapes in geometry. It consists of all points in a plane that are equidistant from a fixed point, called the center. The distance from the center to any point on the circle is called the radius. A circle is perfectly symmetrical, and its basic properties make it an essential concept in various mathematical topics. Some key points about circles are:
  • A circle is defined by its radius.
  • The diameter is twice the radius and passes through the center of the circle.
  • Every point on the circle's edge is equidistant from the center.
Circles appear everywhere in mathematics and real life, making it crucial for students to understand their properties. Whether it's calculating the circumference, area, or arc length, grasping the basics of what defines a circle is essential.
central angle
The central angle is an angle whose vertex is at the center of a circle, and its sides (or rays) extend out to the circumference. This angle plays a significant role in measuring portions of a circle, like sectors and arcs.
Key characteristics of a central angle include:
  • It is measured in degrees or radians.
  • The arc of the circle that it intercepts is directly related to the angle's measure.
  • A full circle has a central angle of 360 degrees or \(2\pi\) radians.
In solving geometric problems like determining arc lengths, knowing how to use the central angle is essential. It helps us break the circle into measurable parts, thus making calculations more manageable.
radius
The radius of a circle is the distance from the center to any point on its circumference. Understanding the radius is crucial because it is a fundamental component of many formulas related to circles, such as the circumference, area, and arc length.
Important aspects of the radius include:
  • The radius is always half of the diameter.
  • All radii of the same circle are equal.
  • It acts as a constant measure that defines the size of the circle.
When computing the arc length, the radius serves as a multiplier in the formula. Hence, understanding its role is vital for accurate calculations and for grasping the general dimensions of the circle.
arc length formula
The arc length formula is one of the essential tools in geometry for finding the measure of a circular arc. It determines the distance along the curved line of the circle's arc.
The formula for arc length \(s\) is:\[ s = r \theta \]where \(r\) is the radius, and \(\theta\) is the central angle in radians.
  • Arc length provides a way to measure only a portion of the circle's circumference.
  • The formula requires the angle to be in radians for accurate results.
  • This calculation is particularly useful in engineering, cartography, and design where curved shapes are prevalent.
By understanding the arc length formula, students can calculate the exact distance between two points along a circle with precision, which is incredibly useful in both academic and practical applications.

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Most popular questions from this chapter

Tires. A car owner decides to upgrade from tires with a diameter of \(24.3\) inches to tires with a diameter of 26.1 inches. If she doesn't update the onboard computer, how fast will she actually be traveling when the speedometer reads \(65 \mathrm{mph}\) ?

In Exercises 1-14, find the exact values of the indicated trigonometric functions using the unit circle. $$ \cos \left(\frac{3 \pi}{4}\right) $$

In Exercises 33-42, find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius \(r\) and angular speed \(\omega\). $$ \omega=\frac{16 \pi \mathrm{rad}}{3 \mathrm{sec}}, r=\frac{7}{3} \mathrm{yd} $$

In Exercises 33-42, find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius \(r\) and angular speed \(\omega\). $$ \omega=\frac{5 \pi \mathrm{rad}}{16 \mathrm{sec}}, r=24 \mathrm{ft} $$

In Exercises 33-42, find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius \(r\) and angular speed \(\omega\). $$ \omega=\frac{\pi \mathrm{rad}}{8 \mathrm{~min}}, r=10.2 \mathrm{in} $$
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