/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 10 Find the measure (in radians) of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 10

Find the measure (in radians) of a central angle \(\theta\) that intercepts an are of length \(s\) on a circle with radius \(r\). \(r=1 \mathrm{m}, s=2 \mathrm{cm}\)

Short Answer

Expert verified
The central angle \( \theta \) is 0.02 radians.

Step by step solution

01

Convert Units if Necessary

Before proceeding, ensure that all measurements are in the same unit. Here, the radius is given as 1 meter and the arc length as 2 centimeters. Convert the arc length into meters: 2 cm = 0.02 m. Now both the radius and arc length are in meters.
02

Use the Formula for Central Angle

The formula to find the central angle \( \theta \) in radians is \[ \theta = \frac{s}{r} \] where \( s \) is the arc length and \( r \) is the radius of the circle.
03

Substitute the Values into the Formula

Substitute the values for \( s \) and \( r \) into the formula: \[ \theta = \frac{0.02}{1} = 0.02 \]
04

Interpret the Result

The central angle \( \theta \) that intercepts the arc with the given length on the circle is 0.02 radians.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Radians
Radians are a way of measuring angles. Unlike degrees, which divide a circle into 360 parts, radians are based on the relationship between a circle's arc length and its radius.
A full circle in radians is equivalent to the ratio of the circle's circumference to its radius. This is approximately 6.28318 radians, but is more commonly known as 2蟺 radians. By using radians, calculations in mathematics, especially those involving trigonometric functions, become much simpler and more convenient.
  • A quarter circle is \(\frac{\pi}{2}\) radians.
  • Half a circle is \(\pi\) radians.
  • And a full circle makes 2\(\pi\) radians.
Understanding radians as a measurement allows us to connect the geometric world with mathematical functions efficiently. They are essential for solving problems involving arcs and circles.
Arc Length
Arc length is a portion of the circumference of a circle. When dealing with circles, knowing arc length helps us understand various properties related to circles and their sectors.
The formula to find the arc length \(s\), in terms of a central angle \(\theta\) in radians and the circle's radius \(r\), is given by:\[ s = \theta \times r \]So, if you know the circle's radius and the central angle in radians, you can easily determine the length of an arc. This relationship is particularly useful in real-world applications, such as designing gears or determining distances along a circular path.
  • The arc length represents a direct relationship between the angle and the radius.
  • An increase in the central angle or the circle's radius will lead to a longer arc.
  • Arc lengths are crucial in understanding the geometry of circles and trigonometry.
Understanding arc length and its calculation ensure effective solving of problems related to circular motion and design.
Circle Radius
The radius of a circle is the distance from the center of the circle to any point on its circumference. The radius is one of the most critical aspects of a circle, defining its size and influencing every property associated with it.
  • The radius is half of the diameter.
  • Any changes in the radius result in proportional changes in circumference and area.
  • The formula for the area of a circle is \(\pi r^2\), and the circumference is \(2\pi r\).
In the context of central angles and arc lengths, the radius forms the basis for calculating how large an angle or how long an arc is relative to the circle as a whole. It links the linear dimensions to angular measures efficiently.
Having the radius allows you to determine other essential properties of the circle and better understand the mathematical concept of circular geometry. Studying how the radius affects other properties is crucial in fields like physics, engineering, and more.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Calculate \(\csc 40^{\circ}\) in the following two ways: a. Find \(\sin 40^{\circ}\) to three decimal places and then divide 1 by that number. Write this last result to five decimal places. b. With a calculator in degree mode, enter \(40,\) sin, \(1 / \mathrm{x},\) and round the result to five decimal places.

Find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius \(r\) and angular speed \(\omega\). $$\omega=\frac{3 \pi \mathrm{rad}}{4 \mathrm{sec}}, r=8 \mathrm{cm}$$

When light passes from one substance to another, such as from air to water, its path bends. This is called refraction and is what is seen in eyeglass lenses, camera lenses, and gems. The rule governing the change in the path is called Snell's law, named after a Dutch astronomer: \(n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2},\) where \(n_{1}\) and \(n_{2}\) are the indices of refraction of the different substances and \(\theta_{1}\) and \(\theta_{2}\) are the respective angles that light makes with a line perpendicular to the surface at the boundary between substances. The figure shows the path of light rays going from air to water. Assume that the index of refraction in air is \(1 .\) (GRAPH CANNOT COPY) If the refraction index for a diamond is \(2.4,\) then to what angle is light refracted if it enters the diamond at an angle of \(30^{\circ} ?\) Round the answer to two significant digits.

In calculus we work with real numbers; thus, the measure of an angle must be in radians. The area of a sector of a circle with radius 3 in. and central angle \(\theta\) is \(\frac{3 \pi}{2}\) in. \(^{2} .\) What is the radian measure of \(\theta ?\)

Find the distance a point travels along a circle over a time \(t,\) given the angular speed \(\omega\) and radius \(r\) of the circle. Round your answers to three significant digits. $$r=5 \mathrm{cm}, \omega=\frac{\pi \mathrm{rad}}{6 \mathrm{sec}}, t=10 \mathrm{sec}$$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.