/*! 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 43 In Exercises 43-52, find the dis... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 43

In Exercises 43-52, find the distance a point travels along a circle \(s\), over a time \(t\), given the angular speed \(\omega\), and radius of the circle \(r\). Round to three significant digits. $$ r=5 \mathrm{~cm}, \omega=\frac{\pi \mathrm{rad}}{6 \mathrm{sec}}, t=10 \mathrm{sec} $$

Short Answer

Expert verified
The point travels approximately 26.18 cm.

Step by step solution

01

Understand the given values

We are given the radius of the circle \( r = 5 \) cm, the angular speed \( \omega = \frac{\pi \text{ rad}}{6 \text{ sec}} \), and the time \( t = 10 \text{ sec} \).
02

Use the formula for arc length

The formula for the arc length \( s \), which is the distance a point travels along a circle, is given by \( s = r \theta \), where \( \theta \) is the angular displacement in radians. Since \( \theta = \omega \cdot t \), we have \( s = r \cdot \omega \cdot t \).
03

Calculate the angular displacement \(\theta\)

Calculate \( \theta \) using \( \theta = \omega \cdot t \). Substituting the given values: \( \theta = \frac{\pi}{6} \cdot 10 = \frac{10\pi}{6} = \frac{5\pi}{3} \) radians.
04

Calculate the arc length \(s\)

Substitute \( r = 5 \) cm and \( \theta = \frac{5\pi}{3} \) radians into the formula \( s = r \theta \) to find \( s = 5 \cdot \frac{5\pi}{3} \). This results in \( s = \frac{25\pi}{3} \) cm.
05

Round the final answer

Calculate \( \frac{25\pi}{3} \) cm to three significant digits. Using \( \pi \approx 3.14159 \), this gives \( s \approx 26.18 \) cm.

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.

Arc Length
Arc length is the measure of the distance a point travels along the edge of a circle. To find this length, we use the formula: \[ s = r \cdot \theta \]where:
  • \( s \) is the arc length
  • \( r \) is the radius of the circle
  • \( \theta \) is the angular displacement in radians
Arc length offers a way to calculate the journey a point takes around a circular path,
allowing us to grasp how far something has traveled over an arc.
In the context of the exercise, once we find the angular displacement \( \theta = \omega \cdot t \),
we plug this into the arc length formula to find our answer. The unit of measure for arc length shares the same unit as the radius,
which in this problem, results in centimeters.
Angular Speed
Angular speed, denoted as \( \omega \), describes how fast an object moves around a circle. It specifies the rate of change of the angular displacement over time and is usually given in radians per second. The relationship between angular speed and arc length can be appreciated through the formula:\[ \omega = \frac{\theta}{t} \]where:
  • \( \omega \) is the angular speed
  • \( \theta \) is the total angle in radians
  • \( t \) is the time in seconds
In our example, the angular speed is \( \frac{\pi}{6} \)
radians per second. This means every 6 seconds, the circle's radius has swept out
an angle equivalent to \( \pi \) radians.
By understanding angular speed, we gain insight into how quickly an object rotates or revolves,
making it particularly useful in fields such as physics and engineering.
Radius of a Circle
The radius of a circle is a crucial measure that stretches from the center of a circle to any point on its border. This length is important because it is a fundamental part of the formula for both arc length and the area of a circle.Knowing the radius allows one to unlock various calculations involving circles:
  • Radius \( r \) is the direct input for the arc length formula: \( s = r \cdot \theta \)
  • It also helps calculate the circumference of a circle: \( C = 2\pi r \)
  • And the area of a circle: \( A = \pi r^2 \)
In this problem, the radius is given as 5 cm, which provides the necessary information to find the arc length.
The radius is a fundamental characteristic of a circle,
affecting all other measurements and properties related to the circle's geometry.
Thus, understanding the radius as part of the circle's anatomy helps in solving various mathematical problems.

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

In Exercises 31-50, use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$ \sin \theta=\frac{\sqrt{3}}{2}, 0 \leq \theta \leq 2 \pi $$

Music. Some people still have their phonograph collections and play the records on turntables. A phonograph record is a vinyl disc that rotates on the turntable. If a 12 -inch-diameter record rotates at \(33 \frac{1}{3}\) revolutions per minute, what is the angular speed in radians per minute?

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 31-50, use the unit circle to find all of the exact values of \(\theta\) that make the equation true in the indicated interval. $$ \cos \theta=0,0 \leq \theta \leq 4 \pi $$

In Exercises 71 and 72, explain the mistake that is made. If a bicycle has tires with radius 10 inches and the tires rotate \(90^{\circ}\) per \(\frac{1}{2}\) second, how fast is the bicycle traveling (linear speed) in miles per hour? Solution: Write the formula for linear speed. \(\quad v=r \omega\) Let \(r=10\) inches and \(\omega=180^{\circ}\) per second. \(\quad v=(10\) in. \()\left(180^{\circ} / \mathrm{sec}\right)\) Simplify. \(\quad v=1800 \mathrm{in} . / \mathrm{sec}\) Let 1 mile \(=5280\) feet \(=63,360\) inches and \(\quad v=\left(\frac{1800 \cdot 3600}{63,360}\right) \mathrm{mph}\) 1 hour \(=3600\) seconds. Simplify. \(\quad v \approx 102.3 \mathrm{mph}\) This is incorrect. The correct answer is approximately \(1.8\) miles per hour. What mistake was made?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.