/*! 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 47 Area of a Sector A central angle... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 47

Area of a Sector A central angle of 2 radians cuts off an arc of length 4 inches. Find the area of the sector formed.

Short Answer

Expert verified
The area of the sector is 4 square inches.

Step by step solution

01

Formula for Area of a Sector

The area of a sector can be found using the formula: \[ A = \frac{1}{2} \times r \times l \] where \( A \) is the area, \( r \) is the radius, and \( l \) is the arc length.
02

Arc Length with Radius

We know the arc length \( l = 4 \) inches and the angle is \( \theta = 2 \) radians. From the arc length formula, \( l = r \theta \), we can find the radius: \( r = \frac{l}{\theta} = \frac{4}{2} = 2 \) inches.
03

Substitute Known Values into Area Formula

Now that we know \( r = 2 \) inches and \( l = 4 \) inches, substitute these values into the area formula: \[ A = \frac{1}{2} \times 2 \times 4 \]
04

Calculate the Area

Simplify the expression to find the area: \[ A = \frac{1}{2} \times 2 \times 4 = 4 \] The area of the sector is 4 square inches.

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.

Area of a Sector
The area of a sector is a segment of a circle's area. Visualize cutting a slice of pizza, where each slice represents a sector. To find the area of this slice, or sector, we use the formula: \ \[ A = \frac{1}{2} \times r \times l \] Here, \( A \) stands for the area of the sector, \( r \) is the radius of the circle from the center to the edge of the sector, and \( l \) is the length of the arc.
  • This formula helps when you know the arc length but need to find the corresponding area.
  • The fraction \( \frac{1}{2} \) arises because the circle's area calculation involves \( \pi r^2 \), and sectors are only a fraction of the whole circle.
Using this approach allows for straightforward plugging in of values to find the area.
Radians
Radians are a unit of angular measure used in trigonometry. Unlike degrees, which separate a circle into 360 parts, radians use \( 2\pi \) to represent a full circle.One radian is the angle made when you wrap the radius around the circle's edge. In simpler terms:
  • A full circle is \( 2\pi \) radians
  • A half circle is \( \pi \) radians
  • Quarter of a circle is \( \frac{\pi}{2} \) radians
Understanding radians is crucial because they simplify many equations in trigonometry and calculus, making calculations more natural compared to degrees.
Arc Length
The arc length is the curved distance along the circle's edge that connects two points. It can be thought of as a rope wrapped along part of a circle's circumference. If you unravel it, the rope would measure the length of the arc.To find the arc length, use the formula: \ \[ l = r \theta \] where \( l \) is the arc length, \( r \) is the circle鈥檚 radius, and \( \theta \) is the angle in radians.
  • This formula reveals how arc length is directly proportional to the angle size when the radius remains constant.
  • For example, doubling \( \theta \) doubles the arc length.
It makes sense when considering that a whole circle (\( 2\pi \) radians) relates to the full circumference.
Formula for Area
Finding the area of a sector employs the formula \( A = \frac{1}{2} \times r \times l \). This results from combining a few geometric principles: understanding the full circle's area and scaling this concept down to parts of the circle - which sectors are.The way it works:
  • The standard area of a circle is \( \pi r^2 \).
  • The formula for a sector modifies this by accounting for the fraction of the circle represented by the arc.
  • Therefore, \( A = \frac{1}{2} \times r \times l \) simplifies our calculations.
By multiplying the radius \( r \) and the arc length \( l \), you've essentially captured the slice's magnitude relative to the whole circle, multiplied by \( \frac{1}{2} \) because it's a linear measure compared to the whole area.

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

Distance and Bearing A plane flying with a constant speed of \(285.5\) miles per hour flies for 2 hours on a course with bearing \(\mathrm{N} 48.7^{\circ} \mathrm{W}\). How far north and how far west does the plane fly?

Use a calculator to find the following. $$ \sec 314^{\circ} 40 $$

Find the exact value of each of the following. $$ \tan 315^{\circ} $$

Find the exact value of each of the following. $$ \csc 330^{\circ} $$

Ski Lift A ski lift operates by driving a wire rope, from which chairs are suspended, around a bullwheel (Figure 6). If the bullwheel is 12 feet in diameter and turns at a rate of 9 revolutions per minute, what is the linear velocity, in feet per second, of someone riding the lift?
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

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.