A circle of a certain radius has the property that if an...

Name: Problem 20, Chapter 6: Functions and Change: A Modeling Approach to College Algebra and Trigonometry
Uploaded: 2020-07-20T11:43:13-08:00
Duration: 1 min 51 s
Channel: Joseph Lentino
Description: A circle of a certain radius has the property that if an angle has its vertex at the center of the circle, then the area it subtends is the same as the arc length. Find the radius of the circle.

A circle of a certain radius has the property that if an angle has its vertex at the center of the circle, then the area it subtends is the same as the arc length. Find the radius of the circle.

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Key Concepts

Radians

Radians are a unit of angular measure used in mathematics, defined such that the angle in radians is the ratio of the length of the arc to the radius of the circle. This concept is crucial because many circle formulas, such as those for arc length and sector area, assume the angle is measured in radians.

Arc Length of a Circle

The arc length of a circle is the distance along the curved line forming the arc. It is calculated using the formula arc length = radius × angle (in radians), which directly links the linear distance of the arc to the angular measure at the circle's center.

Sector Area of a Circle

The sector area of a circle is the area of a 'slice' of the circle, bounded by two radii and the intercepted arc. This area is computed by the formula sector area = 1/2 × radius² × angle (in radians), which shows how the area depends quadratically on the radius and linearly on the central angle.

Equating Sector Area and Arc Length in Problem Solving

When a problem involves setting the sector area equal to the arc length, it requires setting the formula for sector area equal to the formula for arc length and solving for the unknown, typically the radius. This step involves algebraic manipulation and simplification, often canceling common factors, to determine the specific value of the radius that satisfies the given condition.

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