/*! 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 21 Find the area of a sector with a... [FREE SOLUTION] | 91影视

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Chapter 0: Problem 21

Find the area of a sector with a central angle of \(80^{\circ}\) and a radius of 5 in.

Short Answer

Expert verified
The area of the sector is \( \frac{50 \text{pi}}{9} \text{ in}^2 \).

Step by step solution

01

- Identify the given values

The given values are the central angle \( \theta = 80^{\text{circ}} \) and the radius \( r = 5 \text{ in} \).
02

- Convert the central angle to radians

Use the conversion formula \( \text{Radians} = \frac{\theta \times \text{pi}}{180} \). Thus, the angle in radians is \( \theta = \frac{80 \times \text{pi}}{180} = \frac{4 \text{pi}}{9} \).
03

- Use the sector area formula

The area of the sector can be found using the formula \( A = \frac{1}{2} r^2 \theta \).
04

- Plug in the values

Substitute \( r = 5 \text{ in} \) and \( \theta = \frac{4 \text{pi}}{9} \) into the formula: \[ A = \frac{1}{2} (5^2) \frac{4 \text{pi}}{9} = \frac{1}{2} (25) \frac{4 \text{pi}}{9} \].
05

- Simplify the expression

Simplify the expression to find the area: \[ A = \frac{25 \times 4 \text{pi}}{18} = \frac{100 \text{pi}}{18} = \frac{50 \text{pi}}{9} \text{ in}^2 \].

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Key Concepts

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

central angle
In geometry, a central angle is the angle formed by two radii of a circle that meet at its center.
Understanding this concept is crucial for solving problems related to circles and sectors. The central angle \( \theta \) is often given in degrees. In our exercise, the central angle is \(80^{\text{circ}}\).
Central angles are key to various applications such as finding the area of sectors or arc lengths.
radius
The radius of a circle is the distance from the center to any point on its circumference.
It is one of the most fundamental properties of a circle. In the given exercise, the radius \( r \) is 5 inches.
The radius helps determine the size of the circle and plays a crucial role in calculating areas and other properties.
Understanding the radius is essential for applying the sector area formula correctly.
radians conversion
Radians are a unit of angular measure used in many areas of mathematics.
To solve the exercise, we need to convert the central angle from degrees to radians.
The conversion formula is \( \text{Radians} = \frac{\theta \times \text{pi}}{180} \).
Using this formula, an angle of \(80^{\text{circ}}\) converts to radians as follows: \( \theta = \frac{80 \times \text{pi}}{180} = \frac{4 \text{pi}}{9} \).
Converting to radians is important because many formulas, including the sector area formula, use radians.
area formula
The area of a sector is part of the circle's area proportional to its central angle.
The formula for the area of a sector with a central angle \( \theta \) in radians and radius \( r \) is \( A = \frac{1}{2} r^2 \theta \).
In our example, we plug in \( r = 5 \) inches and \( \theta = \frac{4 \text{pi}}{9} \) to get:\( A = \frac{1}{2} (5^2) \frac{4 \text{pi}}{9} \).
Simplifying this expression gives the final area: \( A = \frac{50 \text{pi}}{9} \) square inches.
Understanding how to use this formula allows you to find the area of any sector.

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Most popular questions from this chapter

Use the formulas given in this section to compute the total surface area and the volume of the figure described. All answers should be rounded to the nearest tenth. A closed right circular cylinder of height \(5 \mathrm{cm}\) and radius \(4 \mathrm{cm}\)

If an angle is \(28^{\circ}\) less than three times its complement, how large is the angle?

Answers should be rounded to the nearest tenth unless otherwise indicated. How much water is needed to fill a cylindrical swimming pool of diameter \(4 \mathrm{m}\) to a height of \(1.4 \mathrm{m} ?\)

Two airplanes leave an airport at the same time and at a \(90^{\circ}\) angle from each other. After an hour of flying at the same altitude, one plane is 160 miles from the airport, and the other is 180 miles from the airport. To the nearest tenth of a mile, how far are the planes from each other?

Use the formulas given in this section to compute the total surface area and the volume of the figure described. All answers should be rounded to the nearest tenth. A right circular cylinder, closed at only one end, of height \(4 \mathrm{cm}\) and radius \(5 \mathrm{cm}\)
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