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Chapter 6: Problem 64

A sector of a circle has a central angle of \(60^{\circ} .\) Find the area of the sector if the radius of the circle is \(3 \mathrm{mi}\).

Short Answer

Expert verified
The area of the sector is \( \frac{3}{2} \pi \) square miles.

Step by step solution

01

Understand the formula for the area of a sector

The formula to calculate the area of a sector with central angle \( \theta \) (in degrees) and radius \( r \) is given by \( \text{Area} = \frac{\theta}{360} \times \pi r^2 \). This formula is derived from the proportion of the angle of the sector to the whole angle of the circle, which is \(360^{\circ}\).
02

Plug in the given values

We are given that the central angle \( \theta \) is \( 60^{\circ} \) and the radius \( r \) is \( 3 \) miles. Substitute these values into the formula to find the area of the sector: \[ \text{Area} = \frac{60}{360} \times \pi \times 3^2 \].
03

Simplify the fraction

Simplify the fraction \( \frac{60}{360} \). Divide both the numerator and the denominator by their greatest common divisor, which is \(60\). This gives \( \frac{60}{360} = \frac{1}{6} \).
04

Calculate the area of the sector

Now that we have \( \frac{1}{6} \), substitute it back into the equation: \[ \text{Area} = \frac{1}{6} \times \pi \times 3^2 \]. Calculate \(3^2\), which is \(9\), thus \[ \text{Area} = \frac{1}{6} \times \pi \times 9 \].
05

Final calculation

Multiply the fractions: \( \frac{1}{6} \times 9 = \frac{9}{6} \). Simplify \( \frac{9}{6} \) to \( \frac{3}{2} \). Thus the area is \[ \text{Area} = \frac{3}{2} \pi \].

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Key Concepts

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

Circle Geometry
Circle geometry is a fascinating part of mathematics that deals specifically with shapes and properties related to circles. It is fundamental in understanding the concepts of sectors, which are pie-shaped parts of a circle. A circle itself is a special figure, featuring:
  • Symmetry - Every circle is symmetric around its center.
  • Constant radius - All points on the edge are equidistant from the center.

In circle geometry, it's important to know that the total angle around the center of a complete circle is \(360^{\circ}\). This property is key when calculating areas and understanding other geometric relationships within a circle. When dealing with problems like finding the area of a sector, being comfortable with circle properties helps simplify the problem.
Central Angle
A central angle is the angle formed when two radii of a circle meet at the center. It's an essential element of circle geometry applied frequently in calculations involving sectors. Here's why central angles are important:
  • Central angles help determine the size of the sector they "carve out" in the circle.
  • They are measured in degrees, allowing us to express parts of the circle as fractions of \(360^{\circ}\).

In our exercise, the given central angle is \(60^{\circ}\), which is a sixth of the entire circle (because \( \frac{60}{360} = \frac{1}{6} \)). This fraction is precisely why our formula for the area of a sector includes dividing the central angle by \(360\). It's a straightforward way to find how big the sector is in relation to the whole circle.
Radius
The radius is a line segment from the center of a circle to any point on its circumference. It is a crucial measurement because:
  • The radius is used to calculate the area, circumference, and other properties of a circle.
  • All radii in a circle are the same length, demonstrating the uniform distance from the center to the circle's edge.

In the case of our exercise, with a radius of \(3\) miles, we see how the radius influences the overall size of the circle and thus the size of any sector. The formula to find the area of a sector involves the square of the radius \( (\pi r^2) \) because it scales directly with the size of both the circle and sector. This means that even a small change in the radius can significantly impact the area of the sector. Understanding the role of the radius in circular equations is crucial for accurate calculations.

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Most popular questions from this chapter

Find the distance along an arc on the surface of the earth that subtends a central angle of 1 minute (1 minute \(=\frac{1}{60}\) degree). This distance is called a nautical mile. (The radius of the earth is 3960 mi.)

Calculating Distance Two straight roads diverge at an angle of \(65^{\circ} .\) Two cars leave the intersection at 2: 00 P.M., one traveling at \(50 \mathrm{mi} / \mathrm{h}\) and the other at \(30 \mathrm{mi} / \mathrm{h}\). How far apart are the cars at 2: 30 P.M.?

A winch of radius \(2 \mathrm{ft}\) is used to lift heavy loads. If the winch makes 8 revolutions every \(15 \mathrm{s}\), find the speed at which the load is rising. (IMAGE CAN NOT COPY)

Find the values of the trigonometric functions of \(\theta\) from the information given. $$\cos \theta=-\frac{7}{12}, \quad \theta \text { in Quadrant III }$$

The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in., the wheel sprocket a radius of 2 in., and the wheel a radius of 13 in. The cyclist pedals at 40 rpm. (a) Find the angular speed of the wheel sprocket. (b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.) (IMAGE CAN NOT COPY)
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