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Chapter 4: Problem 81

Find the exact length of the are made by the indicated central angle and radius of each circle. $$\theta=\frac{\pi}{12}, r=8 \mathrm{ft}$$

Short Answer

Expert verified
The exact length of the arc is \(\frac{2\pi}{3}\) feet.

Step by step solution

01

Understanding the Formula

To find the arc length of a circle, we use the formula: \[L = r \, \theta\]where \(L\) is the arc length, \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.
02

Substituting the Values

Given in the problem, the radius \(r = 8\) ft and the central angle \(\theta = \frac{\pi}{12}\). Substitute these values into the formula:\[L = 8 \times \frac{\pi}{12}\]
03

Simplifying the Expression

Now, simplify the expression step-by-step:\[L = 8 \times \frac{\pi}{12} = \frac{8\pi}{12} = \frac{2\pi}{3}\]
04

Final Calculation

Thus, the length of the arc is \(\frac{2\pi}{3}\). Since \(\pi\) is a known constant (approximately 3.14159), this can be left in terms of \(\pi\) or further calculated if needed. For an exact value, it remains as \(\frac{2\pi}{3} \approx 2.094\).

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Key Concepts

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

Central Angle
In the study of circles, the central angle is an angle whose vertex is located at the center of a circle. This angle is crucial as it helps in calculating the arc length and other geometrical properties of the circle. The central angle is measured in radians or degrees.
  • When measured in radians, it directly influences the arc length when multiplied by the radius.
  • One complete circle is equal to an angle of \(2\pi\) radians or 360 degrees.
In the given exercise, the central angle is \(\theta = \frac{\pi}{12}\). This indicates that the angle subtends a small part of the circle due to its small value compared to \(2\pi\). Understanding how central angles interact with other circle properties is essential for grasping circle geometry.
Radius
The radius of a circle is the fixed distance from the center of the circle to any point on the boundary. It plays a key role in calculations concerning the circle鈥檚 area, circumference, and particularly, its arc length. The radius is often symbolized as \(r\). In our exercise, the radius is given as 8 feet.
  • Different radii create different-sized circles, but they all share the property of constant distance from the center to the circle鈥檚 edge.
  • The larger the radius, the larger the circle and its respective segments like arcs and sectors.
When calculating arc length, you multiply the radius by the central angle measured in radians. Comprehending how the radius scales with these measurements is integral to mastering circle-related problems.
Radians
Radians are a unit of angular measure used in many fields of mathematics. Understanding radians is crucial when working in trigonometry and calculus, particularly with circles. Unlike degrees, radians provide a direct relationship to the circle's radius using the formula for arc length.
  • One radian is the angle created when the radius length forms an arc equivalent to the radius along the circle's circumference.
  • There are \(2\pi\) radians in a full circle (360 degrees), making it a natural way to measure angles in the context of circles.
In the exercise, the central angle, \(\theta = \frac{\pi}{12}\), is expressed in radians. This expression allows us to directly calculate the arc length by multiplying with the radius, applying the formula \(L = r\theta\). By mastering radians, students can easily translate angular measurements into linear distances along the circle.
Circle
A circle is a two-dimensional shape defined by all points equidistant from a central point known as the center. This equal distance is known as the radius. Circles are fundamental in geometry and have numerous applications in math and real-world contexts.
  • A circle鈥檚 entire boundary is called the circumference, calculated as \(2\pi r\) where \(r\) is the radius.
  • Sections of the circumference are arcs, defined by their start and end points and the circle's central angles.
  • The area inside the circle is given by the formula \(\pi r^2\).
Grasping the basic properties of circles, such as how the radius and central angle determine arcs, is fundamental. It allows students to solve various problems involving circle geometry, such as finding arc lengths or areas of circular segments.

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Most popular questions from this chapter

When light passes from one substance to another, such as from air to water, its path bends. This is called refraction and is what is seen in eyeglass lenses, camera lenses, and gems. The rule governing the change in the path is called Snell's law, named after a Dutch astronomer: \(n_{1} \sin \theta_{1}=n_{2} \sin \theta_{2},\) where \(n_{1}\) and \(n_{2}\) are the indices of refraction of the different substances and \(\theta_{1}\) and \(\theta_{2}\) are the respective angles that light makes with a line perpendicular to the surface at the boundary between substances. The figure shows the path of light rays going from air to water. Assume that the index of refraction in air is \(1 .\) (GRAPH CANNOT COPY) If light rays hit a glass surface at an angle of \(30^{\circ}\) from the perpendicular and are refracted to an angle of \(18^{\circ}\) from the perpendicular, then what is the refraction index for that glass? Round the answer to two significant digits.

Refer to the following: NASA explores artificial gravity as a way to counter the physiologic effects of extended weightlessness for future space exploration. NASA's centrifuge has a 58 -foot-diameter arm. If two humans are on opposite (red and blue) ends of the centrifuge and they rotate one full rotation every second, what is their linear speed in feet per second?

Determine whether each statement is true or false. If an obtuse triangle is isosceles, then knowing the measure of the obtuse angle and a side adjacent to it is sufficient to solve the triangle.

In calculus, the value \(F(b)-F(a)\) of a function \(F(x)\) at \(x=a\) and \(x=b\) plays an important role in the calculation of definite integrals. Find the exact value of \(F(b)-F(a)\) $$F(x)=2 \tan x+\cos x, a=-\frac{\pi}{6}, b=\frac{\pi}{4}$$

In calculus we work with real numbers; thus, the measure of an angle must be in radians. Determine the angle of the smallest possible positive measure (in radians) that is coterminal with the angle \(750^{\circ}\).
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