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Chapter 1: Problem 78

Find the length of the arc intercepted by the given central angle \(\alpha\) in a circle of radius \(r\). Round to the nearest tenth. $$ \alpha=1, r=4 \mathrm{~cm} $$

Short Answer

Expert verified
4.0 cm

Step by step solution

01

Understand the formula for arc length

The length of an arc (\text{L}) intercepted by a central angle (\theta) in a circle of radius (\text{r}) can be calculated using the formula: \[ L = r \theta \] where \( \theta \) must be in radians.
02

Convert the angle to radians

Since the given angle \( \alpha = 1 \) is presumed to be in radians, no conversion is necessary. If it were in degrees, we would convert it using: \[ \theta = \alpha \times \frac{\pi}{180} \]
03

Plug the values into the formula

Substitute the given values into the arc length formula: \[ L = 4 \text{ cm} \times 1 \text{ rad} \]
04

Calculate the arc length

Perform the multiplication: \[ L = 4 \text{ cm} \]
05

Round to the nearest tenth

Since the result is an integer (4), the arc length does not need further rounding. Hence, the length of the arc is 4.0 cm.

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Key Concepts

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

arc length
Arc length refers to the distance along the curved path of a circle's circumference. It is calculated using a specific formula that involves the circle's radius and the central angle in radians. The formula for arc length (\text{L}) is: \[ L = r \theta \]
Here, \( r \) represents the radius of the circle, and \( \theta \) represents the central angle in radians.
It's important to ensure \( \theta \) is in radians because the relationship between the radius and the central angle directly translates to distance.
Switching \( \theta \) to radians also makes the multiplication straightforward and accurate. In simpler terms, this formula allows us to
central angle in radians
A central angle is an angle whose vertex is the center of a circle and whose sides (or arms) extend out to the circumference. It intercepts an arc on the circle.
The measure of the central angle (\theta) in radians is crucial for determining the arc length. Instead of using degrees, radians provide a direct proportion between the radius and the arc, making calculations more intuitive.
To convert degrees to radians, the formula is: \[ \theta = \frac{\text{degrees} \times \text{蟺}}{180} \]
For instance, an angle of 180 degrees converts to \[ \theta = \frac{180 \times \text{蟺}}{180} = \text{蟺} \text{ radians} \]
In our example, the central angle is already in radians (\theta = 1). No conversion was needed.
circle radius
The radius of a circle (\text{r}) is the distance from the center of the circle to any point on its circumference.
In the formula for arc length, it serves as a key multiplier for the central angle in radians.
For instance, with \( r = 4 \text{cm} \), as in our exercise, each radian of the central angle translates to an arc length of 4 cm multiplied by the value of the angle in radians.
Utilizing the radius in this way directly ties the physical dimensions of the circle to the abstract angle measurement, simplifying the geometry involved.
Substituting the radius value precisely in any formula helps maintain the accuracy and relevance of the geometric computations we perform.

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Most popular questions from this chapter

Fifteen experts are voting to determine the best convertible of the year. The choices are a Porsche Carrera, a Chrysler Crossfire, and a Nissan Roadster. The experts will rank the three cars 1 st, 2nd, and 3 rd. There are three common ways to determine the winner: a. Plurality: The car with the most first-place votes (preferences) is the winner. b. Instant runoff: The car with the least number of preferences is eliminated. Then the ballots for which the eliminated car is first are revised so that the second-place car is moved to first. Finally, the car with the most preferences is the winner. c. The point system: Two points are given for each time a car is ranked first place on a ballot, one point for each time the car appears in second place on a ballot, and no points for third place. When the ballots were cast, the Porsche won when plurality was used, the Chrysler won when instant runoff was used, and the Nissan won when the point system was used. Determine 15 actual votes for which this result would occur.

Find the exact value of each expression for the given value of \(\theta\) Do not use a calculator. $$ \sin (2 \theta) \text { if } \theta=\pi / 8 $$

Find the exact area of the sector of the circle with the given radius and central angle. $$ r=4, \alpha=45^{\circ} $$

Use a calculator to evaluate each expression. Round approximate answers to four decimal places. $$ 1-2 \sin \left(-88.4^{\circ}\right) $$

True or false? Do not use a calculator. $$ \sin \left(91^{\circ}\right)=-\sin \left(89^{\circ}\right) $$
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