Problem 40 - Precalculus

Question

For the following exercises, use to given information to find the length of a circular arc. Round to two decimal places. Find the length of the arc of a circle of radius 12 inches subtended by a central angle of \(\frac{\pi}{4}\) radians.

Step-by-Step Solution

Answer

The arc length is approximately 9.42 inches.

Step 1: Recall the Formula for Arc Length

The formula to find the length of an arc (\(L\)) is given by \( L = r \theta \), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.

Step 2: Identify Given Information

From the problem statement, we know that the radius \(r = 12\) inches and the central angle \(\theta = \frac{\pi}{4}\) radians.

Step 3: Substitute the Values into Formula

Use the formula \(L = r \theta\) and substitute \(r = 12\) and \(\theta = \frac{\pi}{4}\): \[ L = 12 \times \frac{\pi}{4} \].

Step 4: Simplify the Expression

Calculate \(12 \times \frac{\pi}{4}\) by first multiplying and then dividing: \[ L = 12 \times \frac{\pi}{4} = \frac{12\pi}{4} = 3\pi \].

Step 5: Calculate the Numerical Value

To find the approximate length, round \(3\pi\) to two decimal places: use \(\pi \approx 3.14159\), thus \(3\pi \approx 9.42\).

Step 6: Provide the Final Answer

The length of the arc is approximately 9.42 inches, rounded to two decimal places.

Key Concepts

Circle

A circle is a simple closed shape in which all points are equidistant from a fixed point, known as the center. This fixed distance is called the radius. Imagine looking at a wheel or a clock face; these are everyday examples of circles. Circles are fundamental in geometry due to their symmetry and numerous applications in real life.

Circles have special properties:

• The distance around a circle is called its circumference.
• They have an infinite number of lines of symmetry.
• All radii (plural of radius) in a circle are the same length.

Understanding circles is key to learning concepts like arc length, which measures the distance along a portion of the circle's circumference.

Radius

In geometry, the radius of a circle is the distance from the center to any point on its edge. It's a crucial measure because it helps define the size of the circle. Knowing the radius enables calculations of other properties, like the area and circumference of the circle.

Here are a few important points about the radius:

• The diameter of a circle is twice the radius ( \(d = 2r\)).
• The formula for the area of a circle is \(A = \pi r^2\).
• In the arc length formula, radius plays a central role \(L = r \theta\), where \(\theta\) is in radians.

For our specific problem, the radius is 12 inches, helping us determine the arc length of a circle part.

Central Angle

A central angle is an angle whose vertex is the center of the circle, and its sides (or rays) extend to meet the circle's circumference. Central angles play a fundamental role in understanding parts of a circle, such as sectors and arcs.

Here are some essential points about central angles:

• Central angles in a full circle add up to 360 degrees or \(2\pi\) radians.
• They help divide the circle into sectors, much like slicing a pie.
• In radians, the arc length is directly proportional to the central angle. Hence, \(L = r\theta\).

In the exercise, the central angle given is \(\frac{\pi}{4}\) radians, crucial for finding the arc length.

Radians

Radians are a unit of angular measure used predominantly in mathematics. One radian is defined as the angle subtended by an arc of a circle whose length is equal to the circle's radius. Compared to degrees, radians provide a natural way to work with angles in mathematical calculations, especially involving circles and periodic functions.

Here's how radians work:

• A full circle is \(2\pi\) radians (about 6.28 radians).
• Half of a circle is \(\pi\) radians.
• 1 radian ≈ 57.3 degrees.

For our arc length formula, angles must always be in radians. This is why the exercise uses the central angle \(\frac{\pi}{4}\) radians.

Pi

Pi (\(\pi\)) is a mathematical constant representing the ratio of a circle's circumference to its diameter. It's an irrational number, meaning it cannot be exactly expressed as a simple fraction, and its decimal representation is non-repeating and infinite.

• For most practical purposes, \(\pi\) is approximated as 3.14159.
• It's used in various formulas involving circles, such as those for the circumference (\(C = 2\pi r\)) and area (\(A = \pi r^2\)).
• In the arc length formula \(L = r\theta\), \(\theta\) often includes \(\pi\), showing how pi helps measure angles and distances in a circle.

In calculating the arc length of our problem, understanding and using \(\pi\) accurately gives us the precise length in numerical form.