Problem 52 - Algebra and Trigonometry

Question

Find the length of an arc that subtends a central angle of \(45^{\circ}\) in a circle of radius 10 \(\mathrm{m} .\)

Step-by-Step Solution

Answer

The arc length is approximately 7.85 meters.

Step 1: Understand the Arc Length Formula

The formula to find the length of an arc (\(L\)) in a circle when the central angle is given in degrees is \( L = \frac{\theta}{360} \times 2\pi r \), where \(\theta\) is the central angle and \(r\) is the radius of the circle.

Step 2: Substitute the Known Values

We know the central angle \(\theta = 45^{\circ}\) and the radius \(r = 10\, \mathrm{m}\). Substitute these values into the arc length formula: \( L = \frac{45}{360} \times 2\pi \times 10 \).

Step 3: Simplify the Fraction

Simplify the fraction \(\frac{45}{360}\) to \(\frac{1}{8}\) by dividing both the numerator and denominator by 45.

Step 4: Calculate the Arc Length

Now, substitute \(\frac{1}{8}\) into the equation: \(L = \frac{1}{8} \times 2\pi \times 10\). This results in \(L = \frac{1}{8} \times 20\pi\). Simplifying further, we get \(L = 2.5\pi \).

Step 5: Approximate the Final Value

Evaluate \(2.5\pi\) using \(\pi \approx 3.14\). This gives us \(L \approx 2.5 \times 3.14 = 7.85 \ \mathrm{m}\).

Key Concepts

Central Angle

A central angle is an angle whose vertex is located at the center of a circle. It plays a crucial role in determining the arc length of the circle. The central angle helps us understand how much of the circle's circumference is covered by the arc.

• The size of the central angle is measured in degrees.
• A full circle has a central angle of 360 degrees.

In this exercise, the central angle given is 45 degrees. This means that the arc we are interested in forms a segment of the circle that is 45/360 of the entire circumference. It's important to know this ratio because it tells us what fraction of the total circumference the arc constitutes. The larger the central angle, the longer the arc will be, as more of the circle is spanned.

Circle Geometry

Circle geometry is a fascinating area of mathematics that deals with the properties and relationships of circles. Two key elements of circle geometry are the circumference and arc length.

• The circumference is the total distance around the circle.
• An arc is a portion of this circumference.

To find the arc length, we need to know the size of the central angle and the radius. The formula used to calculate the arc length, derived from circle geometry, is \[ L = \frac{\theta}{360} \times 2\pi r \]This equation takes into account the fraction of the circle covered by the central angle and applies it to the total circumference, which is calculated as \(2\pi r\). As a result, the arc length gives us a precise measurement of how long that section of the circle is.

Radius

The radius is a fundamental component in measurements involving a circle. It is defined as the distance from the center of the circle to any point on its edge.

• The radius is always constant for a given circle.
• Doubling the radius doubles the size of the circle.

In our problem, the radius is given as 10 meters. This value is crucial because it directly influences the total circumference of the circle as well as the arc length. The bigger the radius, the longer the circumference, and consequently, the longer the arc for any given central angle. In the arc length formula, the radius is multiplied by \(2\pi\), making it an essential factor in determining how much of the circle is included in the arc.