Problem 88 - Precalculus

Question

Find the exact length of the are made by the indicated central angle and radius of each circle. $$\theta=3^{\circ}, r=1800 \mathrm{km}$$

Step-by-Step Solution

Answer

The exact arc length is \(30\pi\) km.

Step 1: Convert Angular Measurements

To solve this problem, we need the angle in radians because the arc length formula requires it. Convert the given angle from degrees to radians using the conversion factor: \(1^{\circ} = \frac{\pi}{180}\) radians. Therefore, \(\theta = 3^{\circ} = 3 \times \frac{\pi}{180} = \frac{\pi}{60}\) radians.

Step 2: Apply Arc Length Formula

The formula to find the arc length \(L\) is given by \(L = r \cdot \theta\), where \(r\) is the radius and \(\theta\) is the angle in radians. Substitute the values we have: \(r = 1800\) km and \(\theta = \frac{\pi}{60}\). So, \(L = 1800 \times \frac{\pi}{60}\).

Step 3: Simplify the Calculation

Now, compute the calculation: \(L = 1800 \times \frac{\pi}{60} = 30\pi\). This is the exact length of the arc.

Key Concepts

Radian Conversion

When calculating arc length, it is essential to work with radians rather than degrees because the formula for arc length relies on the angle measurement in radians. Radian conversion is a vital step in these calculations. How do we convert from degrees to radians? It's simple! Use the conversion factor:

• One full circle is 360 degrees or equivalent to \(2\pi\) radians.
• Therefore, we use the relation \(1^{\circ} = \frac{\pi}{180}\) radians.

To convert an angle given in degrees to radians, we multiply the angle by \(\frac{\pi}{180}\). For example, if an angle \(\theta\) is \(3^{\circ}\), the conversion to radians becomes \(3 \times \frac{\pi}{180} = \frac{\pi}{60}\). Radians are more convenient for using the arc length formula directly.

Central Angle

The central angle is the angle subtended by an arc at the center of a circle. It is a crucial component for determining different properties associated with that circle, including arc length.Understanding the central angle involves recognizing that it is the part of the circle's total angle, which is linked to the circle's radius and the segment of circumference.

• An angle subtending an entire circle is \(360^{\circ}\), equating to \(2\pi\) radians.
• This is divided proportionately by the angle's measure to calculate arc length.

In arc length problems, calculating the central angle in radians allows us to use straightforward computations to find the arc length. Since the central angle \(\theta = \frac{\pi}{60}\) radians in our example, it precisely gives us the angle needed to compute arc length with a simple formula.

Circle Radius

The radius of a circle is the distance from its center to any point on its circumference. It is a fundamental property that defines the size of the circle. In terms of arc length calculation, the radius directly influences the length of the arc.For an arc length \(L\), the relation involves both the circle's radius \(r\) and the central angle \(\theta\) measured in radians, with the formula \(L = r \cdot \theta\). Hence, the radius acts as a scaling factor for the angle to transform it into a tangible distance around the circle.

• If you imagine a larger circle, the same angle \(\theta\) will cover a longer arc because the circle's circumference is larger due to the increased radius.
• Specifically, in our problem example, the radius \(r = 1800\) km influences the extent of the arc formed by the central angle \(\frac{\pi}{60}\).

Understanding the radius's role helps in visualizing the geometric interpretation that as the radius grows, so does the potential length of arcs defined by a central angle.