Problem 60 - Precalculus Mathematics for Calculus

Question

Find the radius of the circle if an arc of length \(4 \mathrm{ft}\) on the circle subtends a central angle of \(135^{\circ}\).

Step-by-Step Solution

Answer

The radius of the circle is approximately 1.698 ft.

Step 1: Convert Degrees to Radians

First, we need to convert the given central angle from degrees to radians because the formula for the arc length involves the angle in radians. We use the conversion factor \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \).Given angle is \(135^{\circ}\), so:\[ \theta = 135 \times \frac{\pi}{180} = \frac{3\pi}{4} \text{ radians} \]

Step 2: Use the Arc Length Formula

We use the formula for the arc length of a circle, which relates the arc length \(L\), the radius \(r\), and the angle in radians \(\theta\):\[ L = r \theta \]We know the arc length \(L = 4 \text{ ft}\) and \(\theta = \frac{3\pi}{4} \). We need to solve for \(r\).

Step 3: Solve for the Radius

Rearrange the arc length formula to solve for the radius \(r\):\[ r = \frac{L}{\theta} \]Substitute the known values \( L = 4 \text{ ft} \) and \( \theta = \frac{3\pi}{4} \):\[ r = \frac{4}{\frac{3\pi}{4}} = \frac{4 \times 4}{3\pi} = \frac{16}{3\pi} \approx 1.698 \text{ ft} \]

Step 4: Conclusion

The radius of the circle is approximately \(1.698 \mathrm{ft}\).

Key Concepts

Arc Length

The arc length is simply a portion of the circumference of a circle. Imagine "cutting out" a part of a hoop or a circular track. That cut-out section is the arc. The length of this arc depends on two things: how large the circle is and the angle that creates the arc when viewed from the center of the circle. If you want to calculate the arc length, you can use the formula:
\[ L = r \theta \]
where:

• \(L\) is the arc length,
• \(r\) is the radius of the circle,
• \(\theta\) is the central angle in radians.

By rearranging this formula, you can easily calculate the radius if the arc length and the central angle are known. Being able to find the arc length helps solve many everyday geometry problems, from construction to crafting.

Central Angle

The central angle is the angle with its vertex at the center of the circle and its sides extending to the circumference. This angle determines how much of the circle's circumference will make the arc. A larger central angle covers more of the circle, leading to a longer arc, while a smaller angle results in a shorter arc.
Central angles are measured in degrees or radians. In circle geometry, radians are often preferred, simplifying formulas and calculations. For example, in the arc length formula, the angle must be in radians.
Remember, the full circle is covered by a central angle of \(360^{\circ}\) degrees or \(2\pi\) radians. This is a key element that links angles and arcs in the geometry of circles.

• A central angle of \(180^{\circ}\) creates a semicircle.
• A central angle of \(90^{\circ}\) forms a quarter circle, or a quadrant.

Understanding central angles assists in applying various theorems and formulas linked to circles.

Degree to Radian Conversion

Degrees and radians are two different ways of measuring angles. A complete circle is \(360^{\circ}\) or \(2\pi\) radians. For mathematical functions, radians are often more useful.
To convert an angle from degrees to radians, use:
\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]
This conversion is crucial when using the arc length formula, as it requires the angle to be in radians. In practice:

• A \(90^{\circ}\) angle equals \(\frac{\pi}{2}\) radians.
• A \(180^{\circ}\) angle equals \(\pi\) radians.
• A \(360^{\circ}\) angle equals \(2\pi\) radians.

Converting degrees to radians streamlines calculations in any trigonometric function or when determining distances along a circle's circumference.

Circle Geometry

Circle geometry deals with the properties and measures concerning circles, focusing on relationships between angles, lines, and lengths. Circles have a unique property — all points on the circumference are equidistant from the center. This distance is called the radius.
Important terms in circle geometry include:

• The radius, which is the distance from the center of the circle to any point on its circumference.
• The diameter, which is twice the radius and spans from one side of the circle directly across to the other.
• Circumference, the total distance around the circle, calculated as \(2\pi r\).
• Chord, a line segment with its endpoints on the circle.
• The tangent, a line that touches the circle at exactly one point.

These fundamental concepts lay the groundwork for understanding how circles interact with lines and angles, forming the basis of more complex geometrical calculations and theorems.