Problem 60 - Algebra 2

Question

In a circle, an arc of length 8\(\pi \mathrm{cm}\) is intercepted by a central angle of \(\frac{2 \pi}{3}\) radians. What is the radius of the circle? $$ \begin{array}{lllll}{\text { A. } \frac{3 \pi}{16} \mathrm{cm}} & {\text { B. } \frac{16 \pi}{3} \mathrm{cm}} & {\text { C. } \frac{16 \pi^{2}}{3} \mathrm{cm}} & {\text { D. } 12 \mathrm{cm}}\end{array} $$

Step-by-Step Solution

Answer

The radius of the circle is \(\frac{16\pi}{3}\) cm.

Step 1: Identify given information

Identify the given information. The arc length is 8\(\pi\) cm, and the central angle is \(\frac{2\pi}{3}\) radians.

Step 2: Setup the arc length formula

We know that Arc Length = radius x angle (where angle is in radians). Let the radius be r cm. Substituting the given values into the formula, we get: 8\(\pi\) = r x \(\frac{2\pi}{3}\).

Step 3: Solve for radius r

Now solve the equation 8\(\pi\) = r x \(\frac{2\pi}{3}\) for the variable r. Start by dividing both sides of the equation by \(\frac{2\pi}{3}\). Hence, the radius r is given by r = 8\(\pi\) / \(\frac{2\pi}{3}\). Simplify this division to find the value of r.

Step 4: Evaluate the final result

The division results in r = \(\frac{8\pi}{\frac{2\pi}{3}}\) simplifies to r = \(\frac{16\pi}{3}\) cm. Therefore, the radius of the circle is \(\frac{16\pi}{3}\) cm.

Key Concepts

Central Angle

The central angle is vital in measuring parts of a circle. It is the angle formed between two radii of a circle, with its vertex at the center of the circle. This angle helps determine how much of the circle’s circumference is enclosed by the arc formed between these two radii.

The central angle is often measured in radians, which simplifies calculations involving arcs. A key property of the central angle is its direct relation to the arc: the arc length is equal to the circle's radius multiplied by the angle in radians.

Understanding this relationship allows you to easily calculate arc lengths when the central angle and radius are known.

• If you have a central angle and you need the arc length, you multiply the radius by the angle in radians.
• Conversely, if you know the arc length and central angle, you can find the circle's radius by dividing the arc length by the angle in radians.

The central angle gives us a clear insight into the part of the circle captured by the arc.

Circle Radius

The radius of a circle is the constant distance from the center of the circle to any point on its circumference. It is a crucial element in many geometric calculations involving circles, especially when determining properties like area and circumference.

In the context of arc length, the radius is one of the variables in the formula: Arc Length = Radius x Angle (in radians). Here's why the radius is so essential:

• The length of an arc depends directly on the radius; changing the radius will change the arc length for the same central angle.
• Finding the radius from the arc length and angle involves dividing the arc length by the angle in radians.
• In any circle-related problem, knowing the radius provides a basis for finding other measurements like diameter, circumference, and area.

In our exercise, the radius was derived by reconfiguring the arc length formula, showing its foundational role in circular geometry.

Radians Conversion

Radians are an alternative to degrees for measuring angles, especially favorable in mathematical calculations involving circles, thanks to their natural relationship with the circle's radius and circumference.

Unlike degrees, which divide a circle into 360 parts, radians consider the circle's radius as the unit length. A full circle is equivalent to \(2\pi\) radians. This fundamental fact makes them invaluable for geometry and trigonometry.

• To convert degrees into radians, multiply by \(\frac{\pi}{180}\).
• To convert radians back into degrees, multiply by \(\frac{180}{\pi}\).
• Radians simplify formulas: for circular measurements such as arc length and angular velocity.

In practical contexts, like our exercise, understanding radians helps effortlessly link distances (arc lengths) to angles, streamlining the problem-solving process.