Question

The length of an arc of a circle of radius 12 cm is $10\pi$. Find the central angle of this arc.

• A. $160^\circ$
• B. $150^\circ$
• C. $140^\circ$
• D. $130^\circ$

Hint:

Always remember the formula for arc length. If the angle is given in radians, the formula is $L = r\theta_{radians}$. If the angle is in degrees, use $L = \frac{\theta_{degrees}}{360^\circ} \times 2\pi r$. Be careful with units and ensure consistency.

Answer 1

The Correct Option is B

Step 1: Recall the formula for the length of an arc.
The length of an arc ($L$) is given by the formula:
$L = \frac{\theta}{360^\circ} \times 2\pi r$
where $\theta$ is the central angle in degrees and $r$ is the radius of the circle.
Step 2: Identify the given values.
Length of the arc ($L$) = $10\pi$ cm
Radius of the circle ($r$) = 12 cm
Step 3: Substitute the given values into the formula and solve for \( \theta \). \[ 10\pi = \frac{\theta}{360^\circ} \times 2\pi \times 12 \] \[ 10\pi = \frac{\theta}{360^\circ} \times 24\pi \] Divide both sides by \( \pi \): \[ 10 = \frac{\theta}{360^\circ} \times 24 \] Now, isolate \( \theta \): \[ 10 = \frac{24\theta}{360^\circ} \] \[ 10 \times 360^\circ = 24\theta \] \[ 3600^\circ = 24\theta \] \[ \theta = \frac{3600^\circ}{24} \] \[ \theta = 150^\circ \] Step 4: Final Answer.
The central angle of the arc is $150^\circ$. \[ \mathbf{(2)} \quad 150^\circ \]