Question

An arc of length $2.2\text{ cm}$ subtends an angle $\theta$ at the centre of the circle with radius $2.8\text{ cm}$. The value of $\theta$ is

• A. $50^\circ$
• B. $60^\circ$
• C. $45^\circ$
• D. $30^\circ$

Hint:

Notice the decimal relationships during simplification to save time.
Since $17.6$ is exactly $8$ times $2.2$, the calculation simplifies directly to $\frac{360^\circ}{8} = 45^\circ$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question concerns a circle with a radius $r = 2.8\text{ cm}$.
An arc of length $l = 2.2\text{ cm}$ on this circle subtends an angle $\theta$ at the center.
We need to find the value of this angle $\theta$ in degrees.

Step 2: Key Formula or Approach:
The formula relating the length of an arc $l$, the radius $r$, and the central angle $\theta$ (in degrees) is:
\[ l = \frac{\theta}{360^\circ} \times 2\pi r \]
We can rearrange this formula to solve directly for $\theta$:
\[ \theta = \frac{l \times 360^\circ}{2\pi r} \]

Step 3: Detailed Explanation:

• Identify the given parameters:
Arc length, $l = 2.2\text{ cm}$
Radius, $r = 2.8\text{ cm}$
Use $\pi = \frac{22}{7}$

• Set up the equation using the arc length formula:
\[ 2.2 = \frac{\theta}{360^\circ} \times 2 \times \frac{22}{7} \times 2.8 \]

• Simplify the calculation on the right-hand side:
\[ \frac{2.8}{7} = 0.4 \]
\[ 2 \times 22 \times 0.4 = 17.6 \]
So the equation becomes:
\[ 2.2 = \frac{\theta}{360^\circ} \times 17.6 \]

• Isolate the variable $\theta$:
\[ \theta = \frac{2.2 \times 360^\circ}{17.6} \]

• Simplify the fraction:
Observe that $\frac{17.6}{2.2} = 8$.
\[ \theta = \frac{360^\circ}{8} \]
\[ \theta = 45^\circ \]


Step 4: Final Answer:
The value of $\theta$ is $45^\circ$.