The length of an arc of a circle is 22 cm and the radius of the circle is 28 cm. What is the measure of the corresponding central angle in degrees?

Introduction / Context:
This question involves the relationship between the length of an arc, the radius of the circle, and the central angle that subtends the arc. It tests understanding of arc length formulas in degrees and the ability to solve for the angle when arc length and radius are known.

Given Data / Assumptions:

• Arc length s = 22 cm.
• Radius of the circle r = 28 cm.
• Let the central angle be θ degrees.
• We use the degree based arc length formula.
• Use pi approximately equal to 22 / 7 for calculations.

Concept / Approach:
The length of an arc in a circle when the central angle is given in degrees is s = (θ / 360) * 2 * pi * r. Given s and r, we can substitute and solve for θ. This involves straightforward algebra and substitution of the approximate value of pi.

Step-by-Step Solution:
Step 1: Write the arc length formula: s = (θ / 360) * 2 * pi * r.Step 2: Substitute s = 22, r = 28: 22 = (θ / 360) * 2 * pi * 28.Step 3: Simplify the constant part: 2 * 28 = 56, so 22 = (θ / 360) * 56 * pi.Step 4: Using pi = 22 / 7, we have 56 * pi = 56 * (22 / 7) = 8 * 22 = 176.Step 5: So the equation becomes 22 = (θ / 360) * 176.Step 6: Solve for θ: θ = 22 * 360 / 176.Step 7: Compute 22 * 360 = 7920 and then 7920 / 176 = 45 degrees.

Verification / Alternative check:
Verify by substituting θ = 45 degrees back into the formula.Compute s = (45 / 360) * 2 * pi * 28 = (1 / 8) * 56 * pi.Using pi = 22 / 7, s = (1 / 8) * 56 * 22 / 7 = (1 / 8) * 8 * 22 = 22 cm.This matches the given arc length, confirming θ = 45 degrees.

Why Other Options Are Wrong:
Angles like 60 degrees or 75 degrees would produce larger arcs than 22 cm for the same radius.A 90 degree angle would generate an arc equal to one quarter of the full circumference, which is clearly longer than 22 cm for radius 28 cm.An angle of 30 degrees would give an arc shorter than 22 cm.Only 45 degrees produces exactly the given arc length.

Common Pitfalls:
Students sometimes use the radian based arc length formula s = r * θ but incorrectly mix degrees and radians.Another common mistake is to forget the factor of 2 * pi * r in the full circumference, which leads to incorrect equations for arc length.Incorrect use of the approximation for pi or errors in arithmetic when simplifying 56 * pi can also lead to wrong answers.

Final Answer:
The measure of the central angle is 45°.