Question

An arc of a circle is 3π m long, and it subtends an angle of 72∘ at the center of the circle. Find the radius of the circle.

a) 2 m
b) 4 m
c) 3 m
d) 6 m

Answer 1

To find the radius of a circle with an arc length of 3π meters and a subtended angle of 72°, convert the angle to radians, and use the formula for arc length (3π = r ⅔π) to solve for r, resulting in a radius of 4 meters.

Step-by-step explanation:

To find the radius of the circle with an arc length of 3π meters and a central angle of 72°, we can use the relationship between arc length (Δs), radius (r), and the central angle (Δθ) in radians. The formula for arc length is Δs = r Δθ, where Δθ must be in radians. To convert the angle from degrees to radians, we use the conversion factor π radians = 180°. Thus, the angle in radians is Δθ = 72° × (π/180°) = (72/180)π = ⅔π. We can now solve for the radius r:

3π = r × ⅔π,

To isolate r, we divide both sides by ⅔π:

r = (3π) / ⅔π,

r = 3 × (4/1),

r = 4 meters.

The radius of the circle is therefore 4 meters, which corresponds to option b).