A circular piece of wire has a radius of 12 cm. This is cut then bent to form an arc of a circle whose radius is 60 cm. Find the angle subtended at the centre by this arc. Give your answer to the nearest degree.
Please show working out
Thank you so much!
the circumference of the original circle becomes the arc length of the sector.
circumf. of circle = 2πr = 24π
arc = rØ, were Ø is the central angle in radians and r is the radius
so if r = 60
60Ø = 24π
Ø = 24π/60 radians = 2π/5 radians
I know π/5 radians = 36° , so
2π/5 radians = 72°
To find the angle subtended at the centre by the arc, we can use the concept that the ratio of the arc length to the circumference of the whole circle is equal to the ratio of the angle subtended by the arc to the total angle (360 degrees) at the center of the circle.
First, let's find the arc length of the wire before it was bent. Since the wire forms a complete circle with a radius of 12 cm, the circumference of this circle can be calculated using the formula:
C = 2πr.
Substituting the radius (r = 12 cm) into the formula, we get:
C = 2π(12) cm = 24π cm.
Now, let's find the arc length of the new circle formed after bending the wire. Since the radius of the new circle is 60 cm, we can find the arc length using the formula for the length of an arc:
L = (θ/360) × 2πr.
Substituting the given values, we get:
L = (θ/360) × 2π(60) cm = θ/6 cm.
Next, let's find the ratio of the arc length of the new circle to the circumference of the original circle:
L/C = (θ/6) / (24π).
Since the ratio of L to C is equal to the ratio of the angle subtended at the center (θ) to 360 degrees, we can set up the equation:
(θ/6) / (24π) = θ/360.
To solve for θ, we can cross-multiply and simplify the equation as follows:
360(θ/6) = 24πθ,
60θ = 24πθ,
60 = 24π,
π = 60 / (24 × θ),
π ≈ 2.5 / θ.
Now, substitute the approximate value of π (3.14) into the equation:
3.14 ≈ 2.5 / θ.
To find θ, cross-multiply and solve the equation:
3.14θ ≈ 2.5,
θ ≈ 2.5 / 3.14,
θ ≈ 0.796 radians.
To convert radians to degrees, use the conversion factor: 1 radian = 180/π degrees.
θ ≈ 0.796 × (180/π) ≈ 45.64 degrees.
Therefore, the angle subtended at the center by the arc is approximately 46 degrees (to the nearest degree).