The length of an arc can be calculated using the formula:
Length of arc = (angle/360) * 2πr
where r is the radius of the circle.
(a) For a 30° - arc with an area of 144π cm², we first need to find the radius of the circle. The area of the full circle is πr², so the radius is:
πr² = 144π r² = 144 r = 12 cm
Now we can calculate the length of the 30° - arc:
Length of 30° - arc = (30/360) * 2π(12) = π/6 * 24 = 4π cm
Therefore, the length of the 30° - arc is 4π cm.
(b) For a 90° - arc with an area of 144π cm², we can use the same radius as before:
πr² = 144π r² = 144 r = 12 cm
Now we can calculate the length of the 90° - arc:
Length of 90° - arc = (90/360) * 2π(12) = π/2 * 12 = 6π cm
Therefore, the length of the 90° - arc is 6π cm.
