Final answer:
The task involves calculating the base radius of a cone that is formed from bending a 120° sector of a circle with a radius of 14cm. By equating the arc length of the sector to the circumference of the cone's base, we can find the radius of the base of the cone.
Step-by-step explanation:
The student is asking how to find the base radius of a cone that is formed by bending a sector of a circle. The sector has a radius of 14cm and subtends an angle of 120° at its center. To determine the base radius of the cone, we must understand that the circumference of the base of the cone will be equal to the length of the arc of the sector.
First, we find the circumference of the original circle, which is C = 2πr. For a circle with a radius of 14cm, the circumference is C = 2π(14cm).
Since the sector represents 120° of the 360° in a full circle, we calculate the length of the arc by taking the proportion of 120° out of 360° and multiplying it by the total circumference. The arc length, L, is then L = (120/360)×C.
Now we equate the arc length to the circumference of the base of the cone (which is 2πr_base), solve for r_base, and round it to two decimal places as the student requested.
L = 2πr_base
(120/360)×2π(14cm) = 2πr_base
After simplifying, we divide both sides by 2π to find r_base, which is the base radius of the cone.