0580/f/m/42/2021 - the diagram shows a sector OPQ of a circle with Centre O and radius 24cm. Th sector angle is 60. A cone is made from this sector by joining OP to OQ. Calculate the volumes of this cone.

To calculate the volume of the cone, we need to know the radius and the height of the cone.

1. First, we need to calculate the length of the arc OPQ, which will be the circumference of the base of the cone. The length of an arc is given by (θ/360) * 2πr, where θ is the angle of the sector and r is the radius of the circle. Substituting the given values, we get (60/360) * 2π * 24 = 24π cm.

2. Since the circumference of the base of the cone is 2πr, we can equate this to 24π to find the radius of the base of the cone. Solving 2πr = 24π gives r = 12 cm.

3. The slant height of the cone is the radius of the sector, which is 24 cm. We can use the Pythagorean theorem to find the height of the cone. The radius, height, and slant height of a cone form a right triangle, so h = sqrt[(24)^2 - (12)^2] = sqrt[576 - 144] = sqrt[432] = 12sqrt3 cm.

4. Now we can calculate the volume of the cone using the formula V = 1/3πr²h. Substituting the values we found, we get V = 1/3 * π * (12)^2 * 12sqrt3 = 576πsqrt3 cm³.