math

A sector of a circle subtending an angle 300 degrees at the centre is used to form a cone with base radius 6cm. Find the (a.)radius of the circle (b.)volume of the cone (c.)area of the minor sector of the circle

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  1. It often helps to see a physical model of the problem
    Cut out a circular sector with scissors and form it into a cone.
    You will see that the arc length of the sector becomes the circumference of the circular base of the cone

    if the cone is to have a radius of 6 cm, then its circumference is 12π cm

    You should be familiar with the relation,
    arc = rØ for a sector, where Ø is the sector angle in radians
    300° to radians --->Ø/2π = 300/360---> Ø = 10π/6

    12π = r(10π/6)
    r = 12π(6/10π) = 7.2
    So our sector must have a radius of 7.2 cm

    b) If you look at your model you will see that 72 cm becomes the "slant" length of the cone, so using Pythagoras,
    h^2 + 6^2 = 7.2^2 , where h is the height of the cone
    h = √15.84

    volume = (1/3) base x height
    = (1/3)(36π)(√15.84)
    = appr 150.04 cm^3

    c) area of minor sector....
    our needed sector is 5/6 of the circle, so the minor sector would be 1/6 of the area of the circle

    area of minor sector = (1/6)π(7.2^2)
    = appr 27.14 cm^2

    check my arithmetic.

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