#1**+2 **

The slant height of the cone is = BC

The volume of the cone is given by

432pi = pi/3 * r^2 * height

432pi = pi / 3 * 12^2 * height

432 = 144/3 * height multiply both sides by 3/144

3 * 432 / 144 = height

9 cm = height

Now...the slant height of the cone = √ [ r^2 + h^2 ] = √[12^2 + 9^2] =√225 =15 cm = BC

So...the lateral surface area of the cone = the shaded area of the circle =

pi * radius * slant height = pi * 12 * 15 = 180 pi cm^2

And we can set up the following relationship

Total circumference of the circle / Total area of the circle =

Total circumference of shaded area / Total area of shaded area

[2*pi * BC] /[ pi * BC^2] = [2pi * BC * ( D /360)] / [ 180pi ]

Where D is the number of degrees in the arc of the shaded sector

[2 * pi * 15] /[ pi * 15^2 ] =[ 2 * pi * 15] * ( D / 360)] /[180pi]

1 / 15^2 = (D / 360) / 180

1 / 15^2 = D / 64800

64800 / 225 = D = 288°

So...the number of degrees in ABC = 360 - 288 = 72°

CPhill Oct 12, 2018