A circle has a radius of 25. A circular sector, with an angle of \(90\) degrees at the center, is cut from the circle, and then rolled to form a cone. Find the volume of the cone.
The slant height of the cone = the radius of the circle
The circumference of the of the whole circle = 2pi * 25 = 50 pi
Since 90 degrees is taken out - (1/4) of the circumference- then the circumference of the cone = (50)pi * (3/4) = 37.5 pi
The radius of the cone = 1/2 of this = ( 37.5 pi) / (2pi) = 18.75
The height of the cone = sqrt ( slant height ^2 - radius^2 ) = sqrt ( 25^2 - 18.75^2) ≈ 16.54
The volume is (1/3) * pi * ( 18.75)^2 * ( 16.54) ≈ 6089.3 ( cubic units)