spherical sector

Question

3. The lower portion of a perfume dropper is made of solid glass and is in the shape of a spherical cone. The altitude of the conical part is 2 in., and the radius of the base is ¼ in.; find the volume of glass used to make the glass portion of 20000 droppers.

Answer 1

Volume of cone is 1/3(pi)r^2h and volume of hemisphere is 1/2*4/3(pi)r^3, where r=base radius. Total volume=1/3(pi)r^2(h+2r)=1/3(pi)/16*(2+1/2)=5(pi)/96 cubic inches. Volume of 20,000 droppers = 20000*5(pi)/96=3125/3(pi)=3,272.49in^3.