A cone has a surface area of ##600π cm^2## and a base area of ##225π cm^2##. Calculate: the apothem and volume of the cone?

Apothem is ##25## ##cm.## and volume of cone is ##1500pi## ##cm^3##.
In a cone whose radius at base is ##r## and height is ##h##, its apothem is ##a=sqrt(r^2+h^2)##.
While base area in such case is ##pir^2## and curved surface area is ##pira##. Therefore its total surface area is ##pir(r+a)## and volume is ##1/3pir^2h##
As given cone has a base area of ##225pi## ##cm^2##
##pir^2=225pi## and ##r=sqrt225=15## ##cm.##
As total surface are is ##600pi## ##cm^2##, we have
##pir(r+a)=600pi## or ##15(15+a)=600##
i.e. ##15+a=40## and ##a=40-15=25## ##cm.## and
##h=sqrt(25^2-15^2)=sqrt(625-225)=sqrt400=20## ##cm##.
and volume of cone is ##1/3pixx15xx15xx20##
= ##5xx15xx20pi=1500pi## ##cm^3##.