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##.