500π=πrs+A where A=πr^2 the area of the base. If we find r, we can easily calculate A.
A=(500-rs)π. We don't know r the radius, or s, the slant length, but we know h, the height.
r^2=s^2-h^2 (Pythagoras), so A=π(s^2-h^2)=π(s^2-225)=(500-rs)π.
s^2-225=500-rs or s^2+rs=725 and r=(725-s^2)/s=725/s-s
So we have a relationship between s and r, but no absolute values for either.
r=√(s^2-225)=(725-s^2)/s.
Squaring: s^2-225=(725-s^2)^2/s^2; s^4-225s^2=725^2-1450s^2+s^4;
1225s^2=725^2, 35s=725, 7s=145, s=145/7.
r=7*725/145-145/7=7*5-145/7=(245-145)/7=100/7.
A=πr^2=10000π/49=641.14 sq units.
CHECK
h^2+r^2=225+10000/49=(11025+10000)/49=21025/49=(145/7)^2=s^2. OK.
πrs+πr^2=100/7 * 145/7 * π + 10000π/49=14500π/49+10000π/49=24500π/49=500π. OK.