The surface area of a cone is 500pi the height is 15 what is the area of the base.

Question

the formula for surface area is pi times radius times slant height plus pi times radius squared.

Answer 1

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.

Comments

Thanks, Joseph, for looking at my solution and posting your comment.

There are probably other solutions to find the area of the base, but I picked the path that would relate A (the area of the base) to different quantities that could then be equated to find another factor in the solution, namely the slant height then the radius, from which A could easily be calculated. In other words, I could see the problem being based on solving simultaneous equations, a system of equations.