Formula's for the volumes:
cone: v = (pi * x^2 * h) / 3
cilinder: v = pi * y^2 * h
where x = radius of the cone and y = radius of the cilinder, and since h is te same for both (given).
x^2 = 3 * y^2 so x = sqrt(3) * y (1)
Formula's of the surfaces:
S cone = pi * x * sqrt(x^2 + h^2) + pi * x^2 (2)
S cilinder = (2 * p i * y * h) + (2 * pi * y^2) (3)
Subst. (1) into (2) and (3) and setting (2) and (3) equal to each other, don't give the same results when worked out.
So, I think the statement of this word problem is false.