A cone of height 9cm has a volume of n cm cube and a curved surface area of n cm square. Find the vertical angle of the cone

Answer 1

Given:
Height (h) = 9 cm
Volume = n cm cube
Curved surface area = n cm square

Let the radius of the cone be r.

Volume of a cone = 1/3 * π * r^2 * h
n = 1/3 * π * r^2 * 9
n = 3πr^2

Curved surface area of a cone = π * r * l, where l is the slant height of the cone.

l = √(r^2 + h^2)
l = √(r^2 + 81)

n = π * r * √(r^2 + 81)
n/π = r * √(r^2 + 81)
(n/π)^2 = r^2 * (r^2 + 81)
r^2 = (n/π)^2 / (1 + 81)

Vertical angle of the cone can be found as:
tanθ = r/h
tanθ = r/9
tanθ = √(n/π)^2 / (9 * √[(n/π)^2 / (1 + 81)])
tanθ = √(n/π)^2 / (9 √[(n/π)^2 + 81])

Therefore, the vertical angle of the cone is tan^-1(√(n/π)^2 / (9 √[(n/π)^2 + 81])).