To find the vertical angle of the cone, we first need to find the radius of the cone's base.
The volume of a cone is given by the formula V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone. In this case, we know that the height is 9 cm and the volume is n cm^3, so:
n = (1/3) * π * r^2 * 9
n = 3πr^2
r^2 = n/(3π)
r = sqrt(n/(3π))
Next, we find the slant height of the cone. The curved surface area of a cone is given by the formula A = π * r * l, where l is the slant height. In this case, we know that the curved surface area is n cm^2:
n = π * r * l
l = n/(πr)
Now, we can find the vertical angle of the cone. The vertical angle (θ) is the angle between the slant height and the base of the cone, which can be found using the tangent function:
tan(θ) = r/l
tan(θ) = sqrt(n/(3π)) / (n/(π * sqrt(n/(3π)))
tan(θ) = sqrt(n/(3π)) * (π * sqrt(n/(3π))) / n
tan(θ) = π
Therefore, the vertical angle of the cone is tan^(-1)(π).
a cone of height 9 cm has a volume of n cm^3 and a curved surface area of n cm^2. Find the vertical angle of the cone
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