SOLUTION: A cone of height 9cm has a volume of n cm³ and a curved surface area of n cm².find its vertical angle?

Question 1134450: A cone of height 9cm has a volume of n cm³ and a curved surface area of n cm².find its vertical angle?
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
Volume of a right circular cone is
:
1) (1/3) * pi * r^2 * 9 = n cm^3
:
Curved surface area is
:
2) pi * r * square root(r^2 +h^2) = n cm^2
:
Note square root(r^2 +h^2) is slant height
:
Square both sides of = for equation 2 and simplify
:
3) n^2 = (pi^2 * r^4) +(pi^2 *r^2 * h^2)
:
Solve equation 1 for r. since V is constant n and h = 9
:
r = square root(3n/(pi * 9)) = square root(n/(3*pi))
:
Substitute for r in equation 3 and simplify
:
pi^2 * ((n/3*pi)^1/2)^4 +(81 * pi^2 * n/(3*pi)) = n^2
:
n^2/9 +27*pi*n = n^2
:
n^2 +243*pi*n = 9n^2
:
8n^2 -243*pi*n = 0
:
n^2 -30.375*pi*n = 0
:
n * (n -30.375*pi) = 0
:
n = 0, n = 30.375*pi
:
r = square root(30.375 * pi / 3 * pi) = 3.182
:
tan theta = 3.182/9 = 0.3536
:
tan^-1 (0.3536) = 19.47
:
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The vertical angle is 2 * 19.47 = 38.94 degrees
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