show that the semi vertical angle of a right circular cone of a given surface area and maximum value is sin-1 (1/3)
To find the semi-vertical angle of a right circular cone with a given surface area, we first need to derive the equation for the surface area of a cone. The surface area (A) of a cone is given by the formula:
A = πr^2 + πrL
Where:
- π is a mathematical constant approximately equal to 3.14159
- r is the radius of the base of the cone
- L is the slant height of the cone
For a right circular cone, the slant height (L) can be expressed in terms of the radius (r) and the height (h) of the cone using the Pythagorean theorem:
L = √(r^2 + h^2)
Now, let's find the derivative of the surface area equation with respect to the height (h) to maximize the surface area:
dA/dh = d(πr^2 + πrL)/dh
= πr(dL/dh)
The derivative of L with respect to h can be calculated by applying the chain rule:
dL/dh = d(√(r^2 + h^2))/dh
= (1/2)(r^2 + h^2)^(-1/2)(2h)
= h/(√(r^2 + h^2))
Substituting this back into the derivative equation, we get:
dA/dh = πr(h/(√(r^2 + h^2)))
Now, to maximize the surface area, we set the derivative equal to zero and solve for h:
dA/dh = πr(h/(√(r^2 + h^2))) = 0
Since r cannot be zero (otherwise there would be no cone), we can divide both sides of the equation by πrh:
h/(√(r^2 + h^2)) = 0
We can then square both sides of the equation to get rid of the square root:
h^2/(r^2 + h^2) = 0
h^2 = 0
The only solution to this equation is h = 0. However, h cannot be zero since it represents the height of the cone. Therefore, there is no maximum surface area for a cone with a given radius. Thus, it is not possible to prove that the semi-vertical angle of the cone is equal to sin^(-1)(1/3).