Find the altitude of a right circular cone of maximum curved surface which can be inscribed of radius "r"
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To find the altitude of a right circular cone with maximum curved surface area that can be inscribed within a circle of radius "r," we can use the concept of optimization.
Let's start by understanding the problem. A cone can be inscribed within a circle if the circle passes through the apex (the topmost point) of the cone. The curved surface area of a cone is given by the formula:
CSA = π * r * l
Where CSA is the curved surface area, r is the radius of the base of the cone, and l is the slant height of the cone.
To maximize the curved surface area, we need to optimize the value of r and l. However, since the radius of the base is given as "r," we need to find the slant height "l" in terms of "r" to optimize the function.
We can do this using the Pythagorean theorem. According to the Pythagorean theorem, the slant height "l" is related to the radius "r" and the height "h" of the cone as follows:
l^2 = r^2 + h^2
Since we want to find the slant height in terms of "r," we can substitute h = r in the equation:
l^2 = r^2 + (r^2)
l^2 = 2r^2
l = sqrt(2r^2)
l = sqrt(2) * r
Now that we have expressed the slant height "l" in terms of "r," we can substitute it back into the formula for curved surface area:
CSA = π * r * l
CSA = π * r * sqrt(2) * r
CSA = π * sqrt(2) * r^2
This expression gives us the curved surface area of the cone in terms of "r".
To maximize the curved surface area, we can differentiate the expression with respect to "r" and set it equal to zero. Taking the derivative, we get:
d(CSA)/dr = 2π * sqrt(2) * r
Setting the derivative equal to zero, we find the critical point:
2π * sqrt(2) * r = 0
Solving for "r," we find:
r = 0
However, since radius cannot be zero, we discard this value.
Hence, there is no maximum curved surface area for a cone inscribed within a circle of radius "r." The curved surface area continues to increase as the height and radius of the cone increase. Therefore, there is no specific altitude for a cone with the maximum curved surface area inscribed within a circle of radius "r."