A right circular cone has a volume of cubic inches. What shape should it be in order to have the smallest lateral surface area? Find the result of the volume is V cubic inches.
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To find the shape of the right circular cone that has the smallest lateral surface area, we need to consider mathematical optimization principles.
The lateral surface area of a right circular cone is given by the formula:
Lateral Surface Area = πrℓ
where r is the radius of the base and ℓ is the slant height of the cone.
Given that the volume of the cone is V cubic inches, we can use the formula for the volume of a cone to express the radius in terms of the volume:
V = (1/3)πr^2h
where h is the height of the cone.
Now, we want to minimize the lateral surface area of the cone while keeping the volume constant. To do this, we need to find the relationship between the radius and height that satisfies the given volume condition.
Rearranging the volume formula:
r^2h = 3V / π
Taking the square root of both sides:
r = √(3V / πh)
Now we can substitute this value of r into the lateral surface area formula:
Lateral Surface Area = πrℓ
Lateral Surface Area = π * (√(3V / πh)) * ℓ
To minimize the lateral surface area, we need to find the relationship between h and ℓ that gives the smallest value.
Differentiating the lateral surface area formula with respect to ℓ to find the critical points:
d(Lateral Surface Area) / dℓ = 0
(π * (√(3V / πh)) * ℓ)' = 0
Simplifying:
(π * √(3V / πh)) = 0
√(3V / πh) = 0
Since √(3V / πh) cannot equal zero, there are no critical points.
Therefore, the lateral surface area will have a minimum value when there is no solution to d(Lateral Surface Area) / dℓ = 0. In other words, it means the lateral surface area is at its minimum when the slant height (ℓ) is equal to zero.
This means that the shape of the cone with the smallest lateral surface area is a cone with zero slant height, which is a degenerate cone—a cone that degenerates into a flat disk.
Thus, the cone should be flattened into a shape resembling a disk in order to have the smallest lateral surface area.