The circular iron sheet with radius r subtracts a sector, and the remaining sector with center angle of θ is made into a conical funnel. When the value of θ is taken, the volume of the funnel will increase large

The circular iron sheet with radius r subtracts a sector, and the remaining sector with center angle of θ is made into a conical funnel. When the value of θ is taken, the volume of the funnel will increase large

Let R be the bottom radius and H be the height of a conical vessel. Then 2 π r = α R, r = α R / (2 π). H = √ (R & sup2; - R & sup2;) and R be the independent variable. The volume of the vessel v = (1 / 3) π R & sup2; H = (1 / 3) π R & sup2; √ (R & sup2; - R & sup2;) is derived from R: V ′ = (1 / 3) π R [2R & sup2; - 3R & sup2;]

A sector is dug out from a round iron sheet with radius r to make a funnel. When the central angle of the sector is large, the volume of the funnel is the largest
In particular, the expression of the height and the radius of the bottom of the cone are given to show whether there are these equations in the functional formula: v = 1 / 3 * π R * r * H R * r + h * H = R * r & R = 2 π r halo. We have worked out for several hours without any result

When the center angle is (2 √ 6) π / 3 = 294 ° the funnel volume is the largest
MATLAB has not been used, but can be calculated
Let the center angle be a radian, the circle length of crater be ar, the radius of crater be (A / 2 π) and the sectional area of crater be a ^ 2R ^ 2 / 4 π
The funnel height is R √ [1 - (A / 2 π) ^ 2] = (R / 2 π) √ (4 π ^ 2-A ^ 2)
Funnel volume = mouth area × height △ 3 = (R ^ 3 / 24 π ^ 2) a ^ 2 √ (4 π ^ 2-A ^ 2)
When a ^ 2 √ (4 π ^ 2-A ^ 2) is the maximum, the funnel volume is the largest
a^2√(4π^2－a^2) =√(4π^2 a^4－a^6)
Because a

Make a funnel with an upper diameter of 40cm and a height of 12cm. How to calculate the unfolding sector radius
Vertical height: 12cm

The radius of the top of the cone can be calculated by the diameter of the top. Then the radius and height can be used to calculate the radius of the expanded sector. The arc length of the expanded sector is the circumference of the top of the cone. Using the arc length and radius, the radian can be calculated. Knowing the radius and radian, the area can be calculated naturally