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Maximum volume of cylinders that can be made to have a constant surface area.

작성자 Uploader : alice 작성일 Upload Date: 2019-01-02변경일 Update Date: 2019-01-06조회수 View : 323

Type1 Type2
If the surface area of ​​the cylinder is S, the volume is V, the diameter of the cylinder circle is D, and the height of the cylinder is H, the following relation can be established.

S = 2*(π/4)*D^2 + π*D*H

V = (π/4)*D^2*H

Since S is a constant, it can be written as follows.

H = S/(π*D) - (D/2)

Substituting H into the equation of V, it is as follows.

V = (π/4)*D^2*(S/(π*D) - (D/2))

  = (S/4)*D - (π/8)*D^3

Differentiating this with respect to D, the maximum value is obtained.

S/4 - (3*π/8)*D^2 = 0



Thus, the maximum value of the volume is :



On the other hand, the equation in No.1 can be expressed as follows.

S = (3/2) * D ^ 2 * π

When this is solved by substituting it into the surface area formula,

(3/2)*D^2*π = 2*(π/4)*D^2 + π*D*H

D^2 = D*H

Thus,

D = H

That is, when the diameter and the height are the same, the volume of the cylinder becomes the maximum.
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