# Show that the Height of a Cylinder, Which is Open at the Top, Having a Given Surface Area and Greatest Volume, is Equal to the Radius of Its Base. - Mathematics

Sum

Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.

#### Solution

H be the height
V be the volume
S be the total surface area
V = πR2 H

S = πR2 + 2π RH

⇒ H = ("S" - π"R"^2)/(2π"R")

Substituting value of H in V

"V" = 1/2 ("SR" = π"R"^3)

(d"V")/(d"R") = 1/2 ("S" -3π"R"^2)

(d"V")/(d"R") = 0

⇒ 1/2 ("S" -3π"R"^2) = 0

"R" = sqrt("S"/(3π)

(d^2"V")/(d"R"^2) = 1/2 (0 - 6π"R")

= -3πR

(d^2"V")/(d"R"^2) = -3πsqrt("S"/(3π)

= -sqrt3πS < 0

V is greatest when R = sqrt("S"/(3π)

H = ("S" - π xx ("S")/(3π))/(2πsqrt("S"/(3π))

H = ((2S)/3)/(2sqrt((piS)/3))

H = sqrt("S"/(3π)

Hence, proved that radius is equal to the height of the cylinder.

Concept: Simple Problems on Applications of Derivatives
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