Question

Show that the height of a closed right circular cylinder of maximum volume that can be inscribed

Answer 1

hey mate

in two dimensions it looks like a rectangle inscribed in an isosceles triangle. 



let the base of this triangle lie on the x-axis, with the two sides of the triangle symmetrical to the y-axis. 



let the base of the triangle = 2R, height = H 



side of the triangle in quad I has equation y = H - (H/R)x 



base of the cylinder = x 



height of the cylinder = y = H - (H/R)x 



so, cylinder volume is ... 



V =*x*[H - (H/R)x] = H[x- x/R] 



dV/dx = H[2x - 3x/R] 



set dV/dx = 0, x(2 - 3x/R) = 0 



x = 2R/3 ... the cylinder radius for max volume

 

so height of the cylinder=y=H-(H/R)(2R/3)

y=H/3


 

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