A-closed-cylindrical-can-be-is-to-hold-1-liters-of-liquid-How-should-we-choose-the-height-and-radius-to-minimize-the-amount-of-material-needed-to-manufacture-the-can-

Question Number 141368 by bemath last updated on 18/May/21

A c l o s e d c y l i n d r i c a l c a n b e i s t o h o l d

1 l i t e r s o f l i q u i d . H o w s h o u l d w e

c h o o s e t h e h e i g h t a n d r a d i u s

t o m i n i m i z e t h e a m o u n t o f

m a t e r i a l n e e d e d t o m a n u f a c t u r e

t h e c a n ?

Answered by TheSupreme last updated on 18/May/21

y o u h a v e t o m i n i m i z e

S = 2 π R^{2} + π R H

u n d e r t h e c o n s t r a i n

V = π R^{2} H = 1 l t

H = \frac{V}{π R^{2}}

S = 2 π R^{2} + \frac{V}{R}

m i n (S)

\frac{\partial S}{\partial R} = 4 π R - \frac{V}{R^{2}} = 0 \to R =^{3} \sqrt{\frac{V}{4 π}}

R = 4.30 c m

H = 17.21 c m

Commented by iloveisrael last updated on 18/May/21

r \approx 5.4 c m

Answered by iloveisrael last updated on 18/May/21