`100 cm^2` आयतन बाले डिब्बे सभी बंद बेलनकार ( लंब वृतीय ) डिब्बों में से न्यूनतम पृष्ठ क्षेत्रफल वाले डिब्बे की विमाएँ ज्ञात कीजिए

Answer

Radus =`(50)/(pi)^(1/3);Height =2(50)/(pi)^(1/3)`

Explanation

Let r and ha be the radius and height of the cylinder respectively Then volume (V) of the cylinder is given by ltrbgt `V=pir^(2)h=100`
`therefore h=(100)/(pir^(2))`
surface (s) of the cylinder is given by
`S=2pir^(2)+2pirh=2pirr^(2)+(200)/(r)`
`therefore (dS)/(dr)=4pir-(200)/(r^(2)),(d^(2)S)/(dr^(2))=4pir+(400)/(r^(3))`
`(dS)/(dr)=0rarr4pir=(200)/(r^(2))=(50)/(pi)^(1/3)`
For `r=(50)/(pi)^(1/3),(d^(2)S)/(dr^(2))gt0`
`therefore` By second derivation test the surface area is the minimum when the radius of the cylinder is `(50)/(pi)^(1/3)` cm
When r =`(50)/(pi)^(1/3) ,h =(100)/pi((50)/(pi))^(2//3)=2((50)/(pi))^(1//3)`
Hence the requaried dimension of the which has the minimum surface area is given by radius =`(50)/(pi)^(1/3)` cm and height `=2(50)/(pi)^(1//3)cm`