99 Percentile Qs Bank for JEE MainMathematicsApplication of Derivatives
If a cylindrical vessel of given volume V with no lid on the top is to be made from a sheet of metal, then the radius (r) and height (h) of the vessel so that the metal sheet used is minimum, is
Options
- Ar= [3] V , h= [3] V
- Br= V , h= V
- Cr= [3] V , h= [3] V
- Dr= V , h= V
Correct answer
C. r= [3] V , h= [3] V
Step-by-step solution
We have given, V= volume of cylindrical vessel r= radius and h= height As we know for cylindrical vessel v= r^2 h Let S be the area of metal sheet used to form a cylindrical vessel. Then, aligned & S=2 r h+ r^2=2 r V r^2 + r^2 [using Eq. (i) ] & S(r)= 2 V r + r^2 aligned For minimum value of S(r), S^ (r)=0 aligned & -2 V r +2 r=0 2 V r^2 =2 r & r^3= V r= [3] V aligned Now, by using the value of r in Eq. ( i) aligned V & = ( V )^ 2 3 h & h= V ^ 2 / 3 v^ 2 / 3 h= V^ 1 / 3 ^ 1 / 3 = [3] V aligned Hence, the required r