An open box with a square base is to be made out of a given cardboard of area c2 sq. units. Show that the max vol. of the box is c3/(6√3) cubic units.
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1 Answers
106Let the height of the box be h and let the side of the base be a.
Then Total surface area required = 4ha+a2
But total area = c2
So, a2+4ah = c2 ... (i)
Volume = a2h
= a2(c2-a2)/4a
= (ac2-a3)/4 .. (ii)
This is max when dV/da = 0
i.e. c2-3a2=0
==> a = c/√3
from (ii) Vmax = [(c/√3)c2-c3/3√3)]/4
= c3/6√3
