Let the breadth be x, length be 2x and height be h
V=x.2x.h
⇒c=2x2h ....(1)
Area of bottom =2x2 =Area of top
Area of sides =2xh+2xh+xh+xh=6xh.
If R rupees be the cost of material for bottom then for the top and sides is 3R.
∴E=R(2x2)+3R(2x2+6xh)
⇒E=R(8x2+18xh)
or E=R(8x2+18xc2x2)
⇒E=R(8x2+9cx)
where R and c are constants.
dEdx=R(16x−9cx2)
For maximum or minimum,
dEdx=0
∴x=(9c16)1/3
Also, d2Edx2=R(16+18cx3)=+ive
and hence minimum.
∴ dimensions are
[9c16]1/3,2[9c16]1/3 and [32c81]1/3.