Question

A rectangular box is to be made from a sheet of area $1200sq.cm.$ if the length of the box is to be twice its width, find the dimensions of the box so that its volume is maximum.

Answer 1

length $=l$ breadth $=b$ height $=h$ and $l=2b$

Area $\Rightarrow 2lb+2bh+2hl=1200$

$\Rightarrow 2\times 2b\times b+2bh+2h\times 2b=1200$

$4b^2+2bh+4bh=1200$

$6bh=1200-4b^2$

$h=\frac{1200-4b^2}{6b}$

$d\left(v\right)=l\times b\times h=2b^2\times \left[\frac{1200-4b^2}{6b}\right]$

$d\left(v\right)=\frac{1200b-4b^3}{3}$

from volume to ve max, $d^{\prime }\left(v\right)=0$

$\frac{d\left(V\right)}{d\left(B\right)}=\frac{1}{3}[1200-12b^2]=0$

$1200=12b^2$$b=10cm$

$l=2\times b=20cm.h=\frac{1200-4\times {10}^2}{6\times 10}=\frac{40}{3}cm$