Question

A closed (cuboid) box with a square base is to have a volume $2000c.c$. The material for the top and bottom of the box is to cost Rs.$3$ per square $cm$ and the material for the sides is to cost Rs.$1.50$ per square $cm$. If the cost of the materials is to be the least, find the dimensions of the box.

Answer 1

Let $x$ be the length of side of square base.
$y$ be height of cuboid and $c$ be the total cost (in Rs.)
Area of bottom $=x^2$
Area of top $=x^2$
Combined area of top and bottom $=2x^2$
Area of $4$ side$=4xy$
cost of the material for top and bottom $=3\left(2x^2\right)$
$=6x^2$
Cost of the material for the side $=1.5\left(4xy\right)=6xy$
Total cost $c=6x^2+6xy$ ......(1)
Volume of the cost $v=$ area depth $=x^{2}y$
(i.e.,) $x^{2}y=2000$ ....(given)
Eliminating from (1) and (2)
$c\left(x\right)=6x^2+\frac{12000}{x}$
$c^{\prime }\left(x\right)=12x-\frac{12000}{x^2}$
$12x^2-12000=0\to 12(x^2-{10}^2)=0$
$\Rightarrow$ $x=10$ or $x^2+10x+100=0$
$x^2+10x+100=10$ is not possible.
$\therefore$ the critical number is $x=10$
$c^{\prime \prime }\left(x\right)=12+\frac{24000}{x^3}$ $\Rightarrow$ $c^{\prime \prime }\left(10\right)=\frac{24000}{1000}>0$
$\therefore$ $c$ is min at $(10,1800)$
Base length is $10m$ and depth is $20m$.