Question

A tank with rectangular box and rectangular sides, open at the top is to be constructed so that its depth is $2m$ and volume is $8m^3$. If building of tank costs $Rs.70$ per sq metres for the base and $Rs.45$ per square metre for sides. What is the cost of least expensive tank?

Answer 1

Let $l,b$ and $h=2m$ be the length, breadth and depth of the tank respectively.

Volume of the tank $=8m^3$

$\Rightarrow lbh=8$

$\Rightarrow lb\left(2\right)=8$

$\Rightarrow lb=4m^2$

$\Rightarrow b=\frac{4}{l}$

Area of the base $=lb=4m^2$

Total area of the sides $=2lh+2bh=2(l+b)h=4(l+b)m^2$

Total cost of building the tank $=70\times 4+45\times 4(l+b)=280+180(l+b)=280+180(l+\frac{4}{l})$

Now, applying AMGM property, we get that

$l+\frac{4}{l}\ge 2\sqrt{l.\frac{4}{l}}$

$\Rightarrow l+\frac{4}{l}\ge 4$

Therefore, for the cost to be minimum, $l+\frac{4}{l}$ should be minimum, i.e, equal to $4$.

Therefore, minimum cost $=280+180\left(4\right)=280+720=1000$

Hence, minimum cost of building the tank is Rs. $1000$