Question

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

Answer 1

Let $l,b,h$ be the length, breadth and height of the tank respectively.
$\text{Then}h=2\text{m}$
And, volume of tank $=8{\text{m}}^3$
$\Rightarrow l\times b\times h=8\Rightarrow l\times b=4\Rightarrow b=\frac{4}{l}$
Now, area of the base $=lb=4$
And, area of the walls $=2lh+2bh=2h(l+b)$
Therefore, total area is
$A=2h(l+b)+lb\Rightarrow A=4(l+\frac{4}{l})+4$

Differentiating $A$ with respect to $l,$ we get
$\frac{dA}{dl}=4(1-\frac{4}{l^2})+0$
Now, putting $\frac{dA}{dl}=0,$ we get
$4(1-\frac{4}{l^2})=0\Rightarrow 1-\frac{4}{l^2}=0\Rightarrow l^2=4$
$\Rightarrow l=2\left[\text{Since length can't be negative}\right]$
$l=2\Rightarrow b=\frac{4}{l}=2$

$\frac{d^{2}A}{d{l}^2}=\frac{32}{l^3}$
At $l=2,\frac{d^{2}A}{d{l}^2}=\frac{32}{8}=4>0$
Therefore, by second derivative test, area is minimum when $l=2.$
So, we get $l=b=h=2$
Therefore, cost of building base $=70\times lb=70\times 4=$ Rs. $280$
Cost of building walls $=45\times \left(2h\right(l+b\left)\right)=45\times 4\times 4=$ Rs. $720$
Therefore, least cost of tank is Rs. $280+720=$ Rs. $1000$