#### Question

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

#### Solution

Let *l*, *b* and *h* be the length, breadth and height of the tank, respectively.

Height, *h* = 2 m

Volume of the tank = 8 m^{3}

Volume of the tank = l × b × h

∴ *l *× *b* × 2 = 8

\[\Rightarrow lb = 4\]

\[ \Rightarrow b = \frac{4}{l}\]

Area of the base = lb = 4 m^{2}

Area of the 4 walls, *A*= 2*h* (*l* + *b*)

\[\therefore A = 4\left( l + \frac{4}{l} \right)\]

\[ \Rightarrow \frac{dA}{dl} = 4\left( 1 - \frac{4}{l^2} \right)\]

\[\text { For maximum or minimum values of A, we must have }\]

\[\frac{dA}{dl} = 0\]

\[ \Rightarrow 4\left( 1 - \frac{4}{l^2} \right) = 0\]

\[ \Rightarrow l = \pm 2\]

However, the length cannot be negative.

Thus,*l* = 2 m

\[\therefore b = \frac{4}{2} = 2 m\]

\[\text { Now,} \]

\[\frac{d^2 A}{d l^2} = \frac{32}{l^3}\]

\[\text { At }l = 2: \]

\[\frac{d^2 A}{d l^2} = \frac{32}{8} = 4 > 0\]

Thus, the area is the minimum when* l* = 2 m

We have

l = b = h = 2 m

∴ Cost of building the base = Rs 70 × (lb) = Rs 70 × 4 = Rs 280

Cost of building the walls = Rs 2h (l + b) × 45 = Rs 90 (2) (2 + 2)= Rs 8 (90) = Rs 720

Total cost = Rs (280 + 720) = Rs 1000

Hence, the total cost of the tank will be Rs 1000.

#### Notes

The solution given in the book is incorrect. The solution here is created according to the question given in the book.