Question

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth in 2 m and volume is 8m³. If building of tank costs ₹ 70 per sq. metre for the base and ₹ 45 per sq. metre for sides, what is the cost of least expensive tank?

Answer 1

Let l and b be the length and breadth of the tank.

I f C be the  cost of constructing the tank then

Differentiating with respect to I,we get

\(\frac{dC}{dl}=180(1-\frac{4}{l^2})\)  ...(i)

For maxima or minima

\(\frac{dC}{dl}=0\) 

⇒ \(180(1-\frac{4}{l^2})\) 

⇒  l₂ = 4

⇒  l = 2

Differentiating (i) again with respect to l, we get

\(\frac{d^2C}{dl^2}=180+\frac{8}{l^3}\) 

⇒ \(\frac{d^2C}{dl^2}]_{l=2}=181>0\) 

[∵ l = - 2 ]

Here C is minimum when l = 2

∴ b = \(\frac{4}{2}\) = 2

Minimum cost,

= 280 + 180 \((2\,+\,\frac{4}{2})\)

= 280 + 720

= ₹1000.