# A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 2 m and volume is 8 m3. If the 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?

Given$h = 2$

Volume = 8

$\begin{matrix} lbh = V\\ lbh = 8\\ lb = 4 \end{matrix}\qquad \begin{matrix} \text{length} = x \\ \text{breath} = y \end{matrix}$

$xy = 4 \Rightarrow y = \frac{x}{4}$

Let C be the cost of the tank then,

$C = 70xy +45(2\times 2x + (2\times 2y))$

$= 70xy +180 x + 180 y$

$= 70x\frac{4}{x} +180 x + 180 \frac{4}{x}$

$=280 +180 x + \frac{720}{x}\qquad - (i)$

Now, differentiating both sides with respect to $x$, we get

$\frac{\text{d} C}{\text{d}x} = 180 -\frac{720}{x^2}$

For maxima & minima $\Rightarrow \frac{\text{d} C}{\text{d}x} = 0$

$180 -\frac{720}{x^2} = 0 \\\Rightarrow 180 = \frac{720}{x^2} \\\Rightarrow x^2 = \frac{720}{180} \\\Rightarrow x^2 = 4\Rightarrow x =\pm 2$

Again, differentiating both sides w.r.t $x$, we get

$\frac{\text d^2 C}{\text d x^2} = \frac{1440}{x^3}$

$\left .\frac{\text d^2 C}{\text d x^2}\right |_{\text {at }x = 2} = \frac{1440}{8} = 180 > 0$

$\therefore$ When $x =2$, the cost of the tank is minimum

Substituting the value of $x$ in equation (i), we get

$C = 280 + 180\times 2 + \frac{720}{2}$

$= 280 + 180\times 2 + 360$

$= 280 +360 + 360$

$= 1000$

Hence, the cost of the least expensive tank is Rs. 1000.

## Related Chapters

### Preparation Products

##### Knockout KCET 2021

An exhaustive E-learning program for the complete preparation of KCET exam..

₹ 4999/- ₹ 2999/-
##### Knockout KCET JEE Main 2021

It is an exhaustive preparation module made exclusively for cracking JEE & KCET.

₹ 27999/- ₹ 16999/-
##### Knockout NEET Sept 2020

An exhaustive E-learning program for the complete preparation of NEET..

₹ 15999/- ₹ 6999/-
##### Rank Booster NEET 2020

This course will help student to be better prepared and study in the right direction for NEET..

₹ 9999/- ₹ 4999/-