A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 4 meter and volume is 36 cubic meters. If building of the tank costs ₹ \(100\) per square meter for the base and ₹ \(50\) per square meter for the sides, then the cost of least expensive tank is

Let length and breadth of the tank be ' \(x\) ' \(\mathrm{m}\) and ' \(y\) ' \(m\) respectively.
Height of the tank is \(4 \mathrm{~m}\).
Volume \(=36 \mathrm{~m}^3\)
\(
\begin{array}{ll}
\therefore & 4 x y=36 \\
\therefore & x y=9 \\
\therefore & y=\frac{9}{x}
\end{array}
\)
\(\therefore \) Total area of the tank including sides and base \(=x y+2(4 x)+2(4 y)\)
....[From (i) and (ii)]
\(\therefore \mathrm{f}(x)=9+8 x+8\left(\frac{9}{x}\right)\)
\(
=9+8 x+\frac{72}{x}
\)
\(
\therefore \mathrm{f}^{\prime}(x)=8-\frac{72}{x^2}
\)
\(\therefore \mathrm{f}^{\prime}(x)=0 \Rightarrow x=3\)
\(
\Rightarrow y=3
\)
\(\therefore \) Required cost \(=100 \times(3 \times 3)+50\) \(\times(2 \times 4 \times 3+2 \times 4 \times 3) \)
\( =900+2400 \)
\(=\) ₹ \(3300\)