A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth…
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
$₹ 3000$
$₹ 3300$
$₹ 2400$
$₹ 3500$
Solution
Let length and breadth of the tank be ' $x$ ' $\mathrm{m}$ and ' $y$ ' $m$ respectively.
Height of the tank is $4 \mathrm{~m}$.
Volume $=36 \mathrm{~m}^3$
$\begin{array}{ll}
\therefore & 4 x y=36 \\
\therefore & x y=9 \\
\therefore & y=\frac{9}{x}
\end{array}$
$\therefore \quad$ Total area of the tank including sides and base $=x y+2(4 x)+2(4 y)$
....[From (i) and (ii)]
$\therefore \quad \mathrm{f}(x)=9+8 x+8\left(\frac{9}{x}\right)$
$=9+8 x+\frac{72}{x}$
$\therefore \quad \mathrm{f}^{\prime}(x)=8-\frac{72}{x^2}$
$\therefore \quad \mathrm{f}^{\prime}(x)=0 \Rightarrow x=3$
$\Rightarrow y=3$
$\therefore \quad$ Required cost $=100 \times(3 \times 3)+50$
$\begin{aligned}
& \times(2 \times 4 \times 3+2 \times 4 \times 3) \\
= & 900+2400 \\
= & ₹ 3300
\end{aligned}$