A tank with rectangular base and rectangular sides, open at the t… - Vidyadip

Question

A tank with 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 building of tank costs Rs 70 per sq.metres for the base and Rs 45 per sq. metre for sides. What is the cost of least expensive tank ?

✓

Answer

Let x and y be the length and width of rectangular base, v be the volume.

v = 8 (Given)

v = 2xy

8 = 2xy

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

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

$ = x \times \frac{4}{x} \times 70 + 90\left( {x + \frac{4}{x}} \right)$

$ = 280 + 90\left( {x + \frac{4}{x}} \right)$

$\frac{{ds}}{{dx}} = 0 + 90\left( {1 - \frac{4}{{{x^2}}}} \right)$

=$\frac{{{d^2}s}}{{d{x^2}}} = 90\left( {0 + \frac{8}{{{x^3}}}} \right)$

For maximum/minimum

$\frac{{ds}}{{dx}} = 0$

x = 2

${\left( {\frac{{{d^2}s}}{{d{x^2}}}} \right)_{x = 2}}=\frac{{720}}{{{2^3}}} > 0$

s is Minimum at x=2

Minimum cost is given by

$s = 280 + 90\left( {2 + \frac{4}{2}} \right)$

= 280+ 90 (4)

= 280+360

= 640

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