Question

An open tank with a square bottom is to contain 4000 cubic cm . of liquid. The dimensions of the tank so that the surface area of the tank is minimum, is

• A. side = 20 cm, height = 10 cm
• B. side = 10 cm, height = 20 cm
• C. side = 10 cm, height = 40 cm
• D. side = 20 cm, height = 05 cm

Hint:

For an open tank with square base, Surface Area is minimum when $x = 2h$.

Answer 1

The Correct Option is A

Step 1: Volume and Area
$V = x^2 h = 4000 \implies h = \frac{4000}{x^2}$.
Surface Area $S = x^2$ (bottom) $+ 4xh$ (walls).
Step 2: Minimize Area
$S(x) = x^2 + 4x(\frac{4000}{x^2}) = x^2 + \frac{16000}{x}$.
$S'(x) = 2x - \frac{16000}{x^2} = 0 \implies 2x^3 = 16000$.
Step 3: Solve
$x^3 = 8000 \implies x = 20 \text{ cm}$.
$h = \frac{4000}{20^2} = \frac{4000}{400} = 10 \text{ cm}$.
Final Answer: (A)