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: Understanding the Concept:
This is an optimization problem. We need to express the surface area in terms of a single variable using the volume constraint and then find the minimum value using differentiation.
Step 2: Key Formula or Approach:
Let the side of the square base be $x$ and the height be $h$.
Volume $V = x^2 h = 4000 \implies h = \frac{4000}{x^2}$.
Surface area for an open tank $S = x^2 + 4xh$.
Step 3: Detailed Explanation:
Substitute $h$ into the area equation:
\[ S(x) = x^2 + 4x \left( \frac{4000}{x^2} \right) = x^2 + \frac{16000}{x} \] Differentiate with respect to $x$:
\[ \frac{dS}{dx} = 2x - \frac{16000}{x^2} \] For minimum surface area, set $\frac{dS}{dx} = 0$:
\[ 2x = \frac{16000}{x^2} \implies x^3 = 8000 \implies x = 20 \text{ cm} \] Find the height $h$:
\[ h = \frac{4000}{20^2} = \frac{4000}{400} = 10 \text{ cm} \] Step 4: Final Answer:
The dimensions are side = 20 cm and height = 10 cm.