A closed box with a square base and vertical sides is to hold 2000 cm3. The material for the top and the bottom costs Rs. 3 per cm2 and the material for the sides costs Rs. 1.50 per cm2. The least possible cost of the material for the box is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Rs. 1800
Let the base be x cm by x cm and the height h cm. The volume fixes h:
\( \displaystyle x^{2}h = 2000,\ h = \frac{2000}{x^{2}} \)
The top and bottom have area \(2x^{2}\); the four sides have area \(4xh\). The cost is
\( C = 3 \times 2x^{2} + 1.5 \times 4xh \)
\( \displaystyle = 6x^{2} + \frac{12000}{x} \)
Setting the derivative equal to zero:
\( \displaystyle C' = 12x - \frac{12000}{x^{2}} = 0 \)
\( x^{3} = 1000,\ x = 10 \)
Since \(C'' = 12 + \frac{24000}{x^{3}} \gt 0\), this gives the least cost. Then h = 20, and
\( C = 600 + 1200 = 1800 \)
Why the other options are incorrect
Option (1) Rs. 1200 — Incorrect. This is only the cost of the four sides at x = 10; the top and bottom (Rs. 600) are left out.
Option (2) Rs. 2400 — Incorrect. The cost reaches Rs. 1800 at x = 10, so 2400 is not the least cost.
Option (4) Rs. 1500 — Incorrect. This counts only one of the top and bottom (Rs. 300 instead of Rs. 600); the box is closed.