Question

A tank with a rectangular base and rectangular sides, open at the top is made. Depth of the tank is \(4\) m and its volume is \(36\) cubic meters. For making a tank cost of base material used is Rs. \(100\) per sq. meter and that of sides is Rs. \(50\) per sq. meter. Then minimum cost of tank is ..............

• A. Rs. \(1100\)
• B. Rs. \(2200\)
• C. Rs. \(3300\)
• D. Rs. \(4400\)

Hint:

Fix the base area from the volume, then minimise the cost of the sides using AM-GM.

Answer 1

The Correct Option is C

Step 1: Set Up:
Depth \(=4\) m and volume \(=36\ \text{m}^3\), so the base area is \(lb=36/4=9\ \text{m}^2\).

Step 2: Cost Function:
Base cost: \(100\times9=900\) rupees. Side area: \(2(l+b)\times4=8(l+b)\), at 50 per sq. m, giving \(400(l+b)\).
\[ C=900+400(l+b) \]

Step 3: Minimise:
With \(lb=9\), AM-GM gives \(l+b\geq2\sqrt{lb}=6\), with equality at \(l=b=3\).

Step 4: Minimum Cost:
\[ C_{min}=900+400\times6=900+2400=3300 \]
So the minimum cost is Rs. 3300. Option (A) 1100 and (B) 2200 are below the side cost alone.

Final Answer:
The minimum cost is Rs. 3300, option (C). \[ \boxed{\text{(C) Rs. } 3300} \]