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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 meters for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

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Solution

## Let l, b, and h represent the length, breadth, and height of the tank respectively. Then, we have height (h) = 2 m Volume of the tank = 8m3 Volume of the tank = l × b × h ∴ 8 = l × b × 2 Now, area of the base = lb = 4 Area of the 4 walls (A) = 2h (l + b) However, the length cannot be negative. Therefore, we have l = 4. Thus, by second derivative test, the area is the minimum when l = 2. We have l = b = h = 2. ∴Cost of building the base = Rs 70 × (lb) = Rs 70 (4) = Rs 280 Cost of building the walls = Rs 2h (l + b) × 45 = Rs 90 (2) (2 + 2) = Rs 8 (90) = Rs 720 Required total cost = Rs (280 + 720) = Rs 1000 Hence, the total cost of the tank will be Rs 1000.

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