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Question

A tank with rectangular base and rectangular sides, open at the top is to the constructed so that its depth is 2 m and volume is 8 m3. If building of tank cost 70 per square metre for the base and Rs 45 per square matre for sides, what is the cost of least expensive tank?

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Solution

Let l, b and h be the length, breadth and height of the tank, respectively.

Height, h = 2 m

Volume of the tank = 8 m3

Volume of the tank = l × b × h

l × b × 2 = 8

lb=4b=4l

Area of the base = lb = 4 m2

Area of the 4 walls, A= 2h (l + b)

A=4l+4ldAdl=41-4l2For maximum or minimum values of A, we must havedAdl=041-4l2=0l=±2

However, the length cannot be negative.

Thus,
l = 2 m

b=42=2 mNow, d2Adl2=32l3At l=2:d2Adl2=328=4>0

Thus, the area is the minimum when l = 2 m

We have
l = b = h = 2 m

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

Total cost = Rs (280 + 720) = Rs 1000

Hence, the total cost of the tank will be Rs 1000.

Disclaimer: The solution given in the book is incorrect. The solution here is created according to the question given in the book.


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