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


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


then, we have h = 2m


 


Volume of the tank = 8 m3


 


Volume of the tank = l × b × h


 


⇒ 8 = l × b × 2


 


⇒ lb = 4


 


⇒ b =


 


Now, area of the base = lb = 4


 


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


 



 


Now,


 


= 0


 


⇒ l2 = 4


 


⇒ l = ±2


 


Since, length cannot be negative therefore l =2.


 


⇒ b = 2


 


Now,


 


When l =2,


 


Then, by second derivative test, the area is the minimum when l =2.


 


We have, l =b=h=2


 


Therefore, 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.


 


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


 

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