Q24 of 52 Page 1

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

Let the length and width of base of tank be l and b. Given that the height of tank is 2m and it’s volume is 8 m3.


The volume of a cuboid, of length l, width b and height h, is defined by


V(l,b,h)=lbh


Hence, we get 8=lb× 2 or lb=4m2


or


Let the total cost of building tank be T.


Then T(l,b,h)=70lb+45(lh + bh + lh + bh)


=70lb + 90lh + 90bh


Substituting values of h and lb, we get


T=70×4 +180(l + b)


=280+180(l + b)


Since , we get



Differentiating with respect to b, we get




Differentiating with respect to b, we get



For minima, and



⇒ b2=4 or b=2m


,


hence b=2m is a point of minima for function T and T(2)=₹1000 is the least expensive tank.


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