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

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Solution

Let l,b,h be the length, breadth and height of the tank respectively.
Then h=2 m
And, volume of tank =8 m3
l×b×h=8l×b=4b=4l
Now, area of the base =lb=4
And, area of the walls =2lh+2bh=2h(l+b)
Therefore, total area is
A=2h(l+b)+lbA=4(l+4l)+4

Differentiating A with respect to l, we get
dAdl=4(14l2)+0
Now, putting dAdl=0, we get
4(14l2)=014l2=0l2=4
l=2 [Since length can't be negative]
l=2b=4l=2

d2Adl2=32l3
At l=2,d2Adl2=328=4>0
Therefore, by second derivative test, area is minimum when l=2.
So, we get l=b=h=2
Therefore, cost of building base =70×lb=70×4= Rs. 280
Cost of building walls =45×(2h(l+b))=45×4×4= Rs. 720
Therefore, least cost of tank is Rs. 280+720= Rs. 1000

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