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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=8⇒l×b=4⇒b=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)+lb⇒A=4(l+4l)+4 Differentiating A with respect to l, we get dAdl=4(1−4l2)+0 Now, putting dAdl=0, we get 4(1−4l2)=0⇒1−4l2=0⇒l2=4 ⇒l=2 [Since length can't be negative] l=2⇒b=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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