Let a right circular cylinder of radius “R” and height “H” is inscribed in the sphere of given radius “r”.
Let V be the volume of the cylinder.
Then, V = πR2H
Differentiating both sides w.r.t H to get,
For maximum value put dV/dH = 0
Again, differentiating w.r.t H we get,
So, volume is maximum when height of cylinder is .
Substitute in (1) to get,
Let the length and breadth of the tank are L and B.
∴ V = 8
2LB = 8
Total surface area S = Area of base + Area of 4 walls
= LB + 2(B+L).2
The cost of constructing the tank is:
C = 70(LB) + 45(4B + 4L)
Differentiating both sides w.r.t L we get,
For minimization dC/dL = 0,
⇒ L2 = 4
⇒ L = 2
Differentiate (3) w.r.t L to get,
∴ Cost is minimum when L = 2.
B = 2
= 280 + 720
= Rs 1000
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