A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is  `2m` and volume is `8 m^3` . If building of tank costs `Rs. 70` per sq. metre for the base and `Rs. 45` per sq. metre for sides, what is the cost of least expensive tank?

Explanation

Let the length of tank be x metre, breadth by y metre and depth be 2 metre.
`:. ` Volume of tank = 2xy
` rArr 2xy = 8`
` rArr xy = 4 rArr y = 4/x ` ….(1)
`:. ` area of base = xy = 4 and area of four walls
` = 2x + 2y + 2x+ 2y`
` = 4x + 4y = 4 (x+y)`
Given that cost of maunfacturing the base of tank is Rs.`70//m^(2) and" for walls, it is "Rs. 45//m^(2)`.
`:. ` Cost of manufacturing,
`C = Rs. [ 70 xx xy + 45 xx 4(x+y)]`
` = Rs. [70xy + 180 (x+y)]`
` C = 70 xx 4 + 180 (x+4/x)` [ From eqn.(1)]
` = 280 + 180 (x+4/x)`
` rArr (dC)/(dx) = 0 + 180 (1-4/x^(2))= 180 ((x^(2)-4)/x^(2))`
For minimum cost, ` (dC)/(dx) = 0`
` rArr 180 ((x^(2)-4)/x^(2)) = 0 rArr x^(2) = 4`
` rArr x = pm 2`
Now ` (d^(2)C)/(dx^(2)) = 180 xx 8/x^(3)`
at ` x = 2, (d^(2)C)/(dx^(2)) gt 0 `
` :. ` at x = 2, C is minimum.
`:. at x = 2, y = 4/x = 4/2 = 2`
From equation (2), minimum cost of manufacture,
`= Rs. [ 280 + 180(2+4/2)]`
` = Rs. [ 280+720] = Rs. 1000`.