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 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`.