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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 2m and volume is 8m3. 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 ?

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Solution

Let the length and breadth of the tank be x metre and y metre, respectively. The depth (height) is 2m.
Volume of tank =2×x×y=2xy=8
[Given volume of tank =8 m3]
xy=4
y=4x ...(i)

Also, area of the base =xy=x×4x=4
and area of the four sides =2x+2y÷2x+2y=4x+4y=4(x+y)
It is given that, the cost of construction on base is Rs 70 per sq. metre and for side is Rs 45 per sq metre
So, cost of construction, C=Rs[70×xy+45×4(x+y)]
= Rs [70xy + 180(x+y)] ...(ii)
On putting value of y in Eq. (ii) from Eq. (i).we get
C=70×4+180(x+4x)=280+180(x+4x) ...(iii)
On differentiating w.r.t.x, we get dCdx=0+180(14x2)=180(x24x2)
For least expensive, put dCdx=0180(x24x2)=0x2=4x=±2
dCdx changes sign from negative to positive at x = 2.
C is minimum at x = 2
[length of the tank cannot be negative. So, x = - 2 is not consider]
x = 2 and y=4x=42=2
Thus, tank is a cube of side 2m.
Least cost of construction [from Eq. (iii)]
=Rs[280+180(2+42)]=Rs[280+720]=Rs1000


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