An open top rectangular storage tank has a volume of 8 cubic units. Your task, as a designer, is to find the dimensions of tank such that the surface area of the tank is minimised, thus minimising construction material costs.
i)When you have found the dimensions you should have found that the length and width have the same value by symmetry. What is this common value
ii)What is the value of the height of the tank?
iii)Find the minimum surface area of the contruction material required
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Hi,
Let's use the Lagrange technique: Define "x"=width=length "y"=height. Thus (x^2)y=8 -> (x^2)y-8=0. This is the constrainment. The expression we need to minimize is ("z"=surface area) z=(x^2)+4xy. According to the technique we should now build the equation (x^2)+4xy-L[(x^2)y-8]=0 ["L"=Lagrange Variable]. The next step is to use a gradient on the latter equation and to look for critical points on equation with 0. So together with the constrainment we'll have 3 expressions:
(x^2)y=8
2x+4y-2xyL=0
4y-Lx^2=0
You should now solve these equations and substitute the numbers to the "z" equation. The last step is to choose the minimal result.
I'm sorry I couldn't bother to solve the equations.
i) 2 ii) 2 iii) 20 (Bottom (4) + 4 sides (each 4))
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