For the following applied max/min question, only formalize the problem (i.e. ind

Question

For the following applied max/min question, only formalize the problem (i.e. indicate constraint equation(s) and objective equations), first setting up the parameters: "A farmer wants to fence 3 sides of a rectangular yard, adjacent to a river bank (see the picture). The opposite to the bank is to be made of stone that would cost 12 dollars per foot, and the other two sides are to be made of wood with the cost 4 dollars per foot. Assuming that the projected area of the yard is 720 ft^2, find the dimensions that would provide minimal expenses for the farmer": Constraint equation(s):

Explanation / Answer

Given projected area is 720square ft

now let x and y be the length and breadth of the fence

xy= 750square ft since the area of rectangle = xy

now the fence perimeter is given by 2(x+y) -x because river side is not included

=> f(x) = x+2y now y = 720/x

=> f(x) = x+2(750/x) = x +1500/x

to get the minimum dimensions

f'(x) = 0 => 1-1500/x^2 = 0

=> x^2 = 1500 => x = 38.73 and therefore y = 19.36

minimum dimensions are x= 38.73 and y = 19.36

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