## A rectangular area of 36 f t2 is to be fenced off. Three sides will use fencing costing $1 per foot and the remaining side will use fencing

Question

A rectangular area of 36 f t2 is to be fenced off. Three sides will use fencing costing $1 per foot and the remaining side will use fencing costing $3 per foot. Find the dimensions of the rectangle of least cost. Make sure to use a careful calculus argument, including the argument that the dimensions you find do in fact result in the least cost (i.e. minimizes the cost function).

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2021-10-11T00:12:35+00:00
2021-10-11T00:12:35+00:00 1 Answer
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## Answers ( )

Answer:x = 8,49 fty = 4,24 ftStep-by-step explanation:Let x be the longer side of rectangle and y the shorterArea of rectangle = 36 ft² 36 = x* y ⇒ y =36/xPerimeter of rectangle:P = 2x + 2y for convinience we will write it as P = ( 2x + y ) + yC(x,y) = 1 * ( 2x + y ) + 3* yThe cost equation as function of x is:C(x) = 2x + 36/x + 108/xC(x) = 2x + 144/xTaking derivatives on both sides of the equationC´(x) = 2 – 144/x²C´(x) = 0 2 – 144/x² = 0 ⇒ 2x² -144 = 0 ⇒ x² = 72x = 8,49 ft y = 36/8.49 y = 4,24 ftHow can we be sure that value will give us a minimunWe get second derivativeC´(x) = 2 – 144/x² ⇒C´´(x) = 2x (144)/ x⁴so C´´(x) > 0condition for a minimum