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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Jade 2 weeks 2021-10-11T00:12:35+00:00 1 Answer 0

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    2021-10-11T00:14:18+00:00

    Answer:

    x = 8,49 ft

    y  = 4,24  ft  

    Step-by-step explanation:

    Let  x be the longer side of rectangle   and  y  the shorter

    Area of rectangle     =    36 ft²     36  =  x* y   ⇒ y =36/x

    Perimeter of rectangle:

    P  =  2x   +   2y    for convinience we will write it as    P  = ( 2x + y ) + y

    C(x,y)   =  1 * ( 2x  +  y  )  +  3* y

    The cost equation as function of x is:

    C(x)  =  2x  + 36/x   + 108/x

    C(x)  =  2x  + 144/x

    Taking derivatives on both sides of the equation

    C´(x)  = 2  – 144/x²

    C´(x)  = 0         2  – 144/x² = 0       ⇒  2x²  -144 = 0    ⇒  x² =  72

    x = 8,49 ft       y  = 36/8.49    y  = 4,24  ft    

    How can we be sure that value will give us a minimun

    We get second derivative

    C´(x)  = 2  – 144/x²      ⇒C´´(x)  = 2x (144)/ x⁴

    so C´´(x) > 0

    condition for a minimum

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