Answer :
The dimensions for the fence that would minimize the cost are,
[tex]x = \frac{600}{\sqrt{11}} , y = 150\sqrt{11}[/tex].
What is dimensions?
Measurements like length, width, or height are referred to as dimensions. When you discuss an object or location's dimensions, you're referring to its size and proportions.
As given: An auto repair shop plans to fence a rectangular parking lot adjacent to their building of 90,000 square feet.
The fence on the sides (the ones labelled as y) will be chain linked and cost $20 per linear foot.
The fence parallel to the building backs up to a busy street and needs to be reinforced steel, which costs $55 per linear foot.
From this the objective function is,
C(x, y) = 55x + 20y ..(1)
The area of rectangle is,
A = lw
90000 = xy
So,
[tex]x = \frac{90000}{y}[/tex] ..(2)
Plug value of x in equation (1).
[tex]C(x, y)=50(\frac{90000}{y})+20y = c(y)[/tex]
Now to minimize C(y).
Differentiating both sides with respect to y,
[tex]C'(y) = 20 - \frac{50(90000)}{y^2} = 0\\ 20 - \frac{4500000}{y^2} = 0\\\frac{4500000}{y^2} = 20\\ y^2 = \frac{4500000}{20}\\ y = 150\sqrt{11}[/tex]
Plug value of y in equation (2),
[tex]x = \frac{90000}{(150\sqrt{11} )} = \frac{600}{\sqrt{11} }[/tex]
Therefore, the fence's measurements, where x is the fence that runs parallel to the building and y is the fence that runs perpendicular to x,
are, [tex]x = \frac{600}{\sqrt{11} } , y = 150\sqrt{11}[/tex].
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