Respuesta :
Answer:
Length of the rectangle = 9.58 feet and width = 27.14 foot
Step-by-step explanation:
Let the length of the rectangular area is = x feet
and the width of the area = y feet
Area of the rectangle = xy square feet
Or xy = 260
y = [tex]\frac{260}{x}[/tex] -------(1)
Cost to fence the three sides = $3 per foot
Therefore cost to fence one length and two width of the rectangular area
= 3(x + 2y)
Similarly cost to fence the fourth side = $14 per foot
So, the cost of the remaining length = 14x
Total cost to fence = 3(x + 2y) + 14x
Cost (C) = 3(x + 2y) + 14x
C = 3x + 6y + 14x
= 17x + 6y
From equation (1)
C = [tex]17x+\frac{260\times 6}{x}[/tex]
Now we take the derivative,
C' = 17 - [tex]\frac{1560}{x^{2} }[/tex]
To minimize the cost of fencing,
C' = 0
17 - [tex]\frac{1560}{x^{2} }[/tex] = 0
[tex]\frac{1560}{x^{2} }[/tex] = 17
[tex]x^{2} =\frac{1560}{17}[/tex]
[tex]x=\sqrt{\frac{1560}{17} }[/tex]
x = 9.58 foot
and y = [tex]\frac{260}{9.58}[/tex]
y = 27.14 foot
There are several ways to minimize or maximize a function.
The dimensions of the enclosure that is most economical to construct are 27.1 ft by 9.6 ft
Let the sides of the rectangle be x and y.
Such that:
[tex]\mathbf{Area = xy}[/tex]
The area is given as: 260 square feet.
So, we have:
[tex]\mathbf{xy = 260}[/tex]
Make x the subject
[tex]\mathbf{x = \frac{260}y}[/tex]
The perimeter (P) of the rectangle is:
[tex]\mathbf{P=2(x + y)}[/tex]
Expand
[tex]\mathbf{P=2x + 2y}[/tex]
Split
[tex]\mathbf{P=2x + y + y}[/tex]
Three sides cost $3 per foot, while the fourth costs $14 per foot.
So, the cost function is:
[tex]\mathbf{C=3 \times (2x + y) + 14 \times y}[/tex]
[tex]\mathbf{C=6x + 3y + 14y}[/tex]
[tex]\mathbf{C=6x + 17y}[/tex]
Substitute [tex]\mathbf{x = \frac{260}y}[/tex]
[tex]\mathbf{C = 6 \times \frac{260}y + 17y}[/tex]
[tex]\mathbf{C = \frac{1560}y + 17y}[/tex]
Rewrite as:
[tex]\mathbf{C = 1560y^{-1} + 17y}[/tex]
Differentiate
[tex]\mathbf{C' = -1560y^{-2} + 17}[/tex]
Set to 0
[tex]\mathbf{ -1560y^{-2} + 17 = 0}[/tex]
Subtract 20 from both sides
[tex]\mathbf{ -1560y^{-2} = -17}[/tex]
Divide both sides by -1560
[tex]\mathbf{y^{-2} = \frac{-17}{-1560}}[/tex]
Cancel out negatives
[tex]\mathbf{y^{-2} = \frac{17}{1560}}[/tex]
Take inverse of both sides
[tex]\mathbf{y^2 = \frac{1560}{17}}[/tex]
Take the square roots of both sides
[tex]\mathbf{y = \sqrt{\frac{1560}{17}}}[/tex]
[tex]\mathbf{y = 9.6}[/tex]
Recall that:
[tex]\mathbf{x = \frac{260}y}[/tex]
Substitute 9.6 for y
[tex]\mathbf{x=\frac{260}{9.6}}\\[/tex]
[tex]\mathbf{x=27.1}[/tex]
Hence, the dimensions of the enclosure that is most economical to construct are 27.1 ft by 9.6 ft
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