Answer:
The maximum area is 18062.5 ft².
Step-by-step explanation:
Consider the provided information.
A farmer with 850 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle.
The required figure is shown below.
The total length of wire is 850
The perimeter is the sum of all side which is the same as the total length of the wire, therefore
Total fence = Length of 3 parallel wall + Sides of rectangle
[tex]850=3x+2x+2y[/tex]
[tex]850=5x+2y[/tex]
[tex]y=\frac{850-5x}{2}[/tex]
The area of rectangle is A=x×y
Substitute the value of y.
[tex]A=\frac{850-5x}{2}\times x[/tex]
[tex]A=425x-\frac{5x^2}{2}[/tex]
Now differentiate and substitute A'=0 to find the maximum.
[tex]A'=425-5x[/tex]
[tex]425-5x=0[/tex]
[tex]x=85[/tex]
Now substitute the value of x in [tex]y=\frac{850-5x}{2}[/tex]
[tex]y=\frac{850-5(85)}{2}=212.5[/tex]
We need to find the total areas of theses configurations.
[tex]A=85\times 212.5=18062.5[/tex]
Hence, the maximum area is 18062.5 ft².
The area of a shape is the amount of space it occupies.
The maximum area of each pen is 4515.625 square feet, while the maximum area of the farm is 18052.5 square feet
When the farm is divided equally, the farm will have 4 pens of equal dimensions.
See attachment for illustration.
The perimeter (P) of the farm is:
[tex]\mathbf{P = x + x + y + y + y + y + y}[/tex]
[tex]\mathbf{P = 2x + 5y}[/tex]
The perimeter is given as 850.
So, we have:
[tex]\mathbf{2x + 5y = 850}[/tex]
Make 5y the subject
[tex]\mathbf{5y = 850 - 2x}[/tex]
Divide through by 5
[tex]\mathbf{y = \frac{850 - 2x}{5}}[/tex]
The area of the farm is:
[tex]\mathbf{A = xy}[/tex]
Substitute [tex]\mathbf{y = \frac{850 - 2x}{5}}[/tex]
[tex]\mathbf{A = x \times \frac{850 - 2x}{5}}[/tex]
[tex]\mathbf{A = \frac{850x - 2x^2}{5}}[/tex]
Split
[tex]\mathbf{A = \frac{850}{5}x - \frac{2}{5}x^2}}[/tex]
[tex]\mathbf{A = 170x - \frac{2}{5}x^2}}[/tex]
Differentiate with respect to x
[tex]\mathbf{A' = 170 - \frac{4}{5}x}}\\[/tex]
Set to 0
[tex]\mathbf{170 - \frac{4}{5}x} = 0}[/tex]
Collect like terms
[tex]\mathbf{- \frac{4}{5}x} = -170}[/tex]
Cancel out negatives
[tex]\mathbf{ \frac{4}{5}x} = 170}[/tex]
Multiply both sides by 5/4
[tex]\mathbf{ x = 212.5}[/tex]
Recall that:
[tex]\mathbf{A = 170x - \frac{2}{5}x^2}}[/tex]
So, we have:
[tex]\mathbf{A = 170 \times 212.5 - \frac 25 \times 212.5^2}[/tex]
[tex]\mathbf{A = 18062.5}[/tex]
The area of each pen is:
[tex]\mathbf{Pen = \frac{A}{4}}[/tex]
[tex]\mathbf{Pen = \frac{18062.5}{4}}[/tex]
[tex]\mathbf{Pen = 4515.625}[/tex]
Hence, the maximum area of each pen is 4515.625 square feet, while the maximum area of the farm is 18052.5 square feet
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