Question 1201500: A farmer has 6000m of fencing and wishes to create a rectangular field subdivided into 4 congruent plots of land. Determine the dimensions of each plot of land if the area to be enclosed is a maximum.
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39620)   (Show Source):
Answer by math_tutor2020(3817) (Show Source):
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It will depend on what layout the farmer goes with.
Are the plots laid in a single straight line (configuration A)?
Or are the plots in a 2x2 grid (configuration B)?
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Configuration A:
Perimeter = sum of exterior sides
Perimeter = 8x+2y
8x+2y = 6000
2(4x+y) = 6000
4x+y = 6000/2
4x+y = 3000
y = -4x+3000
area of one plot = xy
area of four plots = 4xy
area of four plots = 4x(-4x+3000)
area of four plots = -16x^2+12000x
We wish to max out this area.
Use a graphing tool such as GeoGebra or Desmos to graph the parabola.
The vertex is located at (375, 2250000)
This is the highest point on the parabola.
This means a single plot that has horizontal width of x = 375 meters leads to a total max area of 2,250,000 square meters.
This applies only if configuration A is used.
The y value would be
y = -4x+3000
y = -4*375+3000
y = 1500
Each smaller plot is 375 meters by 1500 meters.
area of one plot = 375*1500 = 562,500 square meters
area of four plots = 4*562500 = 2,250,000 square meters
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Configuration B:
Perimeter = sum of exterior sides
Perimeter = 4x+4y
4x+4y = 6000
4(x+y) = 6000
x+y = 6000/4
x+y = 1500
y = -x+1500
area of one plot = xy
area of four plots = 4xy
area of four plots = 4x(-x+1500)
area of four plots = -4x^2+6000x
Follow similar steps as the previous configuration.
This time the vertex is located at (750,2250000)
This means a horizontal width of x = 750 leads to a max area of 2250000 square meters.
This applies only if configuration B is used.
y = -x+1500
y = -750+1500
y = 750
Each smaller plot is 750 meters by 750 meters.
area of one plot = 750*750 = 562,500 square meters
area of four plots = 4*562500 = 2,250,000 square meters
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Summary:
Configuration A has each smaller plot of 375 meters by 1500 meters.
Configuration B has each smaller plot of 750 meters by 750 meters.
Both configurations lead to a max total area of 2,250,000 square meters when considering all four plots.
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