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The farmer stated that when moving the circular irrigation system along the dimensions of the rectangular field, it will cover the whole area of the field (without overlapping). Verify whether his statement is TRUE.

Mathematics

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Step-by-step solution

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3.1.1

  1. Convert the dimensions of the field from kilometers to meters.

    Length=8 km=8×1000 m=8000 m\text{Length} = 8 \text{ km} = 8 \times 1000 \text{ m} = 8000 \text{ m} Width=4 km=4×1000 m=4000 m\text{Width} = 4 \text{ km} = 4 \times 1000 \text{ m} = 4000 \text{ m}

  2. Calculate the area of the rectangular field using the given formula.

    Area=length×width\text{Area} = \text{length} \times \text{width} Area=8000 m×4000 m\text{Area} = 8000 \text{ m} \times 4000 \text{ m} Area=32,000,000 m2\text{Area} = 32,000,000 \text{ m}^{2}

    The area of the field is 32,000,000 m2\boxed{32,000,000 \text{ m}^{2}}.

    3.1.2
    One source of water for irrigation is a river.
    The answer is River\boxed{\text{River}}.

    3.1.3

  3. Convert the radius from meters to kilometers.

    Radius=250 m=2501000 km=0.25 km\text{Radius} = 250 \text{ m} = \frac{250}{1000} \text{ km} = 0.25 \text{ km}

  4. Calculate the area covered by the circular irrigation system using the given formula.

    Area of circle=3.142×(radius)2\text{Area of circle} = 3.142 \times (\text{radius})^{2} Area=3.142×(0.25 km)2\text{Area} = 3.142 \times (0.25 \text{ km})^{2} Area=3.142×0.0625 km2\text{Area} = 3.142 \times 0.0625 \text{ km}^{2} Area=0.196375 km2\text{Area} = 0.196375 \text{ km}^{2}

    The area that the system can cover is 0.196375 km2\boxed{0.196375 \text{ km}^{2}}.

    3.1.4

  5. Determine the effective diameter of the circular area covered by the system.

    Radius=250 m\text{Radius} = 250 \text{ m} Diameter=2×radius=2×250 m=500 m\text{Diameter} = 2 \times \text{radius} = 2 \times 250 \text{ m} = 500 \text{ m}

  6. Calculate how many times the system needs to be moved along the length and width of the field, assuming each move covers a strip equal to its diameter without overlapping.

    Field length=8 km=8000 m\text{Field length} = 8 \text{ km} = 8000 \text{ m} Field width=4 km=4000 m\text{Field width} = 4 \text{ km} = 4000 \text{ m} Number of moves along length=8000 m500 m=16\text{Number of moves along length} = \frac{8000 \text{ m}}{500 \text{ m}} = 16 Number of moves along width=4000 m500 m=8\text{Number of moves along width} = \frac{4000 \text{ m}}{500 \text{ m}} = 8

  7. Calculate the total number of times the system needs to be moved to cover the entire field.

    Total moves=Moves along length×Moves along width\text{Total moves} = \text{Moves along length} \times \text{Moves along width} Total moves=16×8=128\text{Total moves} = 16 \times 8 = 128

    The number of times he will need to move the circular system is 128\boxed{128}.

    3.1.5

  8. Consider the geometry of covering a rectangular area with circles without overlapping.

    A circular irrigation system covers a circular area. If these circular areas are placed side-by-side without overlapping to cover a rectangular field, there will inevitably be gaps between the circles. Circles cannot perfectly tile a rectangular plane.

  9. Verify the farmer's statement.

    Since circles cannot perfectly cover a rectangular area without leaving gaps, the farmer's statement that moving the circular irrigation system along the dimensions of the rectangular field will cover the whole area of the field (without overlapping) is incorrect.
    The statement is FALSE\boxed{\text{FALSE}}.

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