A garden is 30 meters wide and 40 meters long. There is a pathway that runs along the perimeter inside the garden. The area of the pathway is half of the area of the garden. What is the width of the pathway?


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Answer:

5 m

Step by Step Explanation:
  1. The following figure shows the garden which is 30 meters wide and 40 meters long with a pathway that runs along the perimeter, inside the garden.

    Here, AB = DC = 40 meters,
    AD = BC = 30 meters.
  2. Let us assume that w is the width of the pathway.
    If we subtract twice the width of the pathway from the length and width of the garden, then it is equal to the length and width of the garden without the pathway, respectively.
    Length of the garden without the pathway = 40 - 2w
    Width of the garden without the pathway = 30 - 2w
    Area of the garden without the pathway = (40 - 2w) × (30 - 2w)
    = 1200 - 80w - 60w + 4w2
    = 1200 - 140w + 4w2
  3. Area of the garden = 40 × 30 = 1200 m2
  4. Area of the pathway = Area of the garden - Area of the garden without the pathway
    = 1200 - (1200 - 140w + 4w2)
    = 1200 - 1200 + 140w - 4w2
    = 140w - 4w2 ----(1)
  5. It is given that the area of the pathway is half the area of the garden.
    Therefore, the area of the pathway = 1200 ×  
    1
    2
      = 600 m2 ----(2)
  6. Comparing equations (1) and (2), we get,
    140w - 4w2 = 600
    ⇒ 4w2 - 140w + 600 = 0
    ⇒ w2 - 35w + 150 = 0
    ⇒ w2 - 30w - 5w + 150 = 0
    ⇒ w(w - 30) - 5(w - 30) = 0
    ⇒ (w - 30)(w - 5) = 0
    either, w - 30 = 0or w - 5 = 0
    ⇒ w = 30⇒ w = 5
  7. Since, w has to be less than the width of the garden, hence the width of the pathway is 5 meters.

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