Question

A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109sq. m, then what is the width of the road?

• A. 3m
• B. 2m
• C. 4m
• D. 5m

Answer 1

Answer:

A. 3m

Area and Mensuration - Competitive Exam Focus

• Understanding the Problem: This problem involves calculating the dimensions of a rectangular park with a lawn and two intersecting roads. The key is to relate the total area, the area of the lawn, and the area of the roads.
• Key Concepts:
  • Area of a rectangle = length × width
  • When two roads intersect at the center, the area of overlap (a square) is counted twice if we simply add the areas of the two roads.
• Formulating the Solution:
  • Let the length of the park be $L = 60$ m and the width be $W = 40$ m.
  • Let the width of each road be $x$ meters.
  • The two roads run through the middle, so one road is parallel to the length and the other is parallel to the width.
  • Area of the road parallel to length = $L imes x = 60x$ sq. m.
  • Area of the road parallel to width = $W imes x = 40x$ sq. m.
  • The area where the two roads intersect is a square with side $x$, so its area is $x imes x = x^2$ sq. m.
  • Total area of the roads = (Area of road parallel to length) + (Area of road parallel to width) - (Area of overlap)
    • Total area of roads = $60x + 40x - x^2 = 100x - x^2$ sq. m.
  • Total area of the park = $L imes W = 60 imes 40 = 2400$ sq. m.
  • Area of the lawn = Total area of the park - Total area of the roads
    • Given Area of lawn = 2109 sq. m.
    • So, $2400 - (100x - x^2) = 2109$
    • $2400 - 100x + x^2 = 2109$
    • Rearranging the terms to form a quadratic equation: $x^2 - 100x + 2400 - 2109 = 0$
    • $x^2 - 100x + 291 = 0$
• Solving the Quadratic Equation:
  • The quadratic equation is $x^2 - 100x + 291 = 0$.
  • We need to find two numbers that multiply to 291 and add up to -100.
  • Factoring 291: $291 = 3 imes 97$.
  • So, the numbers are -3 and -97.
  • The equation can be factored as $(x - 3)(x - 97) = 0$.
  • This gives two possible solutions for $x$: $x = 3$ or $x = 97$.
• Interpreting the Solution in Context:
  • The width of the road ($x$) cannot be 97 m, as the width of the park is only 40 m. A road of 97 m width is not physically possible within the park dimensions.
  • Therefore, the only logical solution is $x = 3$ m.
• Competitive Exam Tip: In mensuration problems involving shapes within shapes, always draw a diagram. This helps visualize the overlap and prevents calculation errors. For quadratic equations arising from such problems, quickly check if the roots are physically plausible before selecting the final answer.