Difficulty: Medium
Correct Answer: 3 m
Explanation:
Introduction / Context:
This question tests area subtraction and handling overlap. When two roads cross at the center, their combined area is not simply added because the central square (the overlap region) is counted twice. So we compute total park area, subtract lawn area to get road area, then set up the correct expression for the union of two rectangles with an overlap.
Given Data / Assumptions:
Concept / Approach:
Area covered by road along length = 60*x. Area covered by road along width = 40*x. Overlap at center is a square of area x*x, which gets counted twice in 60x + 40x, so subtract it once. Thus, road area = 60x + 40x - x^2.
Step-by-Step Solution:
Total park area = 60*40 = 2400 m^2
Road area = Total area - Lawn area = 2400 - 2109 = 291 m^2
Set up: 60x + 40x - x^2 = 291
100x - x^2 = 291
x^2 - 100x + 291 = 0
Factor: (x - 3)(x - 97) = 0
x = 3 m or x = 97 m, but 97 m is impossible for a 40 m wide park
So road width x = 3 m
Verification / Alternative check:
If x = 3, road area = 60*3 + 40*3 - 3^2 = 180 + 120 - 9 = 291 m^2. Lawn area = 2400 - 291 = 2109 m^2, exactly as given.
Why Other Options Are Wrong:
1 m: road area becomes 60 + 40 - 1 = 99 m^2, lawn would be 2301 m^2.
2 m: road area becomes 120 + 80 - 4 = 196 m^2, lawn would be 2204 m^2.
4 m: road area becomes 240 + 160 - 16 = 384 m^2, lawn would be 2016 m^2.
5 m: road area becomes 300 + 200 - 25 = 475 m^2, lawn would be 1925 m^2.
Common Pitfalls:
Forgetting to subtract the overlap x^2 once.
Subtracting lawn area from the wrong total.
Accepting an impossible root that does not fit the park dimensions.
Final Answer:
Width of each road = 3 m
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