Difficulty: Medium
Correct Answer: 2000 vehicles per hour and 40 km per hour
Explanation:
Introduction / Context:
The Greenshields linear model assumes speed v decreases linearly with density k: v = v_f * (1 − k/k_j), where v_f is free-flow speed and k_j is jam density. Flow q is the product of speed and density. The model predicts a parabolic flow–density curve with a distinct maximum (capacity) at an intermediate density.
Given Data / Assumptions:
Concept / Approach:
Flow q(k) = k * v = k * v_f * (1 − k/k_j) = v_f * (k − k^2/k_j). The parabola attains its maximum at dq/dk = 0 → k = k_j / 2. At that density, speed is v = v_f / 2 and capacity is q_max = v_f * k_j / 4.
Step-by-Step Solution:
Verification / Alternative check:
Using q_max = v_f * k_j / 4 directly: 80 * 100 / 4 = 2000 veh/h; v* = v_f / 2 = 40 km/h. Both methods agree.
Why Other Options Are Wrong:
Options A and B exaggerate capacity by a factor of 4 (confusing km with m or omitting the 1/4 factor). Option C keeps free-flow speed at capacity (incorrect). Option E picks arbitrary values inconsistent with the linear model.
Common Pitfalls:
Multiplying speed by jam density directly, or assuming capacity occurs at free-flow speed rather than at half of both v_f and k_j in this model.
Final Answer:
2000 vehicles per hour and 40 km per hour
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