A plot of speed-density relationship (linear) of two roads (Road A and Road B) is shown in the figure. If the capacity of Road A is \(C_A\) and the capacity of Road B is \(C_B\), what is \(\frac{C_A}{C_B}\)?
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When dealing with linear speed-density relationships, capacity is directly proportional to both the speed and the density. Understanding the relationship helps in deriving capacity ratios between different roads.
From the speed-density plot, we know that the capacity of a road is given by the product of the maximum flow speed and the density corresponding to that speed. In a linear speed-density relationship:
\[
C = k \cdot u
\]
Where \(C\) is the capacity, \(k\) is the density, and \(u\) is the speed.
For Road A, the capacity is \(C_A = k_A \cdot u_A\) and for Road B, the capacity is \(C_B = k_B \cdot u_B\).
Thus, the ratio of capacities \(\frac{C_A}{C_B}\) is:
\[
\frac{C_A}{C_B} = \frac{k_A \cdot u_A}{k_B \cdot u_B}
\]
\boxed{\frac{k_A u_A}{k_B u_B}}
