Question

The speed of traffic through the Lincoln Tunnel depends on the density of the traffic. Let S be the speed in miles per hour and D be the density in vehicles per mile. The relationship between S and D is approximately S=42-(D/3) for D less than or equal to 100. Find the density that will maximize the hourly flow.

Answer 1

hourly flow = speed*density
\[F = D(42-\frac{D}{3}) = 42D - \frac{D^2}{3}\]
maximize by setting derivative equal to 0
\[\frac{dF}{dD} = 42 - \frac{2D}{3} = 0\]
\[D = 63\]