Consider a side road connecting to a major highway at a stop sign. According to a study by D. R. Drew†, the average delay D, in seconds, for a car waiting at the stop sign to enter the highway is given by D = eqT − 1 − qT q, where q is the flow rate, or the number of cars per second passing the stop sign on the highway, and T is the critical headway, or the minimum length of time in seconds between cars on the highway that will allow for safe entry. We assume that the critical headway is T = 5 seconds. (a) What is the average delay time if the flow rate is 470 cars per hour (0.13 car per second)? (Round your answer to two decimal places.) sec (b) The service rate s for a stop sign is the number of cars per second that can leave the stop sign. It is related to the delay by s = D−1. Use function composition to represent the service rate s as a function of flow rate q. Reminder: (a/b)−1 = b/a. s = (c) What flow rate will permit a stop sign service rate of 4.9742 cars per minute (0.0829 car per second)? (Round your answer to one decimal place.) cars per min
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
Consider a side road connecting to a major highway at a stop sign. According to a study by D. R. Drew†, the average delay D, in seconds, for a car waiting at the stop sign to enter the highway is given by
| eqT − 1 − qT |
| q |
where q is the flow rate, or the number of cars per second passing the stop sign on the highway, and T is the critical headway, or the minimum length of time in seconds between cars on the highway that will allow for safe entry. We assume that the critical headway is T = 5 seconds.
sec
(b) The service rate s for a stop sign is the number of cars per second that can leave the stop sign. It is related to the delay by
s =
(c) What flow rate will permit a stop sign service rate of 4.9742 cars per minute (0.0829 car per second)? (Round your answer to one decimal place.)
cars per min
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