Question

For this exercise, round all regression parameters to three decimal places.

Characteristics of traffic flow include density D, which is the number of cars per mile, and average speed s in miles per hour. Traffic system engineers have investigated several methods for relating density to average speed. One study† considered traffic flow in the north tube of the Lincoln Tunnel and fitted an exponential function to observed data. Those data are partially presented in the table below.

Speed s	Density D
32	34
25	53
20	74
17	88
13	102

(a) Make an approximate exponential model of D as a function of s.
D(s) =

(b) Express using functional notation the density of traffic flow when the average speed is 24 miles per hour.
D

Calculate that density. (Round your answer to the nearest whole number.)
cars/mi

(c) If average speed increases by 1 mile per hour, what can be said about density? (Round your answer to one decimal places.)

The density D decreases by  %.

Answer 1

Now let represents the density of the vehicles in the tunnel as a function of the speed s. Here D is measured in number of cars per mile and the speed s is measured in miles per hour. We use the regression to find the exponential model for the given data.

First enter the data in to TI-84 PLUS as follows:

Press on STAT and then, press on EDIT.

Then enter the data .

It is needed to find function model of line.

Press on STAT and then, press on CALC.

Then Press on ExpReg.

The following is the screen shot.

Then rounding to three decimal places the exponential model is:

Since the density of the vehicles is a function of the speed and is given as therefore when s = 28 miles per hour, then the density is given by,

From part (a),

Therefore

.

Thus,there are almost 44 cars per mile in the tunnel when the average speed is 28 miles per hour.

Now from part (a) we know the density of the vehicles as a function of the average speed is given by,

This shows that the density of the cars is an exponential decay function of the average speed s. The density is measured in the number of cars per mile and the speed is measured in miles per hour.

The decay factor is 0.942

This means when the speed increases by 1 mile per hour, the density gets multiplied by the factor 0.942.

Now the decay factor can be written as

Or

That is,. the percentage decay rate is 5.8%.

Thus, if the average speed of the vehicles is increased by 1 mile per hour, the density of the vehicles in the tunnel decreases by 5.8%.