5 Suppose the gross population density function for an urban

5. Suppose the gross population density function for an urban area is estimated to be:

D(x) = 7500*e^(-0.2x), where e is the base of the natural logarithms and x is distance to downtown in miles, and D(x) is the population density in people per square mile.

a. What is the value of the density gradient for this urban area? Be sure to specify the units in which the gradient is measured.

b. Fill in the blanks in the following table:

Distance          Gross Density

                                    0                      _______

                                    1                      _______

                                    10                    _______

                                    _______          4000

                                    _______          1514

Solution

a) density gradient is change in density due to change distance by 1 mile

dD(x)dx = (-0.2) 7500 e^-0.2x

this is the density gradient and its unit is people per square mile

b) when x=0 , D(x) =7500 e^-0.2(0) =7500

whenx=1,   D(x) = 7500 e^-0.2(1) = 7500*0.82= 6150

when x=10 ,D(x) = 7500 e^-0.2(10)= 7500*0.14 =1050

when D(x) =4000 , x=?    D(x) = 7500 e^-0.2x

                                                 4000 = 7500 e^-0.2x

                                      0.53 = e^-0.2x

taking log both sides

log0.53 = -0.2x

-.63=-0.2x

x= 3.15

when D(x)=1514

1514= 7500 e^-0.2x

0.2 =e^-0.2x

taking log both sides

log 0.2 = -0.2x

-1.61 =-0.2x

x=8.05