Question

Number of Victimizations per 1.000 A model for the number of robberies in a city is given by the equation \( r(t)=10.9-3.2 \ln t \), where \( t \) is years since 1990. Find the relative rate of change

A) r'(t) = -3.2ln(t)
B) r'(t) = -3.2
C) r'(t) = -3.2/t
D) r'(t) = 3.2ln(t)

Answer 1

The relative rate of change of the model for the number of a city's robberies, which follows a logarithmic function, is found by differentiation and is inversely proportional to time; the correct choice is C) r'(t) = -3.2/t.

Step-by-step explanation:

The model given for the number of robberies in a city is a logarithmic function r(t) = 10.9 - 3.2 ∙ ln(t), where t is the number of years since 1990. To find the relative rate of change of this model, we need to take the derivative of r(t) with respect to t, denoted as r'(t).

Using the rules of differentiation for logarithmic functions, the derivative of -3.2 ∙ ln(t) with respect to t is -3.2/t. Therefore, the correct choice for the relative rate of change is:

C) r'(t) = -3.2/t

This result essentially tells us that the relative rate of change of the number of robberies in the city is inversely proportional to the number of years since 1990. the correct choice is C)