Problem 75 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.

Step-by-Step Solution

Answer

The statement 'When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative' makes sense as in an exponential model, a negative growth rate represents a declining population.

Step 1 - Understanding exponential functions for population modeling

Exponential functions are widely used in population modeling. Typically, the model is of the form \(P(t) = P_0e^{kt}\), where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(e\) is the base of the natural logarithm (approximately 2.71828), and \(k\) is the growth rate. If the population is growing, \(k\) is positive; if the population is declining, \(k\) is negative. Thus, the direction of the population change (increase or decrease) is directly linked to the sign of \(k\).

Step 2 - Evaluating the given statement

The statement says, 'When I used an exponential function to model Russia's declining population, the growth rate \(k\) was negative.' This means that the growth rate was less than zero, indicating a decline in the population. Therefore, in the context of the exponential model, a negative growth rate makes sense for a declining population.