Question

p gives the population of a town t years after it was founded. what is the best interpretation for the following statement? the slope of the line tangent to the graph of p at t = 2 is equal to -10. choose 1 answer: a after 2 years, the rate of change of the population decreased at a rate of 10 people per year per year. b after 2 years, the towns population decreased at a rate of 10 people per year.

Explanation:

The slope of the tangent line to a function at a point represents the instantaneous rate of change of the function at that point. Here, \( P(t) \) is the population as a function of time \( t \). The slope of the tangent at \( t = 2 \) being \( - 10 \) means at \( t=2 \) (after 2 years), the population's rate of change is \( - 10 \) people per year (the negative sign indicates a decrease).

Option A refers to the rate of change of the rate of change (which would be the second derivative), but we are dealing with the first derivative (instantaneous rate of change of population), so A is incorrect. Option B correctly interprets the slope of the tangent (instantaneous rate of change) at \( t = 2 \) as the town's population decreasing at a rate of 10 people per year after 2 years.

Answer:

B. After 2 years, the town's population decreased at a rate of 10 people per year.