$P$ gives the population of a town $t$ years after it was founded. $\newline$ What is the best interpretation for the following statement? $\newline$ The slope of the line tangent to the graph of $P$ at $t=2$ is equal to $-10$ . $\newline$ Choose $1$ answer: $\newline$ (A) After $2$ years, the rate of change of the population decreased at a rate of $10$ people per year per year. $\newline$ (B) After $2$ years, the town's population decreased at a rate of $10$ people per year. $\newline$ (C) Over the first two years, the town's population decreased by $10$ people per year. $\newline$ (D) After $2$ years, the town's population decreased at a rate of $10$ people.

Full solution

P

gives the population of a town

t

years after it was founded.

\newline

What is the best interpretation for the following statement?

\newline

The slope of the line tangent to the graph of

P

t=2

is equal to

-10

\newline

Choose

1

answer:

\newline

(A) After

2

years, the rate of change of the population decreased at a rate of

10

people per year per year.

\newline

(B) After

2

years, the town's population decreased at a rate of

10

people per year.

\newline

10

people per year.

\newline

(D) After

2

years, the town's population decreased at a rate of

10

people.

Population Rate of Change: The slope of the line tangent to the graph of $P$ at any point represents the rate of change of the population at that specific time.
Slope at $t=2$ : Since the slope at $t=2$ is $-10$ , this means that at $t=2$ , the population is decreasing at a rate of $10$ people per year.
Interpretation of Slope: The correct interpretation of the slope being $-10$ at $t=2$ is that after $2$ years, the town's population is decreasing at a rate of $10$ people per year. This matches with choice (B).