Question

the population of a certain state can be estimated by the equation p = 70.5t + 19,372.3, where p represents the population of the state in thousands of people t years since 2010. (a) find the slope of the equation. (do not include units in your answer.) m = what does it mean in regard to the population of the state? (include units in your answer.) the population of the state is increasing by (t, p) = ( ) (b) find the vertical - intercept of the equation. give the meaning of the vertical intercept in regard to the population of the state. (include units in your answers as appropriate.) the population of the state was in the year . (t, p) = ( ) (c) find the horizontal intercept of the equation. (round your answers to one decimal place.) give the meaning of the horizontal intercept in regard to the population of the state. (round your second answer down to the nearest year. include units in your answers as appropriate.) the population of the state was in the year .

Explanation:

Step1: Identify the slope - intercept form

The equation of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. In the population equation $P=70.5t + 19372.3$, it is in the form $P=mt + b$ with $t$ as the independent variable (years since 2010) and $P$ as the dependent variable (population in thousands).

Step2: Find the slope

For part (a), the slope $m$ of the equation $P = 70.5t+19372.3$ is $70.5$. It means the population of the state is increasing by 70.5 thousand people per year.

Step3: Find the vertical intercept

For part (b), the vertical intercept occurs when $t = 0$. Substitute $t = 0$ into the equation $P=70.5t + 19372.3$. We get $P=19372.3$. So, $(t,P)=(0,19372.3)$. The meaning is that the population of the state in the year 2010 (when $t = 0$ years since 2010) was 19372.3 thousand people.

Step4: Find the horizontal intercept

For part (c), the horizontal intercept occurs when $P = 0$. Set $P = 0$ in the equation $0=70.5t+19372.3$. Then $70.5t=- 19372.3$, and $t=\frac{-19372.3}{70.5}\approx - 274.8$. Rounding to one decimal place, $t\approx - 274.8$. So, $(t,P)\approx(-274.8,0)$. The meaning is that if the linear trend continued backwards in time, the population would have been 0 approximately 274.8 years before 2010.

Answer:

(a) $70.5$
(b) $(0,19372.3)$; The population of the state in 2010 was 19372.3 thousand people.
(c) $(-274.8,0)$; If the linear trend continued backwards in time, the population would have been 0 approximately 274.8 years before 2010.