Question

The population of a town has been decreasing since 1991. The population of the town was 26,700 people in the year 1995. In the year 2003, the population of the town had decreased to 20,300 people.

i. Find the linear function P=m- t+b that models the town's population P as a function of the number of years, t, since 1991
ii. Find a reasonable domain for the function P
Domain:
iii. Find a reasonable range for the function P
Range:
iv. Interpret the slope of the function P
The population is Select an answer at a rate of
v. In what year will the population of the town be 14,700?
Year:

Answer 1

i. P = -800t + 29900

ii. as an inequality: 0≤t≤37.375; interval notation: [0,37.375]

iii. as an inequality: 0≥P≥29900; interval notation: [0,29900]

iv. the population decreases at a rate of 1600 people every 3 years

v. 2010

Explanation:

i. a) use the points *(4,26700) and **(12,20300) to find the slope (m)

*: the year 1995(4 years after 1991) and the population that year

**: the year 2003(12 years after 1991) and the population that year

m = (20300-26700)/(12-4) = -6400/8 = -800

b) use one of the points and m in (a) to find b

26700 = -800(4) + b

26700 = -3200 + b

29900 = b

ii. starting value for domain will be 0 since 1991 is year 0. Find the ending domain value by using P=0 and solving the equation for t:

0 = -800t + 29900

800t = 29900

t = 37.375

iii. population starts at a high of 29900 people and can't go lower than 0 people.

iv. the slope is given in (population change) / (number of years)

v. make P=14700 and solve the equation for t

14700 = -800t + 29900

800t = 15200

t = 19 (19 years after 1991)