The table shows the projected population of a small town. Let tequals0 correspond to the year 2030. Complete parts (a) through (d) below.
Year, t
0
5
10
15
Population comma p
4700
5250
5800
6350
Question content area bottom
Part 1
a. Graph the data. Does the graph suggest a linear, exponential, or quadratic model? Choose the correct graph below.
A.
0
10
20
4000
8000
t
p
x y graph
B.
0
10
20
4000
8000
t
p
x y graph
C.
0
10
20
4000
8000
t
p
x y graph
D.
0
10
20
4000
8000
t
p
The correct graph is:
B.
0
10
20
4000
8000
t
p
b. Find the rate of change in population with respect to time from one data pair to the next.
The rate of change is
enter your response here per year.
The rate of change in population from one data pair to the next can be calculated by finding the difference in population and dividing by the difference in time.
Rate of change = (p2 - p1) / (t2 - t1)
Using the data pairs (0, 4700) and (5, 5250):
Rate of change = (5250 - 4700) / (5 - 0) = 550 / 5 = 110
Therefore, the rate of change in population with respect to time is 110 per year.
How do the results support your answer to part (a)?
A.
The fact that the rate of change is positive supports the answer from part (a).
B.
The fact that the rate of change is constant supports the answer from part (a).
C.
The fact that the rate of change is an integer supports the answer from part (a).
D.
The fact that the rate of change is negative supports the answer from part (a).
A. The fact that the rate of change is positive supports the answer from part (a).
Since the rate of change is positive, it suggests a linear growth model, which is consistent with the choice of the correct graph A in part (a). Linear models exhibit a constant rate of change, with a positive slope indicating that the population is increasing at a constant rate.
c. Write a function that models the data shown in the table.
pequals
enter your response here
(Simplify your answer. Type an expression using t as the variable.)
To write a linear function that models the data shown in the table, we need to find the equation of the line passing through two points. We can use the point-slope form of a linear equation:
The two points are (0, 4700) and (15, 6350).
The slope (m) can be calculated as:
m = (p2 - p1) / (t2 - t1)
m = (6350 - 4700) / (15 - 0)
m = 1650 / 15
m = 110
Let's use the point-slope form with the point (0, 4700):
p - 4700 = 110(t - 0)
p - 4700 = 110t
p = 110t + 4700
Therefore, the function that models the data shown in the table is: p = 110t + 4700
d. Use the function from part (c) to predict the town's population in 2055.
The function predicts that the town's population in 2055 will be
enter your response here.
(Simplify your answer.)
To predict the town's population in 2055 using the function derived in part (c), we need to substitute t = 2055 into the function p = 110t + 4700:
p = 110(2055) + 4700
p = 225050 + 4700
p = 229750
Therefore, the function predicts that the town's population in 2055 will be 229,750.