Solve. Assume the exercise describes a linear relationship. When writing a linear equation, write the equation in
slope-intercept form.
-In 1860 the population of a midwest city was 20,000. By 1890 it had grown to 24,000. If it continues to grow at the same rate, what will the population be in 1916? Give your answer to the nearest whole number.
A)23,468
B)24,001
C)27,467
D)28,000

Question

Quizplus · Accepted Answer

We can assume that the population growth follows a linear relationship. To find the equation of the line, we need to find the slope and the y-intercept.

Let x be the number of years since 1860. Then, we can find two points on the line: (0, 20,000) and (30, 24,000). (Note that we subtract 1860 from the years to get x-values that are easier to work with.)

Slope = (change in y) / (change in x) = (24,000 - 20,000) / (30 - 0) = 800

Now we can use the point-slope form of a linear equation with either point:

y - 20,000 = 800(x - 0)

Simplifying, we get:

y = 800x + 20,000

To find the population in 1916 (56 years after 1860), we can substitute x = 56 into the equation:

y = 800(56) + 20,000 = 27,468

Rounding to the nearest whole number, we get:

Answer: C (27,467) To find the linear equation, we can use the slope-intercept form, given by:
y = mx + b, 
where y represents the population, x represents the year (with 1860 = 0), m represents the slope or rate of growth, and b represents the initial population in 1860 (20,000).
The slope, m, is found by taking the difference in population over the difference in years:
m = (24,000 - 20,000) / (1890 - 1860) = 80
So the equation becomes:
y = 80x + 20,000
To find the population in 1916 (which is 56 years after 1860), we plug in x = 56:
y = 80(56) + 20,000 ≈ 27,467. 
Therefore, the best choice is C.

Solve Assume the Exercise Describes a Linear Relationship