Answer:
(a) N(t) = 4580t - 9043100
(b) The number of graphic designer jobs is projected to first exceed 300,000 in the year 2038
Explanation:
(a) To find the linear function that relates the number of graphic designer jobs to the year, we can use the slope-intercept form of the equation for a line:
y = mx + b
where y is the number of graphic designer jobs, x is the year, m is the slope of the line, and b is the y-intercept.
We can use the two data points (2010, 276700) and (2020, 322500) to find the slope:
m = (y2 - y1) / (x2 - x1) = (322500 - 276700) / (2020 - 2010) = 4580
Now we can use one of the data points and the slope to find the y-intercept:
y = mx + b
276700 = 4580(2010) + b
b = 276700 - 9319800 = -9043100
Therefore, the linear function that relates the number of graphic designer jobs to the year is:
N(t) = 4580t - 9043100
where t is the year (with t = 2010 corresponding to the year 0).
(b) To find the answer, we need to use the linear function from (a):
N(t) = 4580t - 9043100
We want to find the year when the number of graphic designer jobs first exceeds 300,000. In other words, we want to find the value of t when N(t) is greater than 300,000:
4580t - 9043100 > 300,000
Adding 9043100 to both sides of the inequality, we get:
4580t > 9343100
Dividing both sides by 4580, we get:
t > 2037.54
Since t represents the year, we round up to the next whole year to get:
t = 2038
Therefore, the number of graphic designer jobs is projected to first exceed 300,000 in the year 2038.