Question

There were 276,700 graphic designer jobs in a country in 2010. It has been projected that there will be 322,500 graphic designer jobs in 2020.

(a) Using the data, find the number of graphic designer jobs as a linear function of the year.
N(t) =
(b) Using your model, in what year will the number of graphic designer jobs first exceed 300,000?

Answer 1

(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.

Answer 2

(a)

Let the number of graphic designer jobs be denoted by N(t), where t is the year. We can use the given data to find the equation of the line that represents the number of graphic designer jobs as a linear function of the year.

We have two data points: (2010, 276700) and (2020, 322500). We can use these points to find the slope of the line:

slope = (322500 - 276700) / (2020 - 2010) = 4580

Now we can use the point-slope form of the equation of a line to find the equation of the line:

N(t) - 276700 = 4580(t - 2010)

N(t) = 4580t - 845300

Therefore, the number of graphic designer jobs can be expressed as the linear function N(t) = 4580t - 845300.

(b)

To find the year in which the number of graphic designer jobs first exceeds 300,000, we can set N(t) = 300000 and solve for t:

4580t - 845300 = 300000

4580t = 1145300

t ≈ 250.109

Since t represents the number of years after 1900, we need to add 1900 and the integer part of t to get the year:

1900 + 250 = 2150

Therefore, the number of graphic designer jobs is first projected to exceed 300,000 in the year 2150.