Let x = years after Jason was hired. In his first year (x = 0) y = 1670, and the next year (x = 1) y = 1336. An exponential model has the form y = a(b)^x.
- Find a: a = y(0) = 1670.
- Find b using y(1) = a·b = 1336 ⇒ b = 1336/1670 = 8/10 = 4/5 = 0.8.
So the model is: y = 1670(4/5)^x.
Now find when y < 1000:
1670(4/5)^x < 1000 ⇒ (4/5)^x < 1000/1670 = 100/167.
Take natural logs: x·ln(4/5) < ln(100/167). Since ln(4/5) < 0, dividing by it reverses the inequality:
x > ln(100/167) / ln(4/5) ≈ -0.512824 / -0.223144 ≈ 2.299.
Therefore the number of leavers falls below 1000 after more than about 2.299 years, so the first whole year when it is below 1000 is 3 years after Jason was hired.