The number of medical doctors D (in thousands) in the United States from 1998 through 2006 can be modeled by
D = 431.61 + 121.8t, 8 ¡ t ¡ 16 where t represents the year, with t = 8 corresponding to 1998.¢Ó
(a) In which year did the number of medical doctors reach 860,000?
(b) Use the model to predict in which year the number of medical doctors will reach 1,010,000.
To find the year in which the number of medical doctors reached 860,000, we need to substitute the value of D into the equation and solve for t.
Let's substitute D = 860 into the equation:
860 = 431.61 + 121.8t
Now, let's solve for t:
121.8t = 860 - 431.61
121.8t = 428.39
t = 428.39 / 121.8
t ≈ 3.52
Since t represents the year, with t = 8 corresponding to 1998, we need to add 8 to t:
Year = 8 + 3.52
Year ≈ 11.52
So, in approximately the 11th month of the year following 1998, which is 1998 + 11 = 2009, the number of medical doctors reached 860,000.
To predict in which year the number of medical doctors will reach 1,010,000, we can use the same equation:
D = 431.61 + 121.8t
Let's substitute D = 1,010,000 into the equation:
1,010,000 = 431.61 + 121.8t
Now, let's solve for t:
121.8t = 1,010,000 - 431.61
121.8t = 1,009,568.39
t = 1,009,568.39 / 121.8
t ≈ 8.29
Again, since t represents the year, with t = 8 corresponding to 1998, we add 8 to t:
Year = 8 + 8.29
Year ≈ 16.29
So, in approximately the 16th month of the year following 1998, which is 1998 + 16 = 2014, the number of medical doctors is predicted to reach 1,010,000.