Find the years

Question

The following formula models the number of prison inmates P ,in thousands, who are in U.S. prisons x years after 1980.

p= 2x^2+16x+338

Find the first year in which the population of U.S. prisons was approximately 1028 thousand people.

Answer: The population reached 1028 thousand people in the year ?????

Answer 1

We're looking for the year x in which the population p reached 1028 thousand. Plug in p = 1028 and rearrange into quadratic form. Then factor and solve for x:

 

1028 = 2x² + 16x + 338

0 = 2x² + 16x - 690

0 = 2(x² + 8x - 345)

0 = x² + 8x - 345

0 = (x + 23)(x - 15)

0 = x + 23    OR    0 = x - 15

-23 = x    OR    15 = x

 

Since we're not interested in negative years, x = 15. That means that 15 years after 1980, the prison population was approximately 1028 thousand people. Your answer is 1980 + 15 = 1995.