Question

The population, P in thousands of a resort community is shown by

P(t)= 500t/2t^2+9'


where t is the time in months since city council raised property taxes.

Find the interval on which the population was 40,000 or greater

Answer 1

t ≤ 4.24

Explanation:

P(t) ≥ 40000 implies

500t/(2t²+9) ≥ 40000

Multiplying through by t², we have

500t ≥ 40000(2t²+9)

500t/40000 ≥ 2t²+9

Collecting like terms

0.0125t ≥ 2t²+9

0 ≥ 2t²+ 9 - 0.0125t

2t²+ 9 - 0.0125t ≤ 0

2t²- 0.0125t + 9 ≤ 0

Using the quadratic formula,

`t = \frac{-(-0.0125) +/-\sqrt{(-.0125)^(2) - 4 X 2 X 9} }{2 X 2} \\= (0.0125 +/-√((0.00015625 - 288) )/(4)\\= (0.0125 +/-√(-287.9998) )/(4)\\= (0.0125 +/-16.97i )/(4)\\=0.00313 + 4.24i or 0.00313 - 4.24i`

The factors of the equation are (t - 0.00313 -4.24i) and (t - 0.00313 + 4.24i)

So, (t - 0.00313 -4.24i)(t - 0.00313 + 4.24i) ≤ 0

(t - 0.00313)² - 4.24² ≤ 0

(t - 0.00313)² ≤ 4.24²

taking square-root of both sides,

√(t - 0.00313)² ≤ √4.24²

t - 0.00313 ≤ 4.24

t ≤ 4.24 + 0.00313

t ≤ 4.24313 ≅ 4.24

t ≤ 4.24