Answer:
The intervals in which the population is less than 20,000 include
(0 ≤ t < 0.74) and (11.26 < t ≤ 12)
Step-by-step explanation:
P(t) = 82.5 - 67.5 cos [(π/6)t]
where
P = population in thousands.
t = time in months.
During a year, in what intervals is the population less than 20,000?
That is, during (0 ≤ t ≤ 12), when is (P < 20)
82.5 - 67.5 cos [(π/6)t] < 20
- 67.5 cos [(π/6)t] < 20 - 82.5
-67.5 cos [(π/6)t] < -62.5
Dividing both sides by (-67.5) changes the inequality sign
cos [(π/6)t] > (62.5/67.5)
Cos [(π/6)t] > 0.9259
Note: cos 22.2° = 0.9259 = cos (0.1233π) or cos 337.8° = cos (1.8767π) = 0.9259
If cos (0.1233π) = 0.9259
Cos [(π/6)t] > cos (0.1233π)
Since (cos θ) is a decreasing function, as θ increases in the first quadrant
(π/6)t < 0.1233π
(t/6) < 0.1233
t < 6×0.1233
t < 0.74 months
If cos (1.8767π) = 0.9259
Cos [(π/6)t] > cos (1.8767π)
cos θ is an increasing function, as θ increases in the 4th quadrant,
[(π/6)t] > 1.8767π (as long as (π/6)t < 2π, that is t ≤ 12)
(t/6) > 1.8767
t > 6 × 1.8767
t > 11.26
Second interval is 11.26 < t ≤ 12.
Hope this Helps!!!
