The population of a ski-resort town, as a function of the number of months into the year, can be described by a cosine function. The maximum population of the town is about 15000 people, and the minimum population is about 500 people. At the beginning of the year, the population is at its greatest. After six months, the population reaches its lowest number of people. What is the equation of the cosine function that describes the population of this town?
- 1 month ago
The maximum of the cosine function is at 0. So we choose the beginning of the year as time 0. so our function is 15000 at time 0. that means A*cos(B*0) + C = 15000 = A + C.
At the minimum, the cosine function is -1. This will occur at 180 degrees, so we need the cos(B*6) = cos(180) at the six month mark. This means B = 30. Additionally, at that point, we know that that A*cos(180) + C = 500 = -A + C.
Using A+C = 15000 and -A+C = 500, we see that 2*C = 15500 (add the two functions together). So C = 7750. Plugging that into our second equation, we see that A = 7750-500 = 7250.
So our function is population = 7250*cos(30*M)+7750, where M is the number of months into the year.