Precalculus Homework Help Please

Question

A population of rabbits oscillates 15 above and below an average of 141 during the year, hitting the lowest value in January (t = 0). Find an equation for the population, P, in terms of the months since January, t. =   

What if the lowest value of the rabbit population occurred in April instead?

 =



Answer 1

Since the lowest value occurs at the starting point (t = 0) this most easily fits a negative cosine function, so I'll start with that. The generic function then would be P(t) = -Acos(B(x - C)) + D where:

A = the amplitude (amount of the oscillation from the midline)

B = 2π/period

C = the horizontal shift which in this case is zero since we are using a negative cosine function and that starts at the minimum value when t = 0 which is what we want.

D = the vertical shift (value of the midline)

In this case D (the midline) is 141 and A = 15 (the population oscillates plus/minus 15 from the 141).

B = 2π/12 (which is π/6 when reduced) since the period is 12 months.

Putting these together we get P(t) = -15cos(π/6t) + 141

If the lowest population was delayed until April, then C = 4 (4th month is the low) so the equation becomes:

P(t) = -15cos[π/6(t - 4)] + 141