QuestionA population of rabbits oscillates 33 above and below average during the year, hitting the lowest value in January . The average population starts at 900 rabbits and increases by 130 each year. Find an equation for the population, , in terms of the months since January, .
Studdy Solution
STEP 1
1. The population oscillates sinusoidally with a maximum deviation of 33 rabbits from the average.
2. The lowest population occurs in January, which corresponds to .
3. The average population increases linearly by 130 rabbits per year.
4. is measured in months since January.
STEP 2
1. Determine the sinusoidal component of the population.
2. Determine the linear component of the average population.
3. Combine the sinusoidal and linear components to form the equation for the population.
STEP 3
The population oscillates sinusoidally with an amplitude of 33. Since the lowest point occurs in January, the sinusoidal function should be a cosine function, which starts at its minimum value.
The sinusoidal component can be represented as:
The term accounts for the periodic nature of the oscillation, completing a full cycle every 12 months. The phase shift ensures the cosine function starts at its minimum value.
STEP 4
The average population starts at 900 rabbits and increases by 130 rabbits per year. Since is in months, the increase per month is .
The linear component of the average population is:
STEP 5
Combine the sinusoidal and linear components to form the equation for the population .
The equation is:
The equation for the population in terms of the months since January is:
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