A population of animals oscillates annually from a low of 1300 on January 1st to a high of 2200 on July 1st, and back to a low of 1300 on the following January. Assume that the population is well-approximated by a sine or a cosine function.
a. Find a formula for the population, P, as a function of time, t. Let t represent the number of months after January 1st. Find the amplitude, period and midline.
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To find a formula for the population, P, as a function of time, t, we can use a cosine function as it oscillates between a low and a high value.
A cosine function can be represented by the formula: P = A * cos(B(t - C)) + D, where A is the amplitude, B is the frequency (related to the period), C is the phase shift, and D is the midline.
In this case, the given information suggests that the population oscillates annually. To convert this into months, we need to divide the period by 12 because there are 12 months in a year. The period is the time it takes for one complete cycle. From January 1st to July 1st, the population reaches its high, which is half of a cycle, so the full period is 12 (months) / 1.5 (cycles) = 8 months.
Now, let's calculate the amplitude, period, and midline.
Amplitude:
The amplitude is the distance between the midline and the highest or lowest point on the curve. In this case, the highest point is 2200 and the lowest point is 1300. Since the midline is the average of these two values, the amplitude can be calculated as half of the difference between them: Amplitude = (2200 - 1300) / 2 = 450.
Midline:
The midline is the average value of the highest and lowest points. In this case, the midline is the same as the lowest point, which is 1300.
Now, we have the amplitude (A = 450) and the midline (D = 1300). We need to find the frequency (B) and phase shift (C).
Frequency:
The frequency is related to the period. In this case, we already calculated the period to be 8 months. The formula for frequency is B = (2π) / period. So, B = (2π) / 8 = π / 4.
Phase Shift:
The phase shift is the horizontal shift of the cosine function. From the given information, the population starts at its lowest point, which is January 1st. Considering that January is the first month and February is the second, it means that January has a time value of 1 and July has a time value of 7. To shift the curve, we need to find the horizontal difference between the starting point (January 1st) and the phase shift. In this case, the phase shift is half of a period, which is 8 months / 2 = 4 months. So, the phase shift is C = 1 + 4 = 5.
Now we have all the values to define the formula for the population, P, as a function of time, t.
P = 450 * cos((π / 4)(t - 5)) + 1300
Therefore, the formula for the population as a function of time is P = 450 * cos((π / 4)(t - 5)) + 1300, with an amplitude of 450, a period of 8 months, and a midline of 1300.