The population of a caribou herd varies sinusoidally with a period of one year. The population is at a minimum at the beginning of the year when the population is about 4000 animals. The population is at a maximum of 7500 animals after 6 months. How many caribou are there expected to be at the end of 7 months? Round to the nearest whole number.
help plz
Using a sine curve,
range = 7500-4000 = 3500
sofar p = 1750sin kt , where t is the number of months
period =12 = 2π/k
k = π/6
then p = 1750sin (πt/6)
we need the max to be 7500, not 1750, so we raise the curve
by 7500-1750 = 5750
giving us:
p = 1750sin(πt/6) + 5750
but we want p to be 4000 when t = 0, so we need to shift horizontally
1750sin (π/6(0-d)) + 5750 = 4000
1750sin (π/6(-d)) = -1750
sin (π/6(-d)) = -1
I know sin(-π/2) = -1
π/6(-d) = -π/2
times 6/π
d = 3
p = 1750sin (π/6(t - 3)) + 5750
so when t = 7
p = 1750sin(π/6(7-3)) + 5750 = 7266
To determine the expected population at the end of 7 months, which is 1 month after the population reaches its maximum, we can use a sinusoidal function.
A sinusoidal function is usually written in the form:
f(t) = A * sin(B(t - C)) + D
Where:
- A represents the amplitude (half the difference between the maximum and minimum values),
- B represents the frequency (2π divided by the period),
- C represents the phase shift (where the function starts),
- D represents the vertical shift (the average value between the maximum and minimum values),
- t represents time in months.
Given that the population is at a minimum of 4,000 animals at the beginning of the year (t = 0) and a maximum of 7,500 animals after 6 months (t = 6), we can use these data points to find the values of A, B, C, and D.
First, let's determine the amplitude:
Amplitude = (maximum - minimum) / 2
Amplitude = (7,500 - 4,000) / 2
Amplitude = 3,500 / 2
Amplitude = 1,750
Next, let's find the frequency:
Frequency = 2π / period
Frequency = 2π / 12 months
Frequency = π / 6
Since the population is at its maximum after a 6-month period, we set the phase shift (C) to 6.
To find the vertical shift (D), we calculate the average of the maximum and minimum values:
Vertical shift = (maximum + minimum) / 2
Vertical shift = (7,500 + 4,000) / 2
Vertical shift = 11,500 / 2
Vertical shift = 5,750
Putting it all together, the sinusoidal equation for this caribou herd is:
f(t) = 1,750 * sin((π/6)(t - 6)) + 5,750
Now, let's find the population at the end of 7 months (t = 7):
f(7) = 1,750 * sin((π/6)(7 - 6)) + 5,750
Calculating this equation will give us an estimate of the expected population at the end of 7 months. Round the result to the nearest whole number to get the final answer.