Naturalists find that the populations of some kinds of predatory animals vary periodically. Assume that the population of foxes in a certain forest varies sinusoidally with time. Records started being kept when t = 0 years. A minimum number, 200 foxes, occured when t = 2.9 years. The next maximum, 800 foxes, occurred at t = 5.1 years
I have to find
Foxes are declared to be an endangered species when their population drops below 300. Between what two nonnegative values of "t" were foxes first endangered.
I found one to be about
6.71101496 years i don't know if this is one of the first two how do I deal and how do I find the other one?
In an earlier reply to this same problem to you I had established the equation to be
F = 300sin(5pi/11)(t-4) + 500
I had also shown that this equation satisfies all the data values you gave and it looks like you used it to get your value of 6.7 years.
But remember that the sine is negative in the III and IV quadrants, so I also got a value of t = 7.889 yrs.
Also recall that our period was 4.4, so subtracting 4.4 from any of our answers would produce another set of valid solutions.
6.71 - 4.4 = 2.31
7.889 - 4.4 = 3.49
let's test this
if t = 2.31, F = 300.3
if t = 3.49, F = 300.3
let's take a value between 2.31 and 3.49, how about t=3
F = 203 , which is less than 300
So the foxes were first endangered between the times of 2.3 years and 3.5 years.
wtf is this
my brain cells are bout to explode
To find the other value of "t" when foxes were first endangered, we need to determine the time period between two consecutive minimums in the population cycle.
Given that the minimum number of foxes occurred at t = 2.9 years, and the next maximum occurred at t = 5.1 years, we can calculate the time period between these two points:
Time period = (t2 - t1) = (5.1 years - 2.9 years) = 2.2 years
To find the other value of "t," we need to subtract this time period from the first minimum:
t = t1 - time period = 2.9 years - 2.2 years = 0.7 years
Therefore, the two nonnegative values of "t" when foxes were first endangered are approximately 0.7 years and 6.7 years.
To find the values of "t" when the foxes were first endangered, we need to consider the minimum points of the sinusoidal function. According to the problem, 200 foxes is the minimum number, and foxes are declared endangered when their population drops below 300.
Given that the minimum number of foxes is 200, we can assume the graph of the sinusoidal function will pass through the point (2.9, 200). Using this information, we can establish the equation of the sinusoidal function in the form: foxes = A * sin(B(t - C)) + D, where A represents the amplitude, B represents a constant related to the period, C represents the horizontal shift, and D represents the vertical shift.
We are given the minimum value at t = 2.9, where foxes = 200. Plugging these values into the equation, we get the following equation: 200 = A * sin(B(2.9 - C)) + D.
We are also given the maximum value at t = 5.1, where foxes = 800. Plugging these values into the equation, we get: 800 = A * sin(B(5.1 - C)) + D.
Solving these two equations simultaneously will allow us to determine the values of A, B, C, and D. Once we have these values, we can find the "t" values when the foxes were first endangered.
However, to find the second value of "t" when foxes were first endangered, we can use some intuition and recall that sinusoidal functions repeat their pattern over a period. Since we started recording when t = 0 years, we can expect the graph to repeat after a certain range.
To find that range, we can calculate the period of the sinusoidal function using the formula: Period = 2π / B. Once we have the period, we can add it to the first value of "t" when foxes were first endangered to find the second value.