The population size, P,
of owls (predators) in a certain region can
be modelled by the function
P(t) = 1000 + 100 sin (3.14t/12), where t represents
the time in months. The population size, p,
of mice (prey) in the same region is given by
p(t) = 20 000 + 4000 cos (3.14t/12). State the period, amplitude and vertical displacement for each curve. Graph both functions.
I will do the first, you do the second
P(t) = 1000 + 100 sin(πt/12)
period = 2π/(π/12) = 24 months
amplitude = 100
vertical displacement = 1000
period = 2ð/(ð/12) = 24 months
amplitude = 4000
vertical displacement = 20 000
correct
To determine the period, amplitude, and vertical displacement for each curve, we can look at the general form of the trigonometric functions.
For the owl population function, P(t) = 1000 + 100 sin (3.14t/12):
1. Period: The period of a trigonometric function is the distance between two consecutive identical points on the graph. In this case, the argument of the sine function is (3.14t/12), and since the coefficient of t is 3.14/12, the period can be calculated as 2π divided by this coefficient:
Period = 2π / (3.14/12) = 24 months
2. Amplitude: The amplitude of a trigonometric function represents the maximum displacement from the mean or average value. In this case, the coefficient of the sine function is 100. Therefore, the amplitude is 100.
3. Vertical displacement: The vertical displacement refers to shifting the graph up or down. In this case, the constant term is 1000, so the graph is vertically shifted up by 1000 units.
For the mouse population function, p(t) = 20 000 + 4000 cos (3.14t/12):
1. Period: Following the same reasoning as before, we find that the period is also 24 months.
2. Amplitude: Similarly, the amplitude is given by the coefficient of the cosine function, which is 4000.
3. Vertical displacement: The constant term is 20,000, so the graph is shifted vertically up by 20,000 units.
Now let's graph both functions to visualize them.