In a predator/prey model the predator population is modeled by the function y=1000 sin (2t) +400. What is the smallest population according to this model?
y=1000 sin (2t) +400
sin(anything) lies between -1 and +1
so the smallest value of sin(2t) is -1
so the smallest value of
y = 1000(-1) + 400
= -600
But, it would make not make any sense to have a negative population, so my answer would be zero
thank you!
To find the minimum population in the predator/prey model, we need to find the minimum value of the function y=1000 sin(2t) + 400.
The function y=1000 sin(2t) + 400 is a sinusoidal function, where t represents time. Since the sine function oscillates between -1 and 1, the value of y will also oscillate between 1000 sin(2t) + 400 + 1000 and 1000 sin(2t) + 400 - 1000.
To find the smallest population, we need to find the lowest point of the oscillation, which corresponds to the minimum value of the function. Since sin(2t) has a maximum value of 1 and a minimum value of -1, the minimum value of y occurs when sin(2t) is -1.
Setting sin(2t) = -1, we get:
-1 = sin(2t)
To find the angle whose sine value is -1, we look for the angle in the unit circle that has a reference angle of π/2 (or 90 degrees) and lies in the fourth quadrant. This angle can be denoted as 3π/2.
The general solution for sin(θ) = a is θ = arcsin(a) + 2πn or θ = π - arcsin(a) + 2πn, where n is any integer.
Applying this to our equation:
2t = 3π/2 + 2πn or 2t = π - 3π/2 + 2πn
Simplifying:
t = 3π/4 + πn or t = π/2 - 3π/4 + πn
Now, we need to consider the time range for which the predator population is modeled. Since we don't have any specific time constraints in the question, we'll assume a range of t from 0 to 2π to cover one complete cycle of the sine function.
By substituting different values of n into the equations for t, we can determine the corresponding values of y:
For n = 0:
t = 3π/4 + π(0) = 3π/4
y = 1000 sin(2(3π/4)) + 400 = 1000 sin(3π/2) + 400 = 1000(-1) + 400 = 400
For n = 1:
t = 3π/4 + π(1) = 7π/4
y = 1000 sin(2(7π/4)) + 400 = 1000 sin(7π/2) + 400 = 1000(-1) + 400 = 400
For n = -1:
t = 3π/4 + π(-1) = -π/4
y = 1000 sin(2(-π/4)) + 400 = 1000 sin(-π/2) + 400 = 1000(-1) + 400 = 400
We can continue the same process for other values of n.
From the calculations, we can see that the smallest population according to this model is 400. This minimum population occurs at multiple instances during the time range of 0 to 2π.
Please note that the model provided is a simplified representation of a predator/prey population without considering other external factors that can impact real-life populations.