Environmentalists use sinusoidal functions to model populations of predators and
ID: 3025252 • Letter: E
Question
Environmentalists use sinusoidal functions to model populations of predators and prey in the environment. In a particular study, the population of rabbits was modeled by the function
R(x) = 15000sin(pi/2(x + pi/2))+ 25000
The population of wolves in the same environmental area was modeled by the function
W(x) = 2000sin(pi*x/2) + 5000 In each formula, x represents time in months. Using the graphs of these two equations, make a statement regarding the relationship between the number of rabbits and the number of wolves in this environmental area.
********PLEASE SHOW HOW TO MAKE THE GRAPH IN EXCEL.**********
Explanation / Answer
a)What are the maximum and minimum number of rabbits?
the range of sine is always: -1<sinx<1, where the interval of x is |R
.: The maximum number of rabbits will be at sinx=1 and the minimum at sinx=-1
R(M)=25000+15000(1)=40000 rabbits
R(m)= 25000+15000(-1)=10000 rabbits
b) At what time did the population of rabbits reach its maximum number for the first time?
calculate x when sinx=1, x being that thing that comes after sin in the problem
in the first instance, let's assume /2, since that is when sinx=1
in order for sin(/2(x+/2))=1, /2(x+/2)=/2;
and we will get x=(/2)-1, which is the answer... sort of. (though -something months doesn't sound very conclusive in conversation.)
c) simple matter of replacing x with the equivalent number of units in time. 1yr.=12 months, so x=12.
R(12)=25000 + 15000sin(/2(12+/2))
Too lazy to calculate that, but that's the value.
d) like problem [a] except using the function W.
sinx=1
W(M)=5000+2000(1)=7000 wolves
W(m)=5000+2000(-1)=3000 wolves
The graphs support the relationship that when the wolf population is high, the rabbit population is low, and when the rabbit population is high, the wolf population is low.
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