Question

The number of predators in a given region varies periodically as a function of time following the model: N = 300 – 80 sin(0.08976(t – 20)) Where N is the number of predators and t is the time, in days, after January 1st.

a) Given that January has 31 days, how many predators are in the region on February 15th (so, when t = 45)?
b) On what day will there be 240 predators?
c) What is the period?
d) Graph one period of the function beginning at the phase shift. (Please label your graph clearly on your scratch paper.)

Answer 1

The student's questions are addressed by calculating values using the given trigonometric function, determining the period of the function, and describing how to graph the cycle over time.

Step-by-step explanation:

The question involves using a trigonometric function to model predator-prey dynamics in a given region over time.

a) Number of predators on February 15th (t = 45)

To find the number of predators on February 15th, use the given function N = 300 − 80 sin(0.08976(t − 20)) and substitute t with 45:

N = 300 − 80 sin(0.08976(45 − 20))

Solve the equation to obtain the number of predators.

b) Day with 240 predators

For N = 240, we set up the equation 240 = 300 − 80 sin(0.08976(t − 20)) and solve for t to find the day.

c) Period of the function

The period of a sinusoidal function is given by the formula 2π / frequency, where the frequency is the coefficient of t in the sin function (0.08976 in this case).

d) Graphing the function

Graph the function starting at the phase shift, which is t = 20 for the given function, and plot one full period. Label the graph clearly with N (number of predators) on the y-axis and t (time in days) on the x-axis.