Question

1. recent studies seem to indicate the population of north american porcupine (erethizon dorsatum) varies sinusoidally with the solar (sunspot) cycle due to its effects on earth’s ecosystems. suppose the population of this species in a certain locality is modeled by the function $p(t)=250cos(\frac{2pi}{11}t)+950$, where $p(t)$ represents the population of porcupines in year $t$. use the model to (a) find the period of the function; (b) graph the function over one period; (c) find the maximum and minimum values;

Explanation:

Step1: Recall period formula for cosine function

For a cosine function of the form $y = A\cos(Bt)+C$, the period $T$ is given by $T=\frac{2\pi}{|B|}$. In the function $P(t)=250\cos(\frac{2\pi}{11}t)+950$, $B = \frac{2\pi}{11}$.

Step2: Calculate the period

$T=\frac{2\pi}{\frac{2\pi}{11}}$. When dividing by a fraction, we multiply by its reciprocal, so $T = 2\pi\times\frac{11}{2\pi}=11$.

Step3: Analyze the amplitude and vertical - shift for max and min

The amplitude of the function $P(t)$ is $A = 250$ and the vertical - shift is $C = 950$. The maximum value of a cosine function $y = A\cos(Bt)+C$ is $C + A$ and the minimum value is $C - A$.
Maximum value: $950+250 = 1200$.
Minimum value: $950 - 250=700$.

(a)

The period of the function $P(t)$ is $11$ years.

(b)

To graph $y = 250\cos(\frac{2\pi}{11}t)+950$ over one - period ($0\leq t\leq11$):

• The amplitude is $250$, the vertical shift is $950$, and the period is $11$.
• When $t = 0$, $P(0)=250\cos(0)+950=250\times1 + 950=1200$.
• When $t=\frac{11}{4}$, $P(\frac{11}{4})=250\cos(\frac{\pi}{2})+950=950$.
• When $t=\frac{11}{2}$, $P(\frac{11}{2})=250\cos(\pi)+950=250\times(- 1)+950 = 700$.
• When $t=\frac{33}{4}$, $P(\frac{33}{4})=250\cos(\frac{3\pi}{2})+950=950$.
• When $t = 11$, $P(11)=250\cos(2\pi)+950=1200$. Plot these points and draw a smooth cosine - shaped curve.

(c)

Maximum value: $1200$
Minimum value: $700$

Answer:

(a) 11 years
(b) Graph as described above
(c) Maximum: 1200, Minimum: 700