14. max: 12,000, min: 6500, p = 10. use the information given to write a sinusoidal equation, sketch its…

14. max: 12,000, min: 6500, p = 10. use the information given to write a sinusoidal equation, sketch its graph, and answer the question posed. 15. in geneva, switzerland, the daily temperature in january ranges from an average high of 39°f to an average low of 29°f. (a) find a sinusoidal equation model for the daily temperature; (b) sketch the graph; and (c) approximate the time(s) each january day the temperature reaches the freezing point (32°f). assume t = 0 corresponds to noon. source: 2004 statistical abstract of the united states, table 1331. 19. 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 earths ecosystems. suppose the population of this species in a certain locality is modeled by the function p(t)=250 cos(\$\\frac{2\\pi}{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; and 29. f(t)=24.5 sin\$\\frac{\\pi}{10}(t - 2.5)\$ + 15.5 30. g(t)=40.6 sin\$\\frac{\\pi}{6}(t - 4)\$ + 13.4 chapter 5 an introduction to trigonometric functions 496 (d) estimate the number of years the population is less than 740 animals. source: ilya klvana, mcgill university (montreal), master of science thesis paper, november 2002.

Answer

Explanation:

Step1: Recall the general form of a sinusoidal function

The general form of a sinusoidal function is $y = A\sin(B(x - C))+D$ or $y=A\cos(B(x - C)) + D$, where $|A|$ is the amplitude, $\frac{2\pi}{B}$ is the period, $C$ is the phase - shift and $D$ is the vertical shift.

Step2: Analyze the temperature problem

For the temperature problem in Geneva: The average high is $39^{\circ}F$ and average low is $29^{\circ}F$. The amplitude $A=\frac{39 - 29}{2}=5$. The vertical shift $D=\frac{39 + 29}{2}=34$. Assuming a 24 - hour cycle, if $t = 0$ corresponds to noon, and we use a cosine function (since at $t = 0$ we can assume a non - zero derivative for the temperature function), the general form is $T(t)=A\cos(Bt - C)+D$. The period $P = 24$, so $B=\frac{2\pi}{24}=\frac{\pi}{12}$. Let's assume no phase - shift $C = 0$. The sinusoidal equation is $T(t)=5\cos(\frac{\pi}{12}t)+34$. To find when the temperature reaches the freezing point ($T(t)=32$): [ \begin{align*} 32&=5\cos(\frac{\pi}{12}t)+34\

2&=5\cos(\frac{\pi}{12}t)\ \cos(\frac{\pi}{12}t)&=-\frac{2}{5} \end{align*} ] [ \frac{\pi}{12}t=\cos^{-1}(-\frac{2}{5})+2k\pi\quad\text{or}\quad\frac{\pi}{12}t = 2\pi-\cos^{-1}(-\frac{2}{5})+2k\pi ] [ t=\frac{12}{\pi}\cos^{-1}(-\frac{2}{5})+24k\quad\text{or}\quad t=\frac{12}{\pi}(2\pi-\cos^{-1}(-\frac{2}{5}))+24k ] For $k = 0$, we get the times in the first 24 - hour period.

Step3: Analyze the porcupine population problem

For the porcupine population function $P(t)=250\cos(\frac{2\pi}{11}t)+950$. The amplitude $A = 250$, the vertical shift $D = 950$. The period $T=\frac{2\pi}{B}$, where $B=\frac{2\pi}{11}$, so $T = 11$ years. The maximum value of $P(t)$ occurs when $\cos(\frac{2\pi}{11}t)=1$. Then $P_{max}=250\times1 + 950=1200$. The minimum value of $P(t)$ occurs when $\cos(\frac{2\pi}{11}t)=-1$. Then $P_{min}=250\times(-1)+950 = 700$.

Answer:

For the temperature in Geneva: Sinusoidal equation $T(t)=5\cos(\frac{\pi}{12}t)+34$, times when $T = 32$ are $t=\frac{12}{\pi}\cos^{-1}(-\frac{2}{5})$ and $t=\frac{12}{\pi}(2\pi-\cos^{-1}(-\frac{2}{5}))$ (in the first 24 - hour period). For the porcupine population: Period is 11 years, maximum value is 1200, minimum value is 700.