91Ó°ÊÓ

Height of a Mountain In traveling across flat land, you notice a mountain directly in front of you. Its angle of elevation (to the peak) is \(3.5^{\circ}\). After you drive 13 miles closer to the mountain, the angle of elevation is \(9^{\circ}\). Approximate the height of the mountain.

In Exercises 65-80, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$ \cot 178^{\circ} $$

Respiratory Cycle For a person at rest, the velocity \(v\) (in liters per second) of air flow during a respiratory cycle (the time from the beginning of one breath to the beginning of the next) is given by \(v=0.85 \sin \frac{\pi t}{3}\), where \(t\) is the time (in seconds). (Inhalation occurs when \(v>0\), and exhalation occurs when \(v<0\).) (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function.

Distance A plane flying at an altitude of 7 miles above a radar antenna will pass directly over the radar antenna (see figure). Let \(d\) be the ground distance from the antenna to the point directly under the plane and let \(x\) be the angle of elevation to the plane from the antenna. ( \(d\) is positive as the plane approaches the antenna.) Write \(d\) as a function of \(x\) and graph the function over the interval \(0

In Exercises 65-80, use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$ \tan \left(-\frac{\pi}{9}\right) $$

Predator-Prey Model The population \(C\) of coyotes (a predator) at time \(t\) (in months) in a region is estimated to be $$ C=5000+2000 \sin \frac{\pi t}{12} $$ and the population \(R\) of rabbits (its prey) is estimated to be $$ R=25,000+15,000 \cos \frac{\pi t}{12} $$ (a) Use a graphing utility to graph both models in the same viewing window. Use the window setting \(0 \leq t \leq 100 .\) (b) Use the graphs of the models in part (a) to explain the oscillations in the size of each population. (c) The cycles of each population follow a periodic pattern. Find the period of each model and describe several factors that could be contributing to the cyclical patterns.

Data Analysis: Astronomy The percent \(y\) of the moon's face that is illuminated on day \(x\) of the year 2007 , where \(x=1\) represents January 1 , is shown in the table. (Source: U.S. Naval Observatory) \begin{tabular}{|c|c|} \hline\(x\) & \(y\) \\ \hline 3 & \(1.0\) \\ 11 & \(0.5\) \\ 19 & \(0.0\) \\ 26 & \(0.5\) \\ 32 & \(1.0\) \\ 40 & \(0.5\) \\ \hline \end{tabular} (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. (c) Add the graph of your model in part (b) to the scatter plot. How well does the model fit the data? (d) What is the period of the model? (e) Estimate the moon's percent illumination on March \(12,2007 .\)

Ferris Wheel A Ferris wheel is built such that the height \(h\) (in feet) above ground of a seat on the wheel at time \(t\) (in seconds) can be modeled by $$ h(t)=53+50 \sin \left(\frac{\pi}{10} t-\frac{\pi}{2}\right) $$ (a) Find the period of the model. What does the period tell you about the ride? (b) Find the amplitude of the model. What does the amplitude tell you about the ride? (c) Use a graphing utility to graph one cycle of the model.

Evaluate the sine, cosine, and tangent of the angle without using a calculator. $$ -150^{\circ} $$

In Exercises 81-86, find two solutions of the equation. Give your answers in degrees \(\left(0^{\circ} \leq \theta<360^{\circ}\right)\) and in radians \((0 \leq \theta<2 \pi)\). Do not use a calculator. $$ \text { (a) } \sin \theta=\frac{1}{2} $$