91Ó°ÊÓ

A security camera scans a long straight fence that encloses a section of a military base. The camera is mounted on a post that is located \(5 \mathrm{m}\) from the midpoint of the fence. The camera makes one complete rotation in 60 s. a) Determine the tangent function that represents the distance, \(d\), in metres, along the fence from its midpoint as a function of time, \(t,\) in seconds, if the camera is aimed at the midpoint of the fence at \(t=0\) b) Graph the function in the interval \(-15 \leq t \leq 15\) c) What is the distance from the midpoint of the fence at \(t=10 \mathrm{s},\) to the nearest tenth of a metre? d) Describe what happens when \(t=15 \mathrm{s}\)

A point on an industrial flywheel experiences a motion described by the function \(h(t)=13 \cos \left(\frac{2 \pi}{0.7} t\right)+15\) where \(h\) is the height, in metres, and \(t\) is the time, in minutes. a) What is the maximum height of the point? b) After how many minutes is the maximum height reached? c) What is the minimum height of the point? d) After how many minutes is the minimum height reached? e) For how long, within one cycle, is the point less than \(6 \mathrm{m}\) above the ground? f) Determine the height of the point if the wheel is allowed to turn for \(1 \mathrm{h}\) 12 min.

A rotating light on top of a lighthouse sends out rays of light in opposite directions. As the beacon rotates, the ray at angle \(\theta\) makes a spot of light that moves along the shore. The lighthouse is located \(500 \mathrm{m}\) from the shoreline and makes one complete rotation every 2 min. a) Determine the equation that expresses the distance, \(d,\) in metres, as a function of time, \(t,\) in minutes. b) Graph the function in part a). c) Explain the significance of the asymptote in the graph at \(\theta=90^{\circ}\).

Michelle is balancing the wheel on her bicycle. She has marked a point on the tire that when rotated can be modelled by the function \(h(t)=59+24 \sin 125 t\) where \(h\) is the height, in centimetres, and \(t\) is the time, in seconds. Determine the height of the mark, to the nearest tenth of a centimetre, when \(t=17.5 \mathrm{s}.\)

Sketch the graph of each function over the interval \(\left[-360^{\circ}, 360^{\circ}\right] .\) For each function, clearly label the maximum and minimum values, the \(x\) -intercepts, the \(y\) -intercept, the period, and the range. a) \(y=2 \cos x\) b) \(y=-3 \sin x\) c) \(y=\frac{1}{2} \sin x\) d) \(y=-\frac{3}{4} \cos x\)

A plane flying at an altitude of \(10 \mathrm{km}\) over level ground will pass directly over a radar station. Let \(d\) be the ground distance from the antenna to a point directly under the plane. Let \(x\) represent the angle formed from the vertical at the radar station to the plane. Write \(d\) as a function of \(x\) and graph the function over the interval \(0 \leq x \leq \frac{\pi}{2}\).

The Arctic fox is common throughout the Arctic tundra. Suppose the population, \(F\) of foxes in a region of northern Manitoba is modelled by the function \(F(t)=500 \sin \frac{\pi}{12} t+1000,\) where \(t\) is the time, in months. a) How many months would it take for the fox population to drop to \(650 ?\) Round your answer to the nearest month. b) One of the main food sources for the Arctic fox is the lemming. Suppose the population, \(L,\) of lemmings in the region is modelled by the function \(L(t)=5000 \sin \frac{\pi}{12}(t-12)+10000\) Graph the function \(L(t)\) using the same set of axes as for \(F(t).\) c) From the graph, determine the maximum and minimum numbers of foxes and lemmings and the months in which these occur. d) Describe the relationships between the maximum, minimum, and mean points of the two curves in terms of the lifestyles of the foxes and lemmings. List possible causes for the fluctuation in populations.

a) Graph the function \(f(x)=\cos \left(x-\frac{\pi}{2}\right)\). b) Consider the graph. Write an equation of the function in the form \(y=a \sin b(x-c)+d\). c) What conclusions can you make about the relationship between the two equations of the function?

A mass attached to the end of a long spring is bouncing up and down. As it bounces, its distance from the floor varies sinusoidally with time. When the mass is released, it takes \(0.3 \mathrm{s}\) to reach a high point of 60 cm above the floor. It takes 1.8 s for the mass to reach the first low point of \(40 \mathrm{cm}\) above the floor. a) Sketch the graph of this sinusoidal function. b) Determine the equation for the distance from the floor as a function of time. c) What is the distance from the floor when the stopwatch reads \(17.2 \mathrm{s?}\) d) What is the first positive value of time when the mass is \(59 \mathrm{cm}\) above the floor?

The Canadian National Historic Windpower Centre, at Etzikom, Alberta, has various styles of windmills on display. The tip of the blade of one windmill reaches its minimum height of \(8 \mathrm{m}\) above the ground at a time of 2 s. Its maximum height is \(22 \mathrm{m}\) above the ground. The tip of the blade rotates 12 times per minute. a) Write a sine or a cosine function to model the rotation of the tip of the blade. b) What is the height of the tip of the blade after \(4 \mathrm{s} ?\) c) For how long is the tip of the blade above a height of \(17 \mathrm{m}\) in the first \(10 \mathrm{s} ?\)