Markville Secondary School Smeianu Unit #5 Trigonometric Functions MHF4U Lesson #8: Applications of Trigonometric Functions Learning Goals: By the end of the lesson, I will be able to solve word problems modelled by sinusoidal functions Example 1: P(t) models the distance (in metres) of a swinging pendulum from the place it was released t seconds after it starts to swing. Here, t is entered in radians. P(t)=−5cos(2πt)+5 . What is the first time the pendulum reaches 3.5 m from the place it was released? Round your final answer to the nearest hundredth of a second. Example 2: A Ferris wheel is 4 m off the ground. It has a diameter of 26 m, and rotates once every 32 seconds. You begin the ride sitting in a chair that is 6 m above the ground. a) Determine an equation in terms of sine to model your height, ℎ measured from the ground, in metres at a time 𝑡, in seconds. You may draw a graph to help you. 𝜋 CH 4: ℎ(𝑡) = 13 sin [16 (𝑡 − 8)] + 17 b) How high will you be 10 seconds into the ride? Round to the nearest metre. (about 22m) c) During the first 16 seconds, when will you be 10 m high? Round to two decimal places. (5.11 s. 10.91 s d) Check your answers on the graph. Markville Secondary School Smeianu Unit #5 Trigonometric Functions MHF4U Example 3: Each day, the tide continuously goes in and out, raising and lowering a boat (sinuisoidally) in the harbor. At low tide, the boat is only 2 feet above the ocean floor. And, 6 hours later, at peak high tide, the boat is 40 feet above the ocean floor. a) Write a sine function that describes the boat's distance above the ocean floor as it relates to time. If a graph helps, then draw one! b) For safety, the boat needs 14 feet of depth to sail. If high tide occurs at noon, between what times can the boat go out to sea? Example 4: The following trig function models the position of a rung on a waterwheel: where t = seconds and y = number of feet above water level. a) What is the diameter of the wheel? b) At the top of the wheel, how high is the rung above water level? c) How many rotations per minute does the wheel make? d) What percentage of time does a rung spend underwater? Example 5: A clock with a 14 inch diameter has a minute hand that is 6" long and an hour hand that is 4" in length. Assume a line drawn from the 9 to the 3 represents an elevation of zero. a) Write a sine model showing the position (elevation) of the minute hand tip as it relates to time. What is the elevation of the minute hand tip at 53 minutes past the hour? When is the elevation of the minute hand tip 2 inches below the elevation zero line? Markville Secondary School Smeianu Unit #5 Trigonometric Functions MHF4U b) Write a cosine model describing the position of the hour hand tip related to time. What is the elevation of the hour hand tip at 5:00? When is the elevation 3 inches above the zero elevation line? Homework: Pg. 288 # 10, 13bc, 14, 16, 17, 20, 23, 25,26 Markville Secondary School Smeianu Unit #5 Trigonometric Functions MHF4U Markville Secondary School Smeianu Unit #5 Trigonometric Functions MHF4U Markville Secondary School Smeianu Unit #5 Trigonometric Functions MHF4U Markville Secondary School Smeianu Unit #5 Trigonometric Functions MHF4U Markville Secondary School Smeianu Unit #5 Trigonometric Functions MHF4U
