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This project will involve students in: solving real world problems with the use of Trigonometric Identities (sum and difference of angles identities, double angle and half angle identities). evaluating limits algebraically. Task 1: COMMUNICATION Radio waves that pass through the same place at the same time cause interference. Constructive interference occurs when two waves combine to have greater amplitude than either of the component waves.
A radio transmitter sends out two signals, one for voice communication and another for data. Suppose the equation of the voice wave is v = 10 sin(2t 30˚) and the equation of the data wave is d = 10 cos(2t + 60˚). a. Draw a graph of the waves with the use of a program or software when they are combined. b. Analyze and explain what type of interference results from the combination of the two waves.
Task 2: Aviation Mach number The Mach number of an airplane is the ratio of the speed of a body to the speed of sound in a particular medium, usually the Earth's atmosphere. Mach Number Object speed Speed of Sound
When an airplane travels faster than the speed of sound, a sonic boom is created or the sound waves form a cone behind the airplane (see figures).
If Ѳ is the measure of the angle at the vertex of the cone, M represents the Mach number; the angle is therefore called the Mach angle. Mach number M can be Ѳ determined using the formula sin . M For example, an aircraft flying through air at twice the speed of sound has a Mach number of 2. The Mach number of an aircraft travelling at a given velocity depends on the altitude of the aircraft and other atmospheric conditions that affect the speed of sound near the aircraft.
Examples: A) If Ѳ 75, determine the Mach number and State the type of the sonic [Subsonic, Transonic, Supersonic, High Supersonic, Hypersonic]. B) If Ѳ ,, determine the Mach number and State the type of the sonic [ Subsonic, Transonic, Supersonic, High Supersonic, Hypersonic]. C) Find the angle Ѳ that corresponds to Transonic. Write your answer as exact radians. D) Find the angle Ѳ that corresponds to Mach number 4.5. Write your answer in radians rounded to four decimal places E) The speed of sound is about 761 mph. Determine the speed of an object having the mach numbers in parts C and D. F) Does the vertex angle Ѳ increase or decrease as the Mach number M decreases? Explain your reasoning.
Task 3 EVALUATING LIMITS ALGEBRAICALLY 1. Find these limits using only algebraic techniques. a. lim b. lim c. lim d. lim e. lim √ √
2. Note that whereas direct substitution into the function yields the limit in the first problem above, it results in 0/0 in all of the others. That is why algebraic steps such as factoring are necessary before substituting. The form 0/0 is called an “indeterminate form.” a. Why do you think 0/0 is called an indeterminate form? b. List the other indeterminate forms? c. Is 5/0 an indeterminate form? d. Evaluate lim
Completeness of Tasks 20%
Tasks are totally completed and correct. (100%)
Tasks are partially completed, OR Partially wrong.(75%)
Tasks are partially completed, Tasks are Attempted AND (25% or less) Partially wrong (50%). Student was able to explain the work he/she submitted. Student reflected a shallow understanding of his/her work; she was able to answer some of ...
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