The Mach number M of a supersonic airplane is the ratio of its speed to the speed of sound. When an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. The Mach number is related to the apex angle Θ of the cone by sin(Θ 2) = 1M.
(a) Use a half-angle formula to rewrite the equation in terms of cos Θ.
(b) Find the angle Θ that corresponds to a Mach number of 2.
(c) Find the angle Θ that corresponds to a Mach number of 4.5.
(d) The speed of sound is about 760 miles per hour. Determine the speed of an object with the Mach numbers from parts (b) and (c).