Chapter 5.5, Problem 71E

Match Number

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 $\theta$ of the cone by $\sin \left(\frac{\theta }{2}\right)=\frac{1}{M}$.

(a) Use a half-angle formula to rewrite the equation in terms of $\cos \theta$.

(b) Find the angle $\theta$ that corresponds to a Mach number of 2.

(c) Find the angle $\theta$ 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 part (b) and (c).