Problem 137 - Precalculus with Limits

Question

The mach number \(M\) of an 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 (see figure). The mach number is related to the apex angle \(\theta\) of the cone by \(\sin (\theta / 2)=1 / M.\) (Figure Cant Copy) (a) Find the angle \(\theta\) that corresponds to a mach number of \(1 .\) (b) Find the angle \(\theta\) that corresponds to a mach number of \(4.5 .\) (c) The speed of sound is about 760 miles per hour. Determine the speed of an object with the mach numbers from parts (a) and (b). (d) Rewrite the equation in terms of \(\theta\)

Step-by-Step Solution

Answer

a) When the Mach number is 1, the angle \( \theta \) is 2 radians or approximately 114.59 degrees. b) When the Mach number is 4.5, the angle \( \theta \) is approximately 0.45 radians, roughly 25.84 degrees. c) The speed of an object with a Mach number of 1 is 760 mph, and 3420 mph with a Mach number of 4.5. d) The equation can be rewritten in terms of \( \theta \) as \( \theta = 2M \).

Step 1: Computing the angle for M=1

Given the relationship \(\frac{\theta}{2} = \frac{1}{M}\), we substitute \( M=1 \) into the equation to get \( \theta = 2 \times \frac{1}{M} \). Thus, for \( M=1 \), \( \theta = 2 \) radians or \( 114.59 \) degrees.

Step 2: Computing the angle for M=4.5

With the same formula \(\frac{\theta}{2} = \frac{1}{M}\), we put \( M=4.5 \) to get \( \theta = 2 \times \frac{1}{4.5} \) radians, or \( \theta = 0.45 \) radians (approximately), which is roughly \( 25.84 \) degrees.

Step 3: Calculating the speed for given Mach numbers

The speed of an object can be calculated using the formula \( V = M \times c \), where \( V \) is the speed of the object, \( M \) is the Mach number, and \( c \) is the speed of sound. So, for \( M=1 \) and \( c=760 \) mph, \( V = 1 \times 760 = 760 \) mph. Similarly, for \( M=4.5 \) and \( c=760 \), \( V = 4.5 \times 760 = 3420 \) mph.

Step 4: Rewriting the equation in terms of \( \theta \)

Rearranging the original equation to make \( \theta \) the subject yields \( \theta = 2/\frac{1}{M} \), which can be simplified as \( \theta = 2M \)