Answer:
(a) The Mach number M is defined as the ratio of an airplane's speed to the speed of sound. When an airplane travels at the speed of sound, the Mach number is equal to 1. To find the angle θ that corresponds to a Mach number of 1, we can use the relation between the Mach number and the apex angle of the cone.
The equation relating the Mach number (M) and the apex angle (θ) is given by sin(θ) = 1/M.
For a Mach number of 1, the equation becomes sin(θ) = 1/1 = 1. To find the angle θ, we take the inverse sine (or arcsine) of both sides: θ = arcsin(1).
The arcsine of 1 is 90 degrees or π/2 radians. Therefore, the angle θ that corresponds to a Mach number of 1 is 90 degrees or π/2 radians.
(b) To find the angle θ that corresponds to a Mach number of 2.5, we use the same equation sin(θ) = 1/M.
For a Mach number of 2.5, the equation becomes sin(θ) = 1/2.5. To find the angle θ, we take the inverse sine of both sides: θ = arcsin(1/2.5).
Using a calculator, we find that the arcsine of 1/2.5 is approximately 23.5782 degrees or 0.4115 radians. Rounded to four decimal places, the angle θ that corresponds to a Mach number of 2.5 is 23.5782 degrees or 0.4115 radians.
(c) The speed of sound is approximately 760 miles per hour. To determine the speed of an object corresponding to the Mach numbers in parts (a) and (b), we multiply the Mach number by the speed of sound.
For a Mach number of 1, the speed of the object is 1 * 760 = 760 miles per hour.
For a Mach number of 2.5, the speed of the object is 2.5 * 760 = 1900 miles per hour.
(d) The equation sin(θ) = 1/M can be rewritten as θ = arcsin(1/M). This equation expresses the relationship between the apex angle (θ) and the Mach number (M) using the inverse sine function.