Sonic boom heard in how much distance x?

[leinad0213]

Homework Statement

A supersonic plane passes directly over an observer at point P, flying due west at an altitude of 11.6 km and a speed of Mach 1.60. What is the plane's horizontal distance from point P when the sonic boom is heard?

h = 11.6 km
M = 1.6

Homework Equations

sinθ = Vs/Vo
Vo = 548.8 m/s

The Attempt at a Solution

When I attempted to solve this problem, I assumed T=20C therefore Vs = 343 m/s. This is because the entire homework set has been using this assumption. Next, I drew a triangle from when the aircraft is directly above the observer's head. I solved for θ and got
θ = sin^-1(343/548.8) = 38.68 degrees. Next, I brought time to when the cone hits the observer which makes the triangle tanθ = h/x. I solved for x and got 9.29km. When I put this answer in, it was wrong and I only have 1 try left. Am I missing some kind of conversion or concept? Any help or hints are greatly appreciated.

 

[PeterO]

Homework Statement

A supersonic plane passes directly over an observer at point P, flying due west at an altitude of 11.6 km and a speed of Mach 1.60. What is the plane's horizontal distance from point P when the sonic boom is heard?

h = 11.6 km
M = 1.6

Homework Equations

sinθ = Vs/Vo
Vo = 548.8 m/s

The Attempt at a Solution

When I attempted to solve this problem, I assumed T=20C therefore Vs = 343 m/s. This is because the entire homework set has been using this assumption. Next, I drew a triangle from when the aircraft is directly above the observer's head. I solved for θ and got
θ = sin^-1(343/548.8) = 38.68 degrees. Next, I brought time to when the cone hits the observer which makes the triangle tanθ = h/x. I solved for x and got 9.29km. When I put this answer in, it was wrong and I only have 1 try left. Am I missing some kind of conversion or concept? Any help or hints are greatly appreciated.

The sonic boom is the bunching of sound from the plane, is it not. The sound will take quite a while to travel the 11.6 km straight down to you.
In that time, it will have traveled quite a distance forward at the speed it is travelling.
Indeed, in the time sound traveled 11.6 km, the plane has been traveling 1.6 times as fast as the sound.

Looking at your figures - I think you have the have the wrong function with the triangle. You used sin to find the angle, but tan to use it. Should have been the same function both times I think.