Answer:
a The speed of the plane is [tex]v = 80.851 \ m/s[/tex]
b The altitude of the plane is [tex]Z = 653.27 \ m[/tex]
Explanation:
Let the altitude of the plane is Z
The distance at which the plane hears the horn is [tex]d = \frac{1}{4} * Z[/tex]
The time taken for the sound to reach the plane is [tex]t = 2.02 \ s[/tex]
The speed of sound is [tex]v_s = 343 \ m/s[/tex]
The distance from the point the plane hears the sound to the boat is mathematically evaluated as
[tex]D = v_s * t[/tex]
substituting values
[tex]D = 2.02 *343[/tex]
[tex]D = 692.9 \ m[/tex]
This is the diagonal distance of the boat to the plane (hypotenuse)
The altitude of the plane is the vertical distance of the plane from the boat
and the horizontal distance of the plane from the boat is d
Applying Pythagoras theorem
[tex]D^2 = Z^2 + d^2[/tex]
=> [tex]D^2 = Z^2 +[ \frac{1}{4} * Z ]^2[/tex]
substituting values
[tex]692.9^2 = 1.125 Z^2[/tex]
=> [tex]Z^2 =426764.81[/tex]
=> [tex]Z = 653.27 \ m[/tex]
=> [tex]d = \frac{1}{4} * 653.3[/tex]
=> [tex]d = 163.3 \ m[/tex]
Now the velocity of the plane is
[tex]v = \frac{163.3}{2.02}[/tex]
[tex]v = 80.851 \ m/s[/tex]
