A rescue plane flies horizontally at a constant speed searching for a disabled boat. When the plane is directly above the boat, the boat's crew blows a loud horn. By the time the plane's sound detector receives the horn's sound, the plane has traveled a distance equal to one-third its altitude above the ocean. (a) If it takes the sound 1.92 s to reach the plane, determine the speed of the plane. (Take the speed of sound to be 343 m/s.) ___m/s (b) What is its altitude? ___m

a) Speed of the plane = 108.47 (m/s)

b) Altitude of the plane: 624.77m

Explanation:

Using the speed of sound and the time it took to get to the plane we can determine the distance the sound traveled:

d = (343m/s) * (1.92s) = 658.56m

From the Pythagorean theorem, taking as the altitude and distance traveled two legs of a triangle and the situation traveled by the sound to the plane as the hypotenuse of this, we can establish the following equation:

(658.56m) ² = ((1/3) * h) ² + h²

Being h the altitude of the plane.

From here we can clear the altitude of the plane:

433701.27m² = (10/9) * h²

h² = 433701.27m² * (9/10)

h = (390331.14m²) ^ (1/2)

h = 624.77m

We know that the distance traveled by the plane until the sound is received is one third of the altitude, so this distance will be:

Distance = (1/3) * 624.77m = 208.25m

Considering the time in which the plane traveled this distance we can know the speed of it:

V = (208.25m) / (1.92s) = 108.47 (m/s)