Two spectators at a soccer game see, and a moment later...

Two spectators at a soccer game see, and a moment later hear, the ball being kicked on the playing field. The time delay for spectator $A$ is $0.23 \mathrm{~s}$, and for spectator $B$ it is $0.12$ s. Sight lines from the two spectators to the player kicking the ball meet at an angle of $90^{\circ} .$ How far are (a) spectator $A$ and (b) spectator $B$ from the plaver? (c) How far are the spectators from each other?

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Key Concepts

Right Triangle Geometry

When the lines of sight from two points to an event form a right angle, the situation can be modeled with right triangle geometry. This enables the use of the Pythagorean theorem to relate the distances from the source to each observer and the distance between the observers.

Speed of Sound

This concept refers to the rate at which sound waves travel through a medium such as air. It allows us to calculate the distance sound travels over a given time interval by multiplying the speed of sound by that time delay.

Time Delay and Distance Calculation

The time delay between an event and the hearing of its sound is used to determine how far that sound has traveled. By multiplying the known speed of sound by the delay, one determines the distance from the source to the observer.

Determining Inter-Observer Distance

Using the distances calculated from the sound delay for each observer and the geometric relationship provided by the 90° intersection of the sight lines, one can compute the distance between the two observers through the application of the Pythagorean theorem.

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