To determine the height of the ball at its equilibrium, let's carefully analyze the given equation:
[tex]\[ h = a \sin(b(t - h)) + k \][/tex]
- [tex]\( h \)[/tex]: the height of the ball at any time [tex]\( t \)[/tex].
- [tex]\( a \)[/tex]: the amplitude of the oscillations.
- [tex]\( b \)[/tex]: the angular frequency.
- [tex]\( k \)[/tex]: the equilibrium height, or the mean height around which the ball oscillates.
- [tex]\( t \)[/tex]: the time variable.
The height of the ball at equilibrium is the position where it would rest if it were not oscillating — essentially the average height. In the given equation, this is represented by the term [tex]\( k \)[/tex].
The trigonometric function [tex]\(\sin(b(t - h))\)[/tex] oscillates between -1 and 1. When there is no oscillation, the value of [tex]\(\sin\)[/tex] is zero. At this point, the equation simplifies to [tex]\( h = k \)[/tex].
Thus, the equilibrium height of the ball is [tex]\( k \)[/tex] feet.
