To determine the equilibrium height of the ball suspended from a spring, we need to understand the given equation. The equation for the height, [tex]\( h \)[/tex], of the ball is:
[tex]\[ h = a \sin(b(t - h)) + \mu \][/tex]
Here, [tex]\( a \)[/tex] represents the amplitude of the oscillation, [tex]\( b \)[/tex] is related to the frequency, [tex]\( t \)[/tex] is the time, and [tex]\( \mu \)[/tex] is the vertical shift or the equilibrium position of the height [tex]\( h \)[/tex].
The equilibrium position is the height at which the ball would remain if there were no oscillations, which is in the absence of the sinusoidal component. This reference or average height over time does not depend on the sine function since the sine function oscillates between -1 and 1. Instead, the equilibrium position is represented by the vertical shift [tex]\( \mu \)[/tex].
Therefore, the height of the ball at equilibrium is simply:
[tex]\[ \mu \][/tex]
So, the equilibrium height is [tex]\( \mu \)[/tex] feet. This result corresponds to none of the options given (a, b, h, k). If you meant to provide options including [tex]\( \mu \)[/tex], then [tex]\( \mu \)[/tex] feet should be the correct answer.
However, since no such option is listed, make sure to refer back to the problem statement for any potential rereading or correction. In mathematical terms, the correct equilibrium height is indeed represented by [tex]\( \mu \)[/tex].
