To find the maximum displacement from the equilibrium position for the given simple harmonic motion equation [tex]\( d = 9 \cos \left( \frac{\pi}{2} t \right) \)[/tex], follow these steps:
1. Identify the Amplitude: The equation of simple harmonic motion provided is in the form [tex]\( d = A \cos(\omega t) \)[/tex], where [tex]\( A \)[/tex] is the amplitude and [tex]\( \omega \)[/tex] is the angular frequency. In our case, the equation is [tex]\( d = 9 \cos \left( \frac{\pi}{2} t \right) \)[/tex].
2. Maximum Value of Cosine Function: The cosine function [tex]\( \cos(x) \)[/tex] can take values between -1 and 1. Therefore, its maximum value is 1.
3. Calculate Maximum Displacement: The maximum displacement from the equilibrium position occurs when the cosine function attains its maximum value, which is 1.
[tex]\[
d_{\text{max}} = 9 \cdot 1 = 9
\][/tex]
Thus, the maximum displacement from the equilibrium position is:
[tex]\[
\boxed{9}
\][/tex]
