A uniform beam of mass $m$ and length $L$ leans against a rough wall as shown below. The coefficients of static friction between the beam and the floor, and between the beam and the wall, are $\(\mu\)_{f}=0.50$ and $\(\mu\)_{w}=0.30$ , respectively. Assume the beam is on the verge of slipping and stability is ensured if $\(\theta\) \(\geq\) \(\theta\)_{\(\min\) }$, given by

Calculate the true value of $\(\theta\)_{\(\min\) }$. Then, assuming the wall is frictionless ($\(\mu\)_{w}=0$), the expected value of $\(\theta\) _{\(\min\) }\(\text{ is }\)45^{\(\circ\) }$. Determine the percentage error between the true value and the frictionless wall approximation.