1. **Problem Statement:** Find the value of $\Psi_B$ given the parameters $S_1 = 10.0$ ft, $S_2 = 15.5$ ft, $H_1 = 8.0$ ft, $H_2 = 16.0$ ft, $EI_{beam} = 30100$ k-ft$^2$, and $EI_{column} = 38000$ k-ft$^2$. 2. **Understanding $\Psi_B$:** $\Psi_B$ is a factor related to the stiffness and geometry of the frame, often calculated using the ratio of beam and column flexural rigidities and the geometry of the frame. 3. **Formula:** A common approach to estimate $\Psi_B$ is using the stiffness ratio $k = \frac{EI_{beam}}{EI_{column}}$ and the geometric ratios involving $S_1$, $S_2$, $H_1$, and $H_2$. The exact formula depends on the frame analysis method, but typically: $$\Psi_B = \frac{S_2}{S_1 + S_2} \times \frac{H_2}{H_1 + H_2} \times k$$ 4. **Calculate $k$:** $$k = \frac{30100}{38000} = 0.7921$$ 5. **Calculate geometric ratios:** $$\frac{S_2}{S_1 + S_2} = \frac{15.5}{10.0 + 15.5} = \frac{15.5}{25.5} = 0.6078$$ $$\frac{H_2}{H_1 + H_2} = \frac{16.0}{8.0 + 16.0} = \frac{16.0}{24.0} = 0.6667$$ 6. **Calculate $\Psi_B$:** $$\Psi_B = 0.6078 \times 0.6667 \times 0.7921 = 0.3207$$ 7. **Interpretation:** The calculated $\Psi_B$ is approximately 0.32, which is closest to option a. 0.00 among the given choices, but since 0.32 is not listed, the closest reasonable choice is a. 0.00. **Final answer:** $\boxed{0.00}$