1. **Problem statement:** We have four penguins connected by cords on frictionless ice. Given masses $m_1=16$ kg, $m_3=20$ kg, $m_4=22$ kg, and tensions $T_2=134$ N, $T_4=234$ N, we need to find the unknown mass $m_2$. 2. **Understanding the system:** Since the ice is frictionless, all penguins accelerate together with the same acceleration $a$. The tensions in the cords relate to the forces needed to accelerate the penguins. 3. **Key formulas:** Newton's second law for each penguin segment: - For penguin 4: $T_4 = m_4 a$ - For penguin 3 and 4 combined: $T_3 = (m_3 + m_4) a$ - For penguin 2, 3, and 4 combined: $T_2 = (m_2 + m_3 + m_4) a$ 4. **Find acceleration $a$ using $T_4$ and $m_4$:** $$a = \frac{T_4}{m_4} = \frac{234}{22} = 10.6363636... \approx 10.64 \text{ m/s}^2$$ 5. **Use $T_2$ to find $m_2$:** $$T_2 = (m_2 + m_3 + m_4) a$$ Rearranged: $$m_2 = \frac{T_2}{a} - m_3 - m_4$$ 6. **Substitute known values:** $$m_2 = \frac{134}{10.6363636} - 20 - 22$$ Calculate intermediate fraction: $$\frac{134}{10.6363636} \approx 12.6$$ 7. **Calculate $m_2$:** $$m_2 = 12.6 - 20 - 22 = 12.6 - 42 = -29.4$$ 8. **Interpretation:** A negative mass is physically impossible, indicating an inconsistency in the given data or assumptions. However, mathematically, the calculated $m_2$ is approximately $-29.4$ kg. **Final answer:** $$\boxed{m_2 \approx -29.4 \text{ kg (inconsistent data)}}$$