1. **Stating the problem:** We are given several expressions involving variables $x$, $u$, and $u_1$, and we want to understand and solve for $u$ and $u_1$ based on the given equations. 2. **Given expressions:** - $\frac{x^2}{u'} = \frac{25}{80} < \frac{1}{2}$ - $\frac{1}{2} u = 100$ - $u = 80$ - $u_1^{\frac{1}{2}} \cdot (2u_1)^{\frac{1}{2}} = 25$ 3. **Step 1: Solve for $u$ from $\frac{1}{2} u = 100$** $$\frac{1}{2} u = 100$$ Multiply both sides by 2: $$\cancel{\frac{1}{2}} \times 2 u = 100 \times 2 \Rightarrow u = 200$$ 4. **Step 2: Check the given $u=80$ vs calculated $u=200$** There is a contradiction; the problem states $u=80$ but from the equation $\frac{1}{2} u=100$ we get $u=200$. We will proceed with $u=200$ as it follows from the equation. 5. **Step 3: Simplify $u_1^{\frac{1}{2}} \cdot (2u_1)^{\frac{1}{2}} = 25$** Rewrite the expression: $$u_1^{\frac{1}{2}} \cdot (2u_1)^{\frac{1}{2}} = u_1^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} u_1^{\frac{1}{2}} = 2^{\frac{1}{2}} u_1^{\frac{1}{2} + \frac{1}{2}} = \sqrt{2} u_1$$ So the equation becomes: $$\sqrt{2} u_1 = 25$$ 6. **Step 4: Solve for $u_1$** Divide both sides by $\sqrt{2}$: $$u_1 = \frac{25}{\sqrt{2}} = \frac{25 \sqrt{2}}{2}$$ 7. **Summary:** - From $\frac{1}{2} u = 100$, we find $u = 200$. - From $u_1^{\frac{1}{2}} \cdot (2u_1)^{\frac{1}{2}} = 25$, we find $u_1 = \frac{25 \sqrt{2}}{2}$. **Final answers:** $$u = 200$$ $$u_1 = \frac{25 \sqrt{2}}{2}$$