1. **State the problem:** We need to simplify and evaluate the expression $$T_r = \frac{5^2}{\pi^2 \times \alpha} \ln \left[ \frac{8}{\pi^2} \times \frac{240-60}{105-60} \right]$$ where $\alpha$ is a constant. 2. **Recall the formulas and rules:** - The natural logarithm function is $\ln(x)$. - Simplify inside the logarithm first. - Powers and products should be handled carefully. 3. **Simplify the powers and fractions:** - $5^2 = 25$ - $240 - 60 = 180$ - $105 - 60 = 45$ 4. **Simplify inside the logarithm:** $$\frac{8}{\pi^2} \times \frac{180}{45} = \frac{8}{\pi^2} \times 4 = \frac{32}{\pi^2}$$ 5. **Rewrite the expression:** $$T_r = \frac{25}{\pi^2 \alpha} \ln \left( \frac{32}{\pi^2} \right)$$ 6. **Final expression:** The simplified form is $$T_r = \frac{25}{\pi^2 \alpha} \ln \left( \frac{32}{\pi^2} \right)$$ This is the most simplified form unless a numerical value for $\alpha$ is given. **Answer:** $$T_r = \frac{25}{\pi^2 \alpha} \ln \left( \frac{32}{\pi^2} \right)$$