1. **State the problem:** We need to evaluate the expression $$Tr = \frac{5^2}{\pi^2 \times \alpha} \ln \left[ \frac{8}{\pi^2} \times \frac{240 - 60}{105 - 60} \right]$$ where $\alpha$ is a variable. 2. **Rewrite the formula:** $$Tr = \frac{25}{\pi^2 \alpha} \ln \left[ \frac{8}{\pi^2} \times \frac{180}{45} \right]$$ 3. **Simplify the fraction inside the logarithm:** $$\frac{180}{45} = 4$$ 4. **Substitute back:** $$Tr = \frac{25}{\pi^2 \alpha} \ln \left[ \frac{8}{\pi^2} \times 4 \right] = \frac{25}{\pi^2 \alpha} \ln \left[ \frac{32}{\pi^2} \right]$$ 5. **Explain the logarithm:** The natural logarithm $\ln(x)$ is the power to which $e$ must be raised to get $x$. Here, we take the logarithm of $\frac{32}{\pi^2}$. 6. **Final expression:** $$Tr = \frac{25}{\pi^2 \alpha} \ln \left( \frac{32}{\pi^2} \right)$$ This is the simplified form of the given expression. To find a numerical value, you need to know $\alpha$. **Note:** $\pi$ is approximately 3.1416.