1. **State the problem:** Solve for $y$ in the equation $$5 \times 10^{-4} \times \frac{\pi}{4} \times \cos\left(\frac{\pi y}{2 \times 4}\right) = \frac{\pi}{8} \times 5 \times 10^{4}.$$\n\n2. **Simplify constants on both sides:**\nLeft side constants: $$5 \times 10^{-4} \times \frac{\pi}{4} = \frac{5\pi}{4} \times 10^{-4}.$$\nRight side constants: $$\frac{\pi}{8} \times 5 \times 10^{4} = \frac{5\pi}{8} \times 10^{4}.$$\n\n3. **Rewrite the equation:**\n$$\frac{5\pi}{4} \times 10^{-4} \times \cos\left(\frac{\pi y}{8}\right) = \frac{5\pi}{8} \times 10^{4}.$$\n\n4. **Divide both sides by $\frac{5\pi}{4} \times 10^{-4}$ to isolate cosine:**\n$$\cos\left(\frac{\pi y}{8}\right) = \frac{\frac{5\pi}{8} \times 10^{4}}{\frac{5\pi}{4} \times 10^{-4}} = \frac{\frac{5\pi}{8}}{\frac{5\pi}{4}} \times \frac{10^{4}}{10^{-4}}.$$\n\n5. **Simplify the fraction of constants:**\n$$\frac{\frac{5\pi}{8}}{\frac{5\pi}{4}} = \frac{5\pi}{8} \times \frac{4}{5\pi} = \frac{4}{8} = \frac{1}{2}.$$\n\n6. **Simplify the powers of ten:**\n$$\frac{10^{4}}{10^{-4}} = 10^{4 - (-4)} = 10^{8}.$$\n\n7. **So, the equation becomes:**\n$$\cos\left(\frac{\pi y}{8}\right) = \frac{1}{2} \times 10^{8} = 5 \times 10^{7}.$$\n\n8. **Analyze the result:** The cosine function ranges between $-1$ and $1$, but the right side is $5 \times 10^{7}$, which is much greater than 1.\n\n9. **Conclusion:** There is no real value of $y$ that satisfies this equation because $\cos\left(\frac{\pi y}{8}\right)$ cannot equal $5 \times 10^{7}$.\n\n**Final answer:** No real solution for $y$ exists.