1. The problem is to understand and analyze the given equations: $$y = 1.165 x \cdot 1.023^t$$ and $$1.2 y = x \cdot 1.034^t$$ 2. We want to find the relationship between $x$, $y$, and $t$ or solve for one variable in terms of the others. 3. From the first equation, isolate $y$: $$y = 1.165 x \cdot 1.023^t$$ 4. From the second equation, isolate $y$: $$1.2 y = x \cdot 1.034^t$$ Divide both sides by 1.2: $$y = \frac{x \cdot 1.034^t}{1.2}$$ Show cancellation: $$y = x \cdot \frac{1.034^t}{\cancel{1.2}}$$ 5. Since both expressions equal $y$, set them equal to each other: $$1.165 x \cdot 1.023^t = x \cdot \frac{1.034^t}{1.2}$$ 6. Divide both sides by $x$ (assuming $x \neq 0$): $$1.165 \cdot 1.023^t = \frac{1.034^t}{1.2}$$ Show cancellation: $$\cancel{x} \cdot 1.165 \cdot 1.023^t = \cancel{x} \cdot \frac{1.034^t}{1.2}$$ 7. Multiply both sides by 1.2: $$1.2 \cdot 1.165 \cdot 1.023^t = 1.034^t$$ Calculate $1.2 \times 1.165 = 1.398$: $$1.398 \cdot 1.023^t = 1.034^t$$ 8. Divide both sides by $1.023^t$: $$1.398 = \frac{1.034^t}{1.023^t} = \left(\frac{1.034}{1.023}\right)^t$$ 9. Take the natural logarithm of both sides: $$\ln(1.398) = t \cdot \ln\left(\frac{1.034}{1.023}\right)$$ 10. Solve for $t$: $$t = \frac{\ln(1.398)}{\ln\left(\frac{1.034}{1.023}\right)}$$ Calculate values: $$\ln(1.398) \approx 0.336$$ $$\ln\left(\frac{1.034}{1.023}\right) = \ln(1.01074) \approx 0.0107$$ Therefore: $$t \approx \frac{0.336}{0.0107} \approx 31.4$$ **Final answer:** $$t \approx 31.4$$