1. **State the problem:** Mateo and his cousins invest 25,000 in an account earning 8% interest annually. Mateo proposes the equation $$y = 25,000(0.08)^x$$ to find the amount of money after $$x$$ years. We need to check if this equation is correct. 2. **Understand the formula for compound interest:** The general formula for compound interest is $$y = P(1 + r)^x$$ where: - $$P$$ is the principal amount (initial investment), - $$r$$ is the annual interest rate (as a decimal), - $$x$$ is the number of years, - $$y$$ is the amount after $$x$$ years. 3. **Apply the formula to this problem:** Here, $$P = 25,000$$ and $$r = 0.08$$ (8%). So the correct formula should be: $$y = 25,000(1 + 0.08)^x = 25,000(1.08)^x$$ 4. **Evaluate Mateo's equation after 1 year:** Mateo's equation is $$y = 25,000(0.08)^x$$. After 1 year ($$x=1$$), this gives: $$y = 25,000(0.08)^1 = 25,000 \times 0.08 = 2,000$$ This means the account would have only 2,000 after 1 year, which is less than the initial investment, so this is incorrect. 5. **Evaluate the correct equation after 1 year:** Using the correct formula: $$y = 25,000(1.08)^1 = 25,000 \times 1.08 = 27,000$$ This shows the account grows by 8% to 27,000 after 1 year, which makes sense. 6. **Conclusion:** Mateo's equation uses the interest rate directly as the base, which incorrectly models the amount decreasing each year. The correct equation uses $$1 + r$$ as the base to represent growth. **Final answers for the dropdowns:** - The amount of money in the account after 1 year can be determined by evaluating the expression $$25,000 \times 1.08$$. - The amount of money, $$y$$, in the account after $$x$$ years can be calculated using the equation $$y = 25,000(1.08)^x$$. - Mateo's equation is **not** correct.