1. **State the problem:** Solve the equation $$4.0 = 12.5t - \frac{1}{2}(9.8)t^2$$ for $t$. 2. **Rewrite the equation:** $$4.0 = 12.5t - 4.9t^2$$ 3. **Bring all terms to one side to form a quadratic equation:** $$4.9t^2 - 12.5t + 4.0 = 0$$ 4. **Identify coefficients:** $$a = 4.9, \quad b = -12.5, \quad c = 4.0$$ 5. **Use the quadratic formula:** $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 6. **Calculate the discriminant:** $$\Delta = (-12.5)^2 - 4(4.9)(4.0) = 156.25 - 78.4 = 77.85$$ 7. **Calculate the square root of the discriminant:** $$\sqrt{77.85} \approx 8.82$$ 8. **Find the two possible values for $t$:** $$t_1 = \frac{12.5 + 8.82}{2 \times 4.9} = \frac{21.32}{9.8} \approx 2.18$$ $$t_2 = \frac{12.5 - 8.82}{9.8} = \frac{3.68}{9.8} \approx 0.38$$ 9. **Interpret the solutions:** Both $t \approx 0.38$ seconds and $t \approx 2.18$ seconds satisfy the equation. **Final answer:** $$t \approx 0.38 \text{ s or } t \approx 2.18 \text{ s}$$