1. **State the problem:** We are given the height function of a ball as $x(t) = 160t^2 - 40t$, where $x(t)$ is the height in feet and $t$ is the time in seconds after the ball was kicked. 2. **Goal:** Find the time $t$ when the ball hits the ground. This means finding $t$ such that $x(t) = 0$. 3. **Set the equation to zero:** $$160t^2 - 40t = 0$$ 4. **Factor the equation:** $$t(160t - 40) = 0$$ 5. **Solve each factor:** - $t = 0$ (the initial time when the ball was kicked) - $160t - 40 = 0$ 6. **Solve for $t$ in the second factor:** $$160t - 40 = 0$$ $$160t = 40$$ $$\cancel{160}t = \cancel{40}$$ $$t = \frac{40}{160} = \frac{1}{4} = 0.25$$ 7. **Check the result:** $t=0.25$ seconds is when the height is zero again, but this is too soon for the ball to hit the ground after being kicked (it is likely the initial moment or a mistake in the function). Let's re-examine the original function. 8. **Re-examining the function:** The function is $x(t) = 160t^2 - 40t$. Since the coefficient of $t^2$ is positive, the parabola opens upward, meaning the height increases as $t$ increases, which is unusual for a ball thrown upwards. 9. **Assuming the function models height correctly, the ball hits the ground when $x(t) = 0$ for $t > 0$.** 10. **Solve the quadratic equation:** $$160t^2 - 40t = 0$$ Divide both sides by 40: $$\cancel{40} (4t^2 - t) = 0$$ $$4t^2 - t = 0$$ Factor: $$t(4t - 1) = 0$$ Solutions: $$t = 0$$ or $$4t - 1 = 0$$ $$4t = 1$$ $$t = \frac{1}{4} = 0.25$$ 11. **Interpretation:** The ball hits the ground at $t=0$ (initial kick) and $t=0.25$ seconds. Since the problem asks for the time after being kicked, the answer is $0.25$ seconds. 12. **Check answer choices:** None of the options match $0.25$ seconds exactly. This suggests a possible typo in the function or options. 13. **If the function was $x(t) = -16t^2 + 40t$ (a common projectile formula), then:** Set $x(t) = 0$: $$-16t^2 + 40t = 0$$ Factor: $$t(-16t + 40) = 0$$ Solutions: $$t=0$$ or $$-16t + 40 = 0$$ $$-16t = -40$$ $$t = \frac{40}{16} = 2.5$$ 14. **This matches option B (2.5 seconds).** **Final answer:** The ball hits the ground after **2.5 seconds**.