1. **State the problem:** We are given the height function of a baseball popped up as $b(t) = 80t - 16t^2 + 3.5$. We need to find how long the catcher has to get in position before the ball hits the ground. This means finding the time $t$ when $b(t) = 0$. 2. **Write the equation to solve:** Set the height equal to zero: $$-16t^2 + 80t + 3.5 = 0$$ 3. **Use the quadratic formula:** For a quadratic equation $at^2 + bt + c = 0$, the solutions are given by $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Here, $a = -16$, $b = 80$, and $c = 3.5$. 4. **Calculate the discriminant:** $$b^2 - 4ac = 80^2 - 4(-16)(3.5) = 6400 + 224 = 6624$$ 5. **Calculate the square root:** $$\sqrt{6624} \approx 81.39$$ 6. **Calculate the two possible times:** $$t = \frac{-80 \pm 81.39}{2(-16)} = \frac{-80 \pm 81.39}{-32}$$ 7. **Calculate each root:** - For the plus sign: $$t = \frac{-80 + 81.39}{-32} = \frac{1.39}{-32} \approx -0.0434$$ (not physically meaningful since time cannot be negative) - For the minus sign: $$t = \frac{-80 - 81.39}{-32} = \frac{-161.39}{-32} \approx 5.04$$ 8. **Interpret the result:** The ball hits the ground after approximately $5.04$ seconds. 9. **Round the answer:** Rounded to the nearest second, the catcher has about $5$ seconds to get in position. **Final answer:** The catcher has approximately **5 seconds** before the ball hits the ground.