If a ball is thrown directly upward with a velocity of \( 28 \mathrm{ft} / \mathrm{s} \), its height (in feet) after \( t \) seconds is given by \( y=28 t-16 t^{2} \). What is the maximum height attained by the ball? (Round your answer to the nearest whole number.) Need Help? Watch It Master It

To find the maximum height attained by the ball, we can use the formula for height \( y = 28t - 16t^2 \). The maximum height occurs at the vertex of the parabolic equation, which can be found at \( t = -\frac{b}{2a} \), where \( a = -16 \) and \( b = 28 \). Plugging in the values: \[ t = -\frac{28}{2 \times -16} = \frac{28}{32} = 0.875 \text{ seconds}. \] Now, substitute \( t = 0.875 \) back into the height equation to find the maximum height: \[ y = 28(0.875) - 16(0.875)^2 \approx 12.25. \] So, the maximum height is approximately \( 12 \) feet when rounded to the nearest whole number. To get a bit more enthusiastic about physics, did you know that the trajectory of a ball thrown in the air can be modeled with a simple parabolic equation? It’s not just about how high it goes, but also about understanding gravity’s effects – which is a crucial concept! If you’ve ever seen a basketball arc towards the hoop, you’ve witnessed these principles in action! Also, if you're curious about kinematics beyond this simple example, consider looking into projectile motion. It’s fascinating to explore how angles affect trajectories and how air resistance plays a role in real-world applications, like sports or even space missions! There are plenty of resources, like physics textbooks or online courses, that dive deeper into these concepts with engaging visuals and experiments. Happy learning!