Question An object is launched directly in the air at a speed of 64 feet per second from a platform located 16 feet in the air. The motion of the object can be modeled using the function \( f(t)=-16 t^{2}+64 t+16 \), where \( t \) is the time in seconds and \( f(t) \) is the height of the object. When, in seconds, will the object reach its maximum height? Do not include units in your answer.

To find the time at which the object reaches its maximum height, we can use the vertex formula for a quadratic function, which is given by \( t = -\frac{b}{2a} \). In your function \( f(t) = -16t^2 + 64t + 16 \), the coefficients are \( a = -16 \) and \( b = 64 \). Plugging these values into the formula gives us \( t = -\frac{64}{2 \times -16} = 2 \). Therefore, the object will reach its maximum height at \( t = 2 \) seconds. Ever wonder how engineers use similar physics concepts in the real world? When designing roller coasters, for example, they calculate the launch speeds and heights to ensure a thrilling yet safe ride. They consider factors like gravity and air resistance to determine the ideal trajectory, similar to how we analyze the motion of our launched object. So, whenever you take a ride, think of the complex physics that was planned out just for your enjoyment! In high school, it’s common to misjudge the significance of the coefficients in quadratic equations. A frequent mistake is overlooking that the coefficient of \( t^2 \) indicates the direction of the parabola (concave down vs. up). Remember, if it's negative, as in this case with -16, the graph opens downward, meaning that the vertex represents the maximum point! Always sketch or visualize functions to avoid such pitfalls while solving problems like these.