Title: Solving for Time: When Will the Ball Hit the Ground?

To determine the time it takes for the ball to hit the ground, we need to find the value of \( t \) when the height \( h \) is 0. This means we need to solve the equation: \[ -16t^2 + 30t + 4 = 0 \] This is a quadratic equation of the form \( at^2 + bt + c = 0 \), where \( a = -16 \), \( b = 30 \), and \( c = 4 \). We can solve this using the quadratic formula: \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] First, let's calculate the discriminant (\( b^2 - 4ac \)): \[ b^2 - 4ac = 30^2 - 4(-16)(4) \] \[ = 900 + 256 \] \[ = 1156 \] Now we use the quadratic formula: \[ t = \frac{-30 \pm \sqrt{1156}}{2(-16)} \] \[ t = \frac{-30 \pm 34}{-32} \] This gives us two potential solutions for \( t \): 1. \( t = \frac{-30 + 34}{-32} = \frac{4}{-32} = -\frac{1}{8} \) 2. \( t = \frac{-30 - 34}{-32} = \frac{-64}{-32} = 2 \) The time cannot be negative, so we discard \( t = -\frac{1}{8} \). Therefore, the ball will hit the ground after \( t = 2 \) seconds.