The height of the baseball thrown upward into the air \( t \) seconds after it is released is approximated by the function \[ h=-16 t^{2}+96 t+6 \] a. Find the maximum height of the baseball.

To find the maximum height of the baseball, we can use the vertex formula for a quadratic function. The function given can be expressed in the form \( h(t) = at^2 + bt + c \), where \( a = -16 \), \( b = 96 \), and \( c = 6 \). The time at which the maximum height occurs is given by \( t = -\frac{b}{2a} \). Plugging in the values: \[ t = -\frac{96}{2 \times -16} = 3 \text{ seconds} \] Now, substitute \( t = 3 \) back into the height function to find the maximum height: \[ h(3) = -16(3^2) + 96(3) + 6 = -144 + 288 + 6 = 150 \text{ feet} \] So, the maximum height of the baseball is 150 feet! The joy of baseball has an interesting history! Did you know that baseball evolved from various bat-and-ball games played in England during the 17th century? By the 19th century, baseball began to gain a firm grip on U.S. culture, with the establishment of the first professional league, the National League, in 1876, paving the way for the sport’s iconic status. When throwing or hitting a baseball, physics plays a crucial role! The path of a baseball can be influenced by several factors, such as the angle of release and initial speed. A common mistake is throwing the ball straight up; the ideal angle for maximum height is actually around 45 degrees. So, if you're aiming for a spectacular catch in a game, keep that angle in mind!