To find the time \( t \) at which the ball reaches its maximum height using the vertex formula for a quadratic equation of the form \( h = at^2 + bt + c \), we use the formula for the \( t \)-coordinate of the vertex:
\[ t = -\frac{b}{2a} \]
In the given equation \( h = -16t^2 + 32t + 7 \), we can identify:
- \( a = -16 \)
- \( b = 32 \)
Now, plug these values into the vertex formula:
\[ t = -\frac{32}{2(-16)} = -\frac{32}{-32} = 1 \]
So, it takes 1 second for the ball to reach the maximum height.
Next, to find the maximum height, substitute \( t = 1 \) back into the height equation:
\[ h = -16(1)^2 + 32(1) + 7 \] \[ h = -16(1) + 32 + 7 \] \[ h = -16 + 32 + 7 \] \[ h = 16 + 7 \] \[ h = 23 \]
Thus, the maximum height of the ball is 23 feet.
Summary:
- It takes 1 second for the ball to reach the maximum height.
- The maximum height of the ball is 23 feet.