To find the maximum height of the arrow described by the quadratic equation \( h = -16t^2 + 80t + 25 \), we need to determine the vertex of the parabola represented by this equation. The vertex formula for the time \( t \) at which the maximum height occurs for a quadratic equation in the form \( at^2 + bt + c \) is given by:
\[ t = -\frac{b}{2a} \]
In this equation:
- \( a = -16 \)
- \( b = 80 \)
Plugging in the values:
\[ t = -\frac{80}{2 \times -16} = -\frac{80}{-32} = 2.5 \text{ seconds} \]
Now that we have the time when the arrow reaches its maximum height, we can find the maximum height \( h \) by substituting \( t = 2.5 \) back into the height equation:
\[ h = -16(2.5)^2 + 80(2.5) + 25 \]
Calculating \( (2.5)^2 \) first:
\[ (2.5)^2 = 6.25 \]
So,
\[ h = -16(6.25) + 80(2.5) + 25 \] \[ h = -100 + 200 + 25 \] \[ h = 125 \]
Thus, the maximum height of the arrow is 125 feet.
The answer is: 125 feet.