To find the maximum height of the arrow represented by the equation \( h(t) = -16t^2 + 80t + 25 \), we can use the formula for the vertex of a parabola, since the equation is in the form of a quadratic equation \( h(t) = at^2 + bt + c \).
The vertex of a parabola given by \( h(t) \) occurs at \( t = -\frac{b}{2a} \). In this case:
- \( a = -16 \)
- \( b = 80 \)
Substituting these values into the vertex formula:
\[ t = -\frac{80}{2 \times -16} = -\frac{80}{-32} = \frac{80}{32} = \frac{5}{2} = 2.5 \text{ seconds} \]
Now that we have the time at which the maximum height occurs, we can substitute \( t = 2.5 \) back into the height equation to find the maximum height \( h(2.5) \):
\[ h(2.5) = -16(2.5)^2 + 80(2.5) + 25 \]
Calculating \( (2.5)^2 \):
\[ (2.5)^2 = 6.25 \]
Now substitute this value into the equation:
\[ h(2.5) = -16(6.25) + 80(2.5) + 25 \] \[ = -100 + 200 + 25 \] \[ = 125 \text{ feet} \]
The maximum height of the arrow is \( \boxed{125} \) feet.
1 answer