To determine the time taken by the object to reach its maximum height, we can analyze the given height function:
\[ h(t) = -16t^2 + 40t + 5 \]
This is a quadratic function in the form of \( h(t) = at^2 + bt + c \), where:
- \( a = -16 \)
- \( b = 40 \)
- \( c = 5 \)
The maximum height for a quadratic function occurs at the vertex, and the time \( t \) at which this occurs can be found using the formula:
\[ t = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ t = -\frac{40}{2 \times -16} = -\frac{40}{-32} = \frac{40}{32} = \frac{5}{4} \text{ seconds} = 1.25 \text{ seconds} \]
Now, to find the maximum height reached by the object, we substitute \( t = 1.25 \) back into the height function \( h(t) \):
\[ h(1.25) = -16(1.25)^2 + 40(1.25) + 5 \]
Calculating \( 1.25^2 \):
\[ 1.25^2 = 1.5625 \]
Now substituting this value into the equation:
\[ h(1.25) = -16(1.5625) + 40(1.25) + 5 \] \[ = -25 + 50 + 5 \] \[ = 30 \text{ feet} \]
Thus, the time taken to reach maximum height is \( 1.25 \) seconds, and the maximum height reached by the object is \( 30 \) feet.
Final Answers:
- Time taken to reach maximum height: 1.25 seconds
- Maximum height reached: 30 feet