To determine when the toy rocket will reach its maximum height, we need to focus on the function for the height of the rocket:
[tex]\[ h(t) = -16t^2 + 200t + 50 \][/tex]
This function is a quadratic equation of the form \( h(t) = at^2 + bt + c \), where:
- \( a = -16 \)
- \( b = 200 \)
- \( c = 50 \)
For a parabolic function opening downwards (as indicated by the negative coefficient of \( t^2 \)), the maximum height occurs at the vertex of the parabola. The time \( t \) at which this maximum height occurs can be determined by using the vertex formula for a quadratic equation, which is given by:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Substituting the values of \( a \) and \( b \) into the formula, we get:
[tex]\[ t = -\frac{200}{2 \cdot -16} \][/tex]
Calculating the above, we have:
[tex]\[ t = -\frac{200}{-32} \][/tex]
[tex]\[ t = \frac{200}{32} \][/tex]
[tex]\[ t = 6.25 \][/tex]
Therefore, the toy rocket will reach its maximum height at \( t = 6.25 \) seconds.
The correct answer is:
6.25 seconds
