To solve this problem, we can use knowledge of quadratic functions and their properties. The height of the missile as a function of time, [tex]\( h(t) \)[/tex], is given by the quadratic equation:
[tex]\[
h(t) = -4.9t^2 + 122t + 9
\][/tex]
### Step 1: Determine the time at which the maximum height occurs
For any quadratic function of the form [tex]\( h(t) = at^2 + bt + c \)[/tex], the time at which the maximum (or minimum) height occurs can be found using the vertex formula:
[tex]\[
t = -\frac{b}{2a}
\][/tex]
Here, [tex]\( a = -4.9 \)[/tex] and [tex]\( b = 122 \)[/tex]. Substituting these values into the vertex formula gives:
[tex]\[
t = -\frac{122}{2(-4.9)}
\][/tex]
Simplifying the expression within the fraction:
[tex]\[
t = \frac{122}{9.8}
\][/tex]
[tex]\[
t \approx 12.449
\][/tex]
### Step 2: Calculate the maximum height
To find the maximum height, substitute the time [tex]\( t = 12.449 \)[/tex] back into the original height function [tex]\( h(t) \)[/tex].
[tex]\[
h(12.449) = -4.9(12.449)^2 + 122(12.449) + 9
\][/tex]
Calculating each term separately and then combining:
[tex]\[
h(12.449) \approx -4.9(155.011) + 122(12.449) + 9
\][/tex]
[tex]\[
h(12.449) \approx -759.554 + 1517.778 + 9
\][/tex]
[tex]\[
h(12.449) \approx 768.388
\][/tex]
### Conclusion
The time it takes for the missile to reach its maximum height is approximately:
[tex]\[
\boxed{12.449} \text{ seconds}
\][/tex]
The maximum height of the missile is approximately:
[tex]\[
\boxed{768.388} \text{ meters}
\][/tex]
