7. Consider the flight of an aircraft used to simulate weightlessness. Its flight path can be approximated by the following equation: $h=-10 t^{2}+300 t+9750$ where $\mathrm{h}$ is height in $\mathrm{m}$, and $\mathrm{t}$ is time in seconds.
a) Find the maximum altitude

Step 1 ：The maximum altitude of the aircraft can be found by finding the vertex of the parabola represented by the equation. The x-coordinate (or in this case, the t-coordinate) of the vertex of a parabola given by the equation \(y = ax^2 + bx + c\) is given by \(-\frac{b}{2a}\).

Step 2 ：In this case, \(a = -10\) and \(b = 300\), so we can substitute these values into the formula to find the time at which the maximum altitude is reached.

Step 3 ：Then, we can substitute this time back into the original equation to find the maximum altitude.

Step 4 ：By substituting the values, we get \(t = -\frac{300}{2*(-10)} = 15.0\) seconds.

Step 5 ：Substituting \(t = 15.0\) seconds into the equation \(h = -10*t^2 + 300*t + 9750\), we get \(h = 12000.0000000000\) meters.

Step 6 ：Final Answer: The maximum altitude of the aircraft is \(\boxed{12000}\) meters.