The height in meters of a firework rocket is given by: ℎ(𝑥) = 20𝑥 − 5𝑥2 + 2 , where x=time in seconds after start. Determine the average speed from that to the rocket started to the moment when the height is the highest.

1. Determine the maximum height.
2. Solve \( \frac{d h}{d x} = 0 \).
3. Substitute the time \(x\) back into \(h(x)\).
4. Calculate the duration from \(t = 0\) to the maximum height.
5. Use the average speed formula v_{\text{avg}} = \frac{\Delta h}{\Delta t} .

1. Using derivative: \frac{d}{d x} (20 x - 5 x^2 + 2) = 20 - 10x
2. Setting the derivative to zero: 20 - 10x = 0 \Rightarrow x = 2
3. Substitute \( x = 2 \) into the height function: h(2) = 20(2) - 5(2)^2 + 2 = 40 - 20 + 2 = 22 \, \text{m}
4. Duration from start to maximum height \( \Delta t = 2 - 0 = 2 \, \text{s} \)
5. Average speed:
v_{\text{avg}} = \frac{\Delta h}{\Delta t} = \frac{22 - 2}{2 - 0} = \frac{20}{2} = 10 \, \text{m/s}