Lashonda launches a rocket straight up into the air. The table below gives the height $H(t)$ of the rocket (in meters) at a few times $t$ (in seconds) flight. \begin{tabular}{|c|c|} \hline \begin{tabular}{c} Time $t$ \\ (seconds) \end{tabular} & \begin{tabular}{c} Height $H(t)$ \\ (meters) \end{tabular} \\ \hline 0 & 0 \\ \hline 2.8 & 84 \\ \hline 5.6 & 140 \\ \hline 8.4 & 42 \\ \hline 11.2 & 0 \\ \hline \end{tabular} (a) Find the average rate of change for the height from 0 seconds to 2.8 seconds. Theters per second (b) Find the average rate of change ior the height from 5.6 seconds to 11.2 seconds. $\square$ meters per second

Step 1 ：Given the table of the height of the rocket at different times, we are asked to find the average rate of change for the height from 0 seconds to 2.8 seconds and from 5.6 seconds to 11.2 seconds.

Step 2 ：The average rate of change is calculated by the formula: \[\text{Average rate of change} = \frac{\text{Change in } y}{\text{Change in } x}\] In this case, the 'y' is the height of the rocket and 'x' is the time.

Step 3 ：For part (a), we need to find the average rate of change for the height from 0 seconds to 2.8 seconds. This means we need to subtract the height at 0 seconds from the height at 2.8 seconds and divide by the change in time (2.8 seconds - 0 seconds).

Step 4 ：Substituting the given values into the formula, we get \[\text{Average rate of change} = \frac{84 - 0}{2.8 - 0} = 30\] meters per second.

Step 5 ：For part (b), we need to find the average rate of change for the height from 5.6 seconds to 11.2 seconds. This means we need to subtract the height at 5.6 seconds from the height at 11.2 seconds and divide by the change in time (11.2 seconds - 5.6 seconds).

Step 6 ：Substituting the given values into the formula, we get \[\text{Average rate of change} = \frac{0 - 140}{11.2 - 5.6} = -25\] meters per second.

Step 7 ：Final Answer: The average rate of change for the height from 0 seconds to 2.8 seconds is \(\boxed{30}\) meters per second. The average rate of change for the height from 5.6 seconds to 11.2 seconds is \(\boxed{-25}\) meters per second.