The table below gives the height above the ground, $h$, of a passenger traveling on the Vegas High Roller, currently the largest Ferris wheel in the world with a 520 -foot diameter. ${ }^{1}$ Assume passengers board the wheel at its lowest point ( 30 feet off the ground), time is measured in minutes since boarding, and a full ride takes 30 minutes. Write a difference quotient that represents the passenger's average vertical speed during her trip from the bottom to the top of the wheel, then calculate its value using data from the table. (Recall that heights on the way down are equal to heights on the way up.)
\begin{tabular}{l|c|c|c|c|c|c|c}
\hline$t$ & 0 & 2.5 & 5 & 7.5 & 10 & 12.5 & 15 \\
\hline$h=f(t)$ & 30 & 64.8 & 160 & 290 & 420 & 515.2 & 550 \\
\hline
\end{tabular}

Step 1 ：The table below gives the height above the ground, $h$, of a passenger traveling on the Vegas High Roller, currently the largest Ferris wheel in the world with a 520 -foot diameter. Assume passengers board the wheel at its lowest point ( 30 feet off the ground), time is measured in minutes since boarding, and a full ride takes 30 minutes. Write a difference quotient that represents the passenger's average vertical speed during her trip from the bottom to the top of the wheel, then calculate its value using data from the table. (Recall that heights on the way down are equal to heights on the way up.)

Step 2 ：The average vertical speed is the change in height divided by the change in time. We can calculate this by subtracting the initial height from the final height and dividing by the time taken to reach the top. The initial height is 30 feet at time 0 and the final height is 550 feet at time 15 minutes.

Step 3 ：Let's denote the initial height as \(h_{i}\), the final height as \(h_{f}\), the initial time as \(t_{i}\), and the final time as \(t_{f}\).

Step 4 ：\(h_{i} = 30\) feet, \(h_{f} = 550\) feet, \(t_{i} = 0\) minutes, and \(t_{f} = 15\) minutes.

Step 5 ：The average speed can be calculated using the formula \(\frac{h_{f} - h_{i}}{t_{f} - t_{i}}\).

Step 6 ：Substituting the given values into the formula, we get \(\frac{550 - 30}{15 - 0} = 34.666666666666664\) feet per minute.

Step 7 ：Final Answer: The average vertical speed of the passenger during her trip from the bottom to the top of the wheel is approximately \(\boxed{34.67}\) feet per minute.