A hot air balloon reaches a height of 250 meters two minutes into flight. Ten minutes later, the hot air balloon has reached a height of 725 meters. Which of the following correctly interprets the balloons average rate of change?

Question

1)the balloons height decreases by an average of 59.38 meters per minute from the 2nd to the 10th minute.
2)the balloons height decreases by an average of 47.5 meters per minute from the 2nd to the 12th minute
3)The balloons height increases by an average of 47.5 meters per minute from the 2nd to the 12th minute
4)the balloons height increases by an average of 59.38 meters per minute from the 2nd to the 20th minute.

Answer 1

To calculate the average rate of change in height of the hot air balloon, we need to find the change in height over the change in time.

1. Initial height at 2 minutes: 250 meters
2. Height at 12 minutes (10 minutes later): 725 meters

Now, we can find the change in height:

\[ \text{Change in height} = \text{Final height} - \text{Initial height} = 725 , \text{meters} - 250 , \text{meters} = 475 , \text{meters} \]

Next, we calculate the total time interval:

\[ \text{Time interval} = 12 , \text{minutes} - 2 , \text{minutes} = 10 , \text{minutes} \]

Now we can calculate the average rate of change:

\[ \text{Average rate of change} = \frac{\text{Change in height}}{\text{Time interval}} = \frac{475 , \text{meters}}{10 , \text{minutes}} = 47.5 , \text{meters per minute} \]

Since the balloon is rising, this will be an increase in height.

Among the provided options, the correct interpretation of the balloon's average rate of change is:

3) The balloon's height increases by an average of 47.5 meters per minute from the 2nd to the 12th minute.