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 balloon’s average rate of change?(1 point)

Question

Responses

The balloon’s height decreases by an average of 47.5 meters per minute from the 2nd to the 12th minute.
The balloon’s height decreases by an average of 47.5 meters per minute from the 2nd to the 12th minute.

The balloon’s height increases by an average of 59.38 meters per minute from the 2nd to the 10th minute.
The balloon’s height increases by an average of 59.38 meters per minute from the 2nd to the 10th minute.

The balloon’s height decreases by an average of 59.38 meters per minute from the 2nd to the 10th minute.
The balloon’s height decreases by an average of 59.38 meters per minute from the 2nd to the 10th minute.

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

Answer 1

To determine the average rate of change of the hot air balloon's height from the 2nd minute to the 12th minute, you can use the formula for average rate of change:

\[ \text{Average Rate of Change} = \frac{\text{Change in Height}}{\text{Change in Time}} \]

1. Initial and final heights:

   • Height at 2 minutes: 250 meters
   • Height at 12 minutes (2 minutes + 10 minutes): 725 meters

2. Change in height: \[ \text{Change in Height} = 725 , \text{meters} - 250 , \text{meters} = 475 , \text{meters} \]

3. Change in time: \[ \text{Change in Time} = 12 , \text{minutes} - 2 , \text{minutes} = 10 , \text{minutes} \]

4. Average rate of change: \[ \text{Average Rate of Change} = \frac{475 , \text{meters}}{10 , \text{minutes}} = 47.5 , \text{meters per minute} \]

Since the height of the balloon increases over this time period, the average rate of change is positive.

Thus, the correct interpretation is:

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