The altitude of a hot air balloon is changing at a constant rate. The table shows the linear relationship between y, the altitude of the airplane in feet, and x, the time in minutes.

Airplan's Altitude
 
Time, x
(minutes)	Altitude, y
(feet)
1.5	34,000
3	43,000
5.25	56,500

Complete the statement that describes the rate of change of the altitude of the airplane with respect to time.

Choose the correct answer from each drop-down menu to complete the sentence.

The altitude of the airplane (Select)  at a rate of (Select)  feet per minute.
 

To solve this problem, we need to calculate the rate of change of the altitude with respect to time. This can be done by using the formula for the slope between two points on a line, which is the "change in y" divided by the "change in x."

Given the table:
- (1.5, 34,000)
- (3, 43,000)
- (5.25, 56,500)

### Step 1: Calculate the rate of change (slope) using any two points.

Let's use the first two points (1.5, 34,000) and (3, 43,000):
\[
\text{Rate of change} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{43,000 - 34,000}{3 - 1.5} = \frac{9,000}{1.5} = 6,000 \, \text{feet per minute}
\]

### Step 2: Verify with the other points

Now, check with the next pair of points (3, 43,000) and (5.25, 56,500):
\[
\frac{y_2 - y_1}{x_2 - x_1} = \frac{56,500 - 43,000}{5.25 - 3} = \frac{13,500}{2.25} = 6,000 \, \text{feet per minute}
\]

Since the rate is the same for both pairs of points, the rate of change is constant.

### Conclusion:

The altitude of the airplane **increases** at a rate of **6,000** feet per minute.

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Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the slope between any two points on a line?
2. What are other methods for finding the rate of change in different scenarios?
3. How would the rate of change differ if the altitude was decreasing?
4. What if the relationship between altitude and time wasn't linear—how could we analyze it?
5. How can we predict the altitude at a given time using the slope?

**Tip:** When dealing with a linear relationship, the slope represents the constant rate of change between the two variables.

The altitude of a hot air balloon is changing at a constant rate. The table shows the linear relationship between y, the altitude of the airplane in feet, and x, the time in minutes.

Airplane's Altitude:

Time, x (minutes)	Altitude, y (feet)
1.5	34,000
3	43,000
5.25	56,500

Complete the statement that describes the rate of change of the altitude of the airplane with respect to time.

The altitude of the airplane (Select) at a rate of (Select) feet per minute.

Learn how to calculate the rate of change of an airplane's altitude using linear functions and the slope formula. This example involves determining the altitude change per minute using data from a table. Suitable for students in Grades 7-10.

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