https://file.aiquickdraw.com/mathbot/file/1730085874575sk2dozw7.jpg

Let's analyze and solve each question based on the provided information.

### Given Information:
1. **Balloon A**:
   - Formula: \( d = 280t \)
   - \( d \) is the distance in centimeters, and \( t \) is the time in seconds.

2. **Balloon B**:
   - Table of distances for various times:
     - \( t = 0 \): \( d = 0 \)
     - \( t = 1 \): \( d = 290 \)
     - \( t = 2 \): \( d = 580 \)
     - \( t = 3 \): \( d = 870 \)

3. **Balloon C**:
   - Graph showing a linear relationship between distance and time, with a slope indicating a rate of rise.

### Questions and Solutions:

#### (a) What is the rate of each balloon's rise?
The rate of rise is the rate of change of distance with respect to time, essentially the slope or speed in cm/s.

1. **Balloon A**:
   - Given \( d = 280t \), the rate of rise is the coefficient of \( t \), which is **280 cm/s**.

2. **Balloon B**:
   - From the table, we can calculate the rate of rise by finding the difference in distance over time:
     - From \( t = 1 \) to \( t = 2 \): \((580 - 290) / (2 - 1) = 290\)
   - The rate of rise is constant at **290 cm/s**.

3. **Balloon C**:
   - Observing the graph, the line passes through points such as (0, 0) and (1, 310).
   - The rate of rise is therefore \( 310 \, \text{cm/s} \).

**Answer**:
   - Balloon A: 280 cm/s
   - Balloon B: 290 cm/s
   - Balloon C: 310 cm/s

#### (b) Which balloon is rising the fastest?
The fastest balloon is the one with the highest rate of rise.

- **Balloon C** has the highest rate at **310 cm/s**.

**Answer**: Balloon C.

#### (c) How much does each balloon rise in 2 seconds?

1. **Balloon A**:
   - Using \( d = 280t \):
     - \( d = 280 \times 2 = 560 \, \text{cm} \)

2. **Balloon B**:
   - From the table, at \( t = 2 \), \( d = 580 \, \text{cm} \).

3. **Balloon C**:
   - Using the rate of 310 cm/s, in 2 seconds:
     - \( d = 310 \times 2 = 620 \, \text{cm} \)

**Answer**:
   - Balloon A: 560 cm
   - Balloon B: 580 cm
   - Balloon C: 620 cm

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Let me know if you need further details or have any questions about these calculations.

### Further Questions:
1. How would the results change if the rate for each balloon increased by 10%?
2. Can you create a general formula for a balloon's height over time given a constant rise rate?
3. What if Balloon C's rate was not constant? How would you determine the rate at each second?
4. How long would it take for each balloon to reach a height of 1000 cm?
5. How do these concepts apply in physics when discussing velocity and acceleration?

### Tip:
When analyzing graphs, remember that the slope (rate of change) represents the speed or rate of rise in distance-time graphs.

Three hot air balloons are rising at constant rates. For each, the distance risen depends on time. Find the rate of each balloon's rise, identify the fastest balloon, and determine how much each balloon rises in 2 seconds.

Learn how to calculate the rate of rise for three hot air balloons using linear equations, tables, and graphs. Identify the fastest balloon and determine the distance each balloon rises in a given time. Suitable for Grades 6-8.

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