https://file.aiquickdraw.com/mathbot/file/1726753291245694009ih.jpg

Let's break down the problem in the image step by step.

### Given:
1. **Red balloon**:
   - Starts at a height of 100 meters.
   - Increases by 3% every minute.

2. **Blue balloon**:
   - Starts at a height of 120 meters.
   - Increases by 4% every minute.

3. **Green balloon**:
   - Starts at a height that is twice that of the red balloon (i.e., \(2 \times 100 = 200\) meters).
   - Descends at half the rate of the blue balloon's ascent rate, meaning the descent rate is \( \frac{4\%}{2} = 2\% \) per minute.

### Equations:
- For the **red balloon**, its height after \(t\) minutes is given by:
  \[
  r(t) = 100 \times (1.03)^t
  \]
  (since its height increases by 3% per minute).

- For the **blue balloon**, its height after \(t\) minutes is given by:
  \[
  b(t) = 120 \times (1.04)^t
  \]
  (since its height increases by 4% per minute).

- For the **green balloon**, which starts at a height of 200 meters and descends by 2% every minute, its height after \(t\) minutes is given by:
  \[
  g(t) = 200 \times (0.98)^t
  \]
  (since the green balloon decreases in height by 2% per minute).

### Task 1: Complete the equation for \(g(t)\):
The equation for the green balloon is:
\[
g(t) = 200 \times (0.98)^t
\]
This matches what is asked in the first part of the problem.

### Task 2: Compare the heights of the three balloons after 10 minutes.
We can now calculate the heights after 10 minutes for each balloon.

1. **Red balloon**: 
   \[
   r(10) = 100 \times (1.03)^{10}
   \]

2. **Blue balloon**: 
   \[
   b(10) = 120 \times (1.04)^{10}
   \]

3. **Green balloon**: 
   \[
   g(10) = 200 \times (0.98)^{10}
   \]

Let me calculate these values for you.After 10 minutes, the heights of the balloons are:

- Red balloon: approximately 134 meters.
- Blue balloon: approximately 178 meters.
- Green balloon: approximately 163 meters.

The difference between the highest and the lowest balloon is approximately **43 meters**.

### Do you have any questions or would you like more details? 

Here are 5 related questions you could consider:

1. How would the heights of the balloons change after 20 minutes?
2. If the ascent/descent rates changed, how would that affect the final heights?
3. What if the balloons started at different initial heights?
4. Could we calculate when two of the balloons would have the same height?
5. How would the problem change if the rates of increase were not constant?

**Tip:** When calculating percentage changes over time, always make sure to express the rate as a factor (e.g., 3% increase becomes 1.03).

Two hot air balloons, one red and one blue, took off from different platforms and began ascending at a constant rate. The red balloon took off from a platform that was at a height of 100 meters, and the blue balloon took off from a platform that was at a height of 120 meters. Each minute, the red balloon's height increased by 3%, and the blue balloon's height increased by 4%. When the red and blue balloons began their ascent, a green balloon that was twice as high as the red balloon started descending. The green balloon's height decreased at a rate half as fast as the blue balloon's ascent rate. Complete the equation for g(t), the height of the green after t minutes, and compare the heights of the three balloons after 10 minutes.

Explore how the heights of three balloons change over time using exponential growth and decay formulas. The red balloon starts at 100 meters and increases by 3% per minute, the blue balloon starts at 120 meters and increases by 4% per minute, while the green balloon, starting at 200 meters, descends by 2% per minute. Calculate their heights after 10 minutes and find the difference between the highest and lowest balloons.

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