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The problem you're dealing with involves two planes climbing at different rates and asks when their altitudes will be the same.

**Step-by-step solution:**

Let \( t \) represent the time in seconds when the two planes will reach the same altitude.

### Information given:
- Plane A's initial altitude: 5000 feet
- Plane B's initial altitude: 4225 feet
- Plane A's rate of climb: 35.25 feet per second
- Plane B's rate of climb: 50.75 feet per second

We can express the altitude of each plane as a function of time \( t \).

- **Plane A's altitude**: \( 5000 + 35.25t \)
- **Plane B's altitude**: \( 4225 + 50.75t \)

At the point where their altitudes are equal, we can set these two equations equal to each other:

\[
5000 + 35.25t = 4225 + 50.75t
\]

### Solve for \( t \):

1. Subtract 35.25t from both sides:

\[
5000 = 4225 + 15.5t
\]

2. Subtract 4225 from both sides:

\[
775 = 15.5t
\]

3. Divide by 15.5 to solve for \( t \):

\[
t = \frac{775}{15.5} = 50 \text{ seconds}
\]

So, it will take **50 seconds** for the planes to be at the same altitude.

### Find their altitude at that time:

Plug \( t = 50 \) into either altitude equation. We'll use Plane A's equation:

\[
\text{Altitude} = 5000 + 35.25 \times 50 = 5000 + 1762.5 = 6762.5 \text{ feet}
\]

So, the planes will be at the same altitude of **6762.5 feet** after **50 seconds**.

---

Would you like further clarification or any additional details?

Here are some related questions for further exploration:
1. What if Plane A climbed at a faster rate than Plane B? How would the result change?
2. How would the solution change if both planes started at the same altitude but climbed at different rates?
3. What if the planes were descending instead of ascending?
4. Can you calculate how much higher Plane B will be after 100 seconds compared to Plane A?
5. How can you set up this problem using graphing techniques?

**Tip:** When solving for the same altitude, always align the units (time, altitude rate) correctly to avoid errors.

An air traffic controller is tracking two planes. Plane A is at 5000 feet, gaining altitude at 35.25 feet per second, and Plane B is at 4225 feet, gaining altitude at 50.75 feet per second. How many seconds until they reach the same altitude, and what will their altitude be?

This problem involves solving linear equations to determine when two planes, with different starting altitudes and climbing rates, will reach the same altitude. Suitable for Grades 8-10, this detailed solution explores rates of change and linear equality.

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