An air traffic controller is tracking two planes. To start, Plane A is at an altitude of 3000 feet and Plane B is at an altitude of 1623 feet. Plane A is gaining altitude at 30.25 feet per second and Plane B is gaining altitude at 55.75 feet per second. How many seconds will pass before the planes are at the same altitude? What will their altitude be when they're at the same altitude?

To solve this problem, we need to determine when the altitudes of Plane A and Plane B will be the same and find the altitude at that moment.

### Step 1: Set up the equations
Let \( t \) represent the time in seconds after the planes start climbing.

- **Plane A's altitude** after \( t \) seconds:
  \[
  \text{Altitude of Plane A} = 3000 + 30.25t
  \]

- **Plane B's altitude** after \( t \) seconds:
  \[
  \text{Altitude of Plane B} = 1623 + 55.75t
  \]

### Step 2: Set the altitudes equal to each other
We need to find the time \( t \) when both planes have the same altitude, so we set the two altitude expressions equal:
\[
3000 + 30.25t = 1623 + 55.75t
\]

### Step 3: Solve for \( t \)
First, subtract \( 30.25t \) from both sides to get:
\[
3000 = 1623 + 25.5t
\]

Next, subtract 1623 from both sides:
\[
1377 = 25.5t
\]

Now, solve for \( t \) by dividing both sides by 25.5:
\[
t = \frac{1377}{25.5} \approx 54
\]

So, it will take approximately 54 seconds for the planes to reach the same altitude.

### Step 4: Find the altitude at that time
Substitute \( t = 54 \) back into the equation for the altitude of either plane (both will give the same result since their altitudes are equal at this time):

For Plane A:
\[
\text{Altitude} = 3000 + 30.25(54) = 3000 + 1633.5 = 4633.5 \text{ feet}
\]

For Plane B:
\[
\text{Altitude} = 1623 + 55.75(54) = 1623 + 3010.5 = 4633.5 \text{ feet}
\]

### Final Answer:
- The planes will be at the same altitude after approximately \( \boxed{54} \) seconds.
- Their altitude at that moment will be \( \boxed{4633.5} \) feet.

Would you like more details or have any questions?

### Related Questions:
1. How would the result change if Plane A was gaining altitude faster than Plane B?
2. How long would it take for Plane A to be 1000 feet higher than Plane B?
3. If Plane A started at 2000 feet instead of 3000 feet, how would this affect the time until they reach the same altitude?
4. What would happen if both planes started at the same altitude but with different rates of ascent?
5. Can you express the time and altitude using a different method, like graphing the equations?
6. What if Plane B was losing altitude instead of gaining it? How would that change the equation?
7. How can you verify the solution using a different approach or by checking calculations?
8. If Plane A starts at a different time than Plane B, how would you adjust the equations?

### Tip:
When solving problems involving rates of change, setting up equations based on initial values and rates can help you find when two quantities will be equal.

Learn how to calculate when two planes, starting at different altitudes and gaining altitude at different rates, will be at the same altitude. This problem covers algebraic concepts and linear equations involving rates of change. Suitable for Grades 7-9 students.

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