Question
Scientists released a weather balloon from a raised platform at 4:00 p.m. The weather balloon rose at a constant speed. At 4:05 pm, the weather balloon's altitude was 1,482 meters. At 4:09, the weather balloon had reached an altitude of 2,626 meters.
How many meters did the weather balloon rise each minute?
___
How high was the launch platform?
___
Complete the equation that describes the relationship between the altitude of the weather balloon in meters, A, and the elapsed time in minutes, t.
A = ___ t + ___
How many meters did the weather balloon rise each minute?
___
How high was the launch platform?
___
Complete the equation that describes the relationship between the altitude of the weather balloon in meters, A, and the elapsed time in minutes, t.
A = ___ t + ___
Answers
GPT-4o mini
To determine how many meters the weather balloon rose each minute, we can first calculate the elevation change between the two given times.
1. **Calculate the altitude increase:**
- At 4:05 p.m., the altitude was 1,482 meters.
- At 4:09 p.m., the altitude was 2,626 meters.
- The change in altitude is:
\[
2,626 \text{ m} - 1,482 \text{ m} = 1,144 \text{ m}
\]
2. **Calculate the time elapsed:**
- From 4:05 to 4:09, 4 minutes elapsed.
3. **Calculate the rise per minute:**
- The rise per minute can be determined by dividing the change in altitude by the time elapsed:
\[
\frac{1,144 \text{ m}}{4 \text{ min}} = 286 \text{ m/min}
\]
So, the weather balloon rose **286 meters per minute**.
---
Next, to determine how high the launch platform was, we need to find the altitude at t = 0 (when the balloon was released at 4:00 p.m.).
1. **Let’s establish a linear function based on the information we have:**
We know:
- At t = 5 minutes (4:05 p.m.), A = 1,482 m
- At t = 9 minutes (4:09 p.m.), A = 2,626 m.
2. **Using the rise rate we found (286 m/min):**
- The altitude A can be expressed as:
\[
A = 286t + h
\]
where \( h \) is the height of the launch platform.
3. **Substituting the information into the equations:**
For t = 5:
\[
1,482 = 286(5) + h \implies 1,482 = 1,430 + h \implies h = 1,482 - 1,430 = 52 \text{ meters}
\]
So, the height of the launch platform is **52 meters**.
---
Finally, we can write the equation that describes the relationship between the altitude A of the weather balloon in meters and the elapsed time t in minutes.
Based on our calculations:
- The rise rate is 286 meters per minute.
- The height of the launch platform is 52 meters.
Thus, the equation is:
\[
A = 286t + 52
\]
**Final Answers:**
- The weather balloon rose **286 meters per minute**.
- The height of the launch platform was **52 meters**.
- The equation is \( A = 286t + 52 \).
1. **Calculate the altitude increase:**
- At 4:05 p.m., the altitude was 1,482 meters.
- At 4:09 p.m., the altitude was 2,626 meters.
- The change in altitude is:
\[
2,626 \text{ m} - 1,482 \text{ m} = 1,144 \text{ m}
\]
2. **Calculate the time elapsed:**
- From 4:05 to 4:09, 4 minutes elapsed.
3. **Calculate the rise per minute:**
- The rise per minute can be determined by dividing the change in altitude by the time elapsed:
\[
\frac{1,144 \text{ m}}{4 \text{ min}} = 286 \text{ m/min}
\]
So, the weather balloon rose **286 meters per minute**.
---
Next, to determine how high the launch platform was, we need to find the altitude at t = 0 (when the balloon was released at 4:00 p.m.).
1. **Let’s establish a linear function based on the information we have:**
We know:
- At t = 5 minutes (4:05 p.m.), A = 1,482 m
- At t = 9 minutes (4:09 p.m.), A = 2,626 m.
2. **Using the rise rate we found (286 m/min):**
- The altitude A can be expressed as:
\[
A = 286t + h
\]
where \( h \) is the height of the launch platform.
3. **Substituting the information into the equations:**
For t = 5:
\[
1,482 = 286(5) + h \implies 1,482 = 1,430 + h \implies h = 1,482 - 1,430 = 52 \text{ meters}
\]
So, the height of the launch platform is **52 meters**.
---
Finally, we can write the equation that describes the relationship between the altitude A of the weather balloon in meters and the elapsed time t in minutes.
Based on our calculations:
- The rise rate is 286 meters per minute.
- The height of the launch platform is 52 meters.
Thus, the equation is:
\[
A = 286t + 52
\]
**Final Answers:**
- The weather balloon rose **286 meters per minute**.
- The height of the launch platform was **52 meters**.
- The equation is \( A = 286t + 52 \).