Question: User: The Problem: The Deep-Sea Submersible
A marine biology team is operating a deep-sea submersible to map the ocean floor. The
submersible's onboard computer has a total storage capacity of $1.5 \times 10^3$ Gigabytes (GB).
Before the dive begins, the computer already contains $2^8$ GB of operating system files and old data.
During the dive, the following events occur:
1. The submersible records high-resolution sonar data for $4.5$ hours. The data is collected at a rate
of $1.2 \times 10^2$ GB per hour.
2. To save space, the onboard computer immediately runs a compression algorithm on the newly
collected sonar data, reducing its size to exactly $\frac{2}{3}$ of its original size.
3. The crew also runs a disk cleanup, which deletes a batch of corrupted temporary files. The size
of the deleted files is equivalent to $\sqrt{625} \times 10^1$ GB.
Your Tasks:
Part 1: Mathematical Modeling (Criterion A & C)
Write a single mathematical expression that represents the remaining storage space (in GB) on the
submersible's computer at the end of the dive.
Part 2: Evaluation and Communication (Criterion A & C)
Evaluate your expression step-by-step using the correct order of operations.
• Clearly show how you convert between scientific notation, powers, square roots, and standard
form.
• State the final remaining storage space in standard form.
Part 3: Real-Life Application and Justification (Criterion D)
The team plans to do a second dive the next day. They estimate they will collect exactly $1.15 \times 10^3$
GB of uncompressed data, which will go through the exact same compression algorithm.
• Will they have enough remaining storage space for this second dive?
• Discuss whether your mathematical conclusion makes complete sense in a real-life context. (Hint:
Think about how computer storage actually works in reality, or potential risks during the mission).
1. **State the problem:**
We want to find the remaining storage space on the submersible's computer after recording and compressing sonar data, and deleting corrupted files.
2. **Define variables and given data:**
- Total storage capacity: $1.5 \times 10^3$ GB
- Existing data: $2^8$ GB
- Recording time: $4.5$ hours
- Data rate: $1.2 \times 10^2$ GB/hour
- Compression ratio: $\frac{2}{3}$ (new size is two-thirds original)
- Deleted files size: $\sqrt{625} \times 10^1$ GB
3. **Write the expression for new data collected before compression:**
$$\text{New data} = 4.5 \times 1.2 \times 10^2$$
4. **Apply compression to new data:**
$$\text{Compressed data} = \frac{2}{3} \times (4.5 \times 1.2 \times 10^2)$$
5. **Write the expression for remaining storage space:**
$$\text{Remaining} = 1.5 \times 10^3 - 2^8 - \frac{2}{3} \times (4.5 \times 1.2 \times 10^2) + \sqrt{625} \times 10^1$$
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6. **Evaluate each term step-by-step:**
- Calculate $2^8$:
$$2^8 = 256$$
- Calculate new data before compression:
$$4.5 \times 1.2 \times 10^2 = 4.5 \times 120 = 540$$
- Calculate compressed data:
$$\frac{2}{3} \times 540 = \frac{2}{3} \times 540 = 360$$
- Calculate deleted files size:
$$\sqrt{625} = 25$$
$$25 \times 10^1 = 25 \times 10 = 250$$
- Calculate total storage capacity:
$$1.5 \times 10^3 = 1500$$
7. **Substitute values into the remaining storage expression:**
$$\text{Remaining} = 1500 - 256 - 360 + 250$$
8. **Simplify step-by-step:**
$$1500 - 256 = 1244$$
$$1244 - 360 = 884$$
$$884 + 250 = 1134$$
So, the remaining storage space is $1134$ GB.
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9. **Part 3: Second dive storage check**
- Estimated uncompressed data for second dive:
$$1.15 \times 10^3 = 1150$$
- Compressed size for second dive:
$$\frac{2}{3} \times 1150 = \frac{2}{3} \times 1150 = 766.67$$ (approx.)
- Compare with remaining storage:
$$1134 - 766.67 = 367.33$$ GB remaining after second dive data storage.
**Conclusion:** They have enough storage space for the second dive.
10. **Real-life context discussion:**
- Although mathematically there is enough space, real computer storage may have overhead, fragmentation, or reserved space.
- Compression efficiency might vary.
- Unexpected data or errors could reduce available space.
- It's prudent to have a safety margin and not fill storage completely.
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**Final answers:**
- Remaining storage after first dive: $1134$ GB
- Enough space for second dive: Yes, with approximately $367.33$ GB left