Problem 71 - Algebra 1

Question

Evaluate the expression. Use estimation to check your answer. Susmarine DEPTH A submarine is at a depth of 725 feet below sea level. Five minutes later, it is at a depth of 450 feet below sea level. What is the change in depth of the submarine? Did it go up or down?

Step-by-Step Solution

Answer

The change in depth is 275 feet (since 725 - 450 = 275). Since this is a positive number, it indicates the submarine ascended during this timeframe.

Step 1: Calculating the Change in Depth

We know that the initial depth of the submarine is 725 feet and the final depth is 450 feet. In order to find the change, we can subtract the final depth from the initial depth such that: Change in Depth = Initial Depth - Final Depth = 725 - 450

Step 2: Interpreting the Result

If the result is a positive number, it indicates that the submarine ascended, if it's a negative number it indicates the submarine descended.

Key Concepts

Subtraction

Subtraction in algebra is the process used to calculate the difference between two numbers or expressions. It is a fundamental operation in mathematics, often represented by the minus sign (-). In the case of our submarine problem, subtraction is used to find out how much the depth has changed. Subtraction requires identifying which number to subtract from which to ensure accurate results.

To figure out the change in depth for the submarine, the formula is straightforward: Initial depth minus Final depth. In mathematical terms, for our problem, this is expressed as: \[725 - 450 \]This step-by-step calculation will lead us to the change in depth.

• Initial Depth: The starting point of your measurement (725 feet in this case).
• Final Depth: The ending point of your measurement (450 feet in this example).
• Change in Depth: The result of subtracting the final depth from the initial depth.

Once you conduct the subtraction, the result gives a clear picture of how much the submarine moved: whether it went further down or came up towards the surface.

Estimation

Estimation is a mathematical technique used to get an approximate answer or prediction without needing a precise calculation. It helps to quickly check whether the detailed calculation might be reasonable. This is useful when you need a quick sanity check or when the exact value is not necessary.

For the submarine problem, you can use estimation to check your work by rounding the numbers involved. Instead of dealing precisely with 725 and 450, consider simplifying these figures for a faster calculation:

• Round 725 to 700
• Round 450 to 500

Now, perform the subtraction with these rounded numbers: \[700 - 500 = 200 \]This estimated result is close to the precise calculation and serves as a quick check that our answer makes sense. Estimation doesn’t replace an exact answer but confirms that your subtraction is in the right ballpark.

Interpretation of Results

After performing subtraction and possibly a quick estimation, interpreting the results is crucial. Understanding what your calculated or estimated number signifies in the context of the problem is the final task.

For the submarine, interpreting a result of \[725 - 450 = 275 \]helps us know the kind of movement involved. If the change in depth is positive, it means the submarine has ascended, moving closer to the sea level from a deeper point.
Such interpretation is essential because it connects mathematical operations to real-world phenomena. It aids in answering qualitative questions like "Did it go up or down?" In our problem, since 275 is a positive number, we conclude that the submarine ascended by 275 feet.
Being able to interpret results within the context allows you to communicate findings clearly and make sense of mathematical answers in practical scenarios.