Question

A research submarine is attached to a crane on a ship The submarine is currently at a depth of 147 feet. If the crane raises the submarine 69 feet, what is the elevation of the submarine?

Short answer

Answer: -78 feet

Step-by-step solution

Understanding the problem and given information

Initially, the submarine is at a depth of 147 feet below sea level. The crane raises the submarine 69 feet, so we want to find the final elevation of the submarine. Note that the elevation will be negative since the submarine is still below sea level.

Calculate the final depth

To find out the remaining depth of the submarine after being raised by 69 feet, subtract the distance raised from the initial depth: Final depth = Initial depth - Distance raised Final depth = 147 - 69

Calculate the final depth

Now, compute the final depth by subtracting the values: Final depth = 147 - 69 Final depth = 78 feet

Calculate the elevation

The elevation is the negative of the depth since the submarine is still below sea level: Elevation = -Final depth Elevation = -78 feet

State the result

The elevation of the submarine after being raised 69 feet by the crane is -78 feet.

Key concepts

Subtraction

Subtraction is the mathematical process of taking away one number from another. It is a fundamental operation in arithmetic that you use in everyday life, often without even realizing it.
In this exercise, we see subtraction in action when calculating the new position of the submarine. When the submarine moves vertically, subtraction helps us determine how much it ascends from a certain point. For instance:

• The initial depth of the submarine is 147 feet below sea level.
• We subtract 69 feet, the distance it was raised, to find how deep it still is.

Thus, subtraction comes into play as: \(147 - 69 = 78\). This shows that the submarine's new depth is 78 feet below sea level.

Negative Numbers

Negative numbers are values less than zero, often used to represent loss, decrease, or below a starting point. In many real-world scenarios, especially dealing with depths or elevations, these numbers become very handy.
For example, when discussing the submarine's elevation in this problem, a depth below sea level is represented as a negative number. Once we know the depth, the elevation becomes negative as well:

• Since the submarine's depth remains 78 feet below sea level, its elevation is \(-78\) feet.

Understanding negative numbers is crucial for situations involving below-sea-level calculations or bank overdrafts. They help us express values that go beyond zero in the opposite direction.

Depth Calculation

Depth calculation involves determining how deep or high something is based on its position relative to a reference point, often sea level or ground level. This involves combining basic arithmetic operations and understanding of directions, like up or down.
In the case of the submarine:

• The starting depth is 147 feet below sea level.
• After the crane raises it by 69 feet, we calculate the remaining depth.

This calculation is important to understand whether the object approaches or moves away from the reference point. After subtracting, we know the submarine is still 78 feet deep.

Problem-Solving Steps

Problem-solving is crucial in mathematics and everyday life. Breaking down a problem into manageable pieces makes it easier to tackle. Here's a simple structured approach applied in this exercise:

• Understanding the Problem: Know what you are being asked. Here, find the submarine's new elevation after being raised.
• Gathering Given Information: Identify key details like initial depth (147 feet) and the raised distance (69 feet).
• Performing Calculations: Subtract to find new depth and convert to elevation.
• Stating the Conclusion: Present the final answer clearly, which in this case is that the submarine's elevation is -78 feet.

Following these steps helps ensure no part of the problem is overlooked and helps in arriving at a precise solution.