- The initial position is identified as \(-750\) due to being below sea level.
- The total change in position is calculated by multiplying the rate of rise (50 meters/hour) by the time (4 hours), giving \(4 \times 50\).
- Adding the change to the initial position results in the expression \(-750 + 4 \times 50\).
- Matching this with the provided choices confirms that option c is the correct representation. Final answer: \(\boxed{c}\).
Explanation
- Analyzing the Problem and Defining Initial State
The problem asks us to determine the expression that represents the final position of a submarine relative to sea level. We define sea level as 0 meters. Positions below sea level are represented by negative integers. The initial depth of the submarine is given as 750 meters below sea level, so the initial position is \(-750\). Over a 4-hour period, the submarine rises at a rate of 50 meters per hour. Since the submarine is rising (moving toward sea level), this movement represents a positive change in the submarine's position.
- Calculating the Change in Position and Final Expression
To find the total change in position, we multiply the rate of change by the total time. The rate of change is 50 meters per hour, and the duration is 4 hours. Thus, the total upward displacement is \(\Delta P = 50 \times 4\). The submarine's final position relative to sea level, denoted as \(P_f\), is the sum of its initial position and the total change:
\[P_f = \text{Initial Position} + \text{Total Change}\]
Substituting the values we have, the final position expression becomes:
\[P_f = -750 + (50 \times 4)\]
This is equivalent to \(-750 + 4 \times 50\).
- Conclusion and Selection of the Correct Option
We now compare our derived expression, \(-750 + 4 \times 50\), with the given options:
A\(-750 - 4 \times 50\)
B\((-750 + 4) \times 50\)
C\(-750 + 4 \times 50\)
D\(750 + 4 \times (-50)\)
Comparing these, we see that option (c) matches our expression perfectly. Therefore, the correct expression representing the new position of the submarine is \(-750 + 4 \times 50\). The result is \(\boxed{c}\).
Examples
Understanding how to represent vertical changes using positive and negative integers is essential for navigation in aviation and marine science. For example, if an aircraft is cruising at an altitude of 30,000 feet and needs to descend at a rate of 1,000 feet per minute for 5 minutes, the pilot can calculate the final altitude as \(30,000 - (1,000 \times 5) = 25,000\) feet. Mastering these calculations allows for accurate planning of maneuvers and safe operations in three-dimensional spaces.