- Laurie's descent is defined by 5-foot increments (\(-5\) feet per increment).
- An instrument check happens every 20 feet, which is equivalent to 4 increments of 5 feet (\(4 \times (-5)\)).
- After 3 checks, the total depth is \(3 \times 4 \times (-5)\).
- This simplifies to \((4)(-5)(3) = -60\).
- Final answer: \(\boxed{(4)(-5)(3)}\).
Explanation
- Analyzing the Depth Components
To determine Laurie's maximum depth, we break down her descent into smaller, quantifiable components. Laurie dives deeper in increments of 5 feet, and since descending means moving away from the surface, we represent this as a depth change of \(-5\) feet. She checks her instruments every time her depth totals 20 feet. Since 20 feet is comprised of four 5-foot increments, the total depth change for one instrument check is 4 increments of -5 feet, which is \(4 \times (-5) = -20\) feet. Finally, Laurie checks her instruments 3 times, meaning she performs this 20-foot descent three separate times. Therefore, her total depth is the number of checks multiplied by the depth change per check: \(3 \times (-20)\). Substituting \(-20\) with \(4 \times (-5)\), we get the expression \(3 \times 4 \times (-5)\), which can also be written as \((4)(-5)(3)\).
- Selecting the Correct Expression
Evaluating the given options, we can see if they result in the total depth of \(-60\) feet:
- \((4)(-5)(3) = 4 \times (-5) \times 3 = -20 \times 3 = -60\).
- \((-4)(-5)(-3) = -4 \times (-5) \times (-3) = 20 \times (-3) = -60\).
- \((-5)(20)(3) = -5 \times 20 \times 3 = -300\).
- \((-5)(-20)(-3) = -5 \times (-20) \times (-3) = -300\).
Both \((4)(-5)(3)\) and \((-4)(-5)(-3)\) yield -60. However, following the derivation based on the number of increments per check (4), the increment depth (-5), and the number of checks (3), the expression \((4)(-5)(3)\) most directly represents the logic of the problem structure: \((4 \text{ increments}) \times (-5 \text{ feet/increment}) \times (3 \text{ checks})\).
Examples
This problem illustrates how complex physical movements can be modeled using simple arithmetic multiplication. In fields like oceanography, navigation, or construction engineering, it is often necessary to track total progress or displacement by breaking down movements into recurring, manageable intervals. Understanding how signs and factors represent physical changes—like depth, altitude, or budget tracking—is essential for accurately modeling real-world processes.