- Identify positions relative to the lobby: Julia is at \(+40\) and Brady is at \(-24\).
- Use the distance formula between two points on a line: \(\text{Distance} = |a - b|\).
- Substitute the values: \(|40 - (-24)|\).
- Evaluate the expression: \(|40 + 24| = 64\). The correct option is D.
- Final answer: \(\boxed{D}\)
Explanation
- Analyzing the Positions of Julia and Brady
To determine the distance between two positions on a vertical coordinate system (like elevator floors), we use the concept of a reference point. Let the hotel lobby be the origin, marked at \(0\). Positions above the lobby are represented by positive numbers, and positions below the lobby are represented by negative numbers.
Julia's position is \(40\) feet above the lobby, so we represent her location as \(+40\).
Brady's position is \(24\) feet below the lobby, so we represent his location as \(-24\).
- Calculating the Distance
The distance between two points \(a\) and \(b\) on a one-dimensional coordinate line is given by the absolute value of the difference between their coordinates: \(\text{Distance} = |a - b|\).
In this scenario, we have \(a = 40\) and \(b = -24\). Substituting these values into the distance formula:
\(\text{Distance} = |40 - (-24)|\)
Subtracting a negative number is equivalent to adding its positive counterpart, so this simplifies to:
\(\text{Distance} = |40 + 24| = |64| = 64\).
Thus, the expression that calculates the distance between them is \(|40 - (-24)|\).
- Conclusion and Matching the Options
Comparing our result to the given options:
A \(-|40|-|24|=-64\)
B \(-|40|+|24|=-16\)
C \(|40-24|=16\)
D \(|40-(-24)|=64\)
The expression that matches our derived formula is D. Therefore, the distance between Julia and Brady is \(64\) feet, and the correct expression is \(|40 - (-24)|\).
Examples
Calculating distance between vertical positions is a fundamental skill in many fields, such as architecture and engineering. For example, if an architect needs to calculate the total length of an elevator shaft that spans floors above and below a ground level, they must add the absolute distances from the ground floor. This ensures that all measurements are positive and accurate, just like finding the total distance between Julia and Brady.