Problem 59 - Beginning and Intermediate Algebra

Question

Solve. See Example 8 . The deepest canyon in the world is the Great Canyon of the Yarlung Tsangpo in Tibet. The bottom of the canyon is 17,657 feet below the surrounding terrain, called the rim. If you are standing 1230 feet above the bottom of the canyon, how far from the rim are you?

Step-by-Step Solution

Answer

16,427 feet.

Step 1: Understand the Problem

You are standing 1230 feet above the bottom of the canyon. The canyon's bottom is 17,657 feet below the rim. You need to determine how far from the rim you are.

Step 2: Set Up the Equation

To find out how far you are from the rim, start by considering the position of the canyon's bottom as a point of reference at 0 feet, and thus the depth of the canyon is -17,657 feet. If you are 1230 feet above this bottom point, you can express this as: Position = -17,657 + 1230.

Step 3: Perform the Calculation

Now, compute: Position = -17,657 + 1230 = -16,427 feet. This means you are 16,427 feet below the rim.

Step 4: Express the Final Answer

Since the question asks for the distance between you and the rim, we take the absolute value of the position calculated. Distance = |-16,427| = 16,427 feet.

Step 5: Verify the Accuracy

Re-evaluate the situation and re-calculate if necessary to ensure the solution is correct: 1. Bottom of the canyon = -17,657 feet. 2. Your position = -17,657 + 1,230. Both lead to the conclusion that the distance to the rim is 16,427 feet.

Key Concepts

Depth Calculations

When tackling problems involving depth calculations, it's crucial to understand the context and the relative positioning involved. In this exercise, we have a canyon with a known depth from the rim to the bottom. The depth is given as 17,657 feet below the rim. This negative value signifies that the bottom of the canyon is beneath the surface level or the rim.

When you stand at a certain height above this point, say 1,230 feet, you measure your location making sure not to confuse it with absolute heights. Your height above the canyon then becomes a reference point to help calculate placement in relation to the rim.

Thus, solving for depth in this context is similar to calculating a difference in height, as you'll end up using subtraction to find how deep you remain below the rim.

Absolute Value

The absolute value concept is fundamental in ensuring that the final answer to any problem, especially physical distances, remains positive. In an algebraic problem-solving context, especially involving heights or depths, values can often be negative. This was the case when determining how far you were from the canyon rim.

The calculated position was originally -16,427 feet, indicating a position below the starting reference — the rim. However, since distance cannot be negative in real-world scenarios, we use the absolute value to transform this negative number into a positive one.

This gives us the absolute distance, making it straightforward to interpret. Here, |−16,427| becomes 16,427 feet, correctly displaying the distance in positive terms.

Equation Setup

Equation setup in algebraic problem-solving involves identifying the right mathematical expressions to describe the situation. Establishing the correct equation ensures the solution is based on a solid understanding of the problem.

In this case, the depth of the canyon is the critical factor known and serves as our reference. By setting the bottom of the canyon at 0, we can show positions relative to this point. The position you occupy is expressed in terms of addition or subtraction relative to this baseline figure of -17,657 feet.

Such an equation setup (Position = -17,657 + 1,230) ensures calculations proceed logically, accurately mapping the elevation difference and making the resulting position clear: how far below the rim you are. The effectiveness of this equation setup relies on grasping both the situational context and the role of mathematical operations in bridging known and unknown quantities.