Problem 83 - Prealgebra

Question

The Bright Angel trail at Grand Canyon National Park ends at Indian Garden, \(3,060\) feet below the trailhead. Write this as a negative number with respect to the trailhead.

Step-by-Step Solution

Answer

The location of Indian Garden is -3,060 feet relative to the trailhead.

Step 1: Understanding the Sentence

The problem states that Indian Garden is 3,060 feet below the trailhead. To represent this with a negative number, recognize that being 'below' is equivalent to having a negative sign associated with the height or depth.

Step 2: Assigning the Correct Sign

Since Indian Garden is below the starting point (the trailhead), assign a negative sign to the number: Indian Garden = -3,060 feet.

Step 3: Writing the Result

Now, write the result as the problem asks for the elevation of Indian Garden relative to the trailhead using a negative number. Therefore, the position of Indian Garden can be expressed as: -3,060 feet.

Key Concepts

Integer Representation

When we talk about integers, we refer to the family of whole numbers, both positive and negative, including zero. Integers are used widely in mathematics and everyday life to represent values without fractions or decimals. Let's think of them as stepping stones for understanding more complex mathematical concepts. Integers can include values like \(1, 2, 3\) for positive numbers, and \(-1, -2, -3\) for negative numbers. Zero is also included as a neutral integer, neither positive nor negative.
In the context of problems involving elevation or depth, we use integers to convey whether something is at a higher position (positive integers) or a lower position (negative integers) compared to a reference point. In the case of the Bright Angel Trail, using a negative integer to describe the height of Indian Garden below the trailhead helps us easily understand its position relative to where the trail begins.

Elevation Difference

The concept of elevation difference comes into play when describing heights in relation to a specific reference point, such as sea level or, in the case of our exercise, a trailhead.
If you're standing at the top of a hill, and your friend is at the bottom, their relative position to you can be described as "below" your position, which is a negative difference if considering upwards as positive.
In the exercise about the Bright Angel Trail, the difference in elevation between the trailhead and Indian Garden is 3,060 feet, but since Indian Garden is below the trailhead, this difference is expressed as a negative integer: \(-3,060\) feet. This negative sign signifies descending or being below the reference starting point in elevation, showing the difference in a very clear way.

Understanding Negative Values

Negative numbers can sometimes feel confusing, especially when thinking about real-world uses. Understanding what negative values mean and how they function is important. Generally, negatives denote reversal or opposition. For depth and elevation, they signify positions 'below' a set reference.
Using negative values allows us to represent these below-reference scenarios clearly and effectively. For example, when you are below sea level, you can think of it as your position being negative in respect to sea level. It's like saying you're at a position 'in the negative', or down from where you started.
Even though negative values might initially seem unnatural when representing physical spaces or things, remember that it is just a useful convention. Negative numbers help organize and communicate positions and differences in a straightforward, standardized way that lets everyone understand the concept similarly.