Problem 121 - Beginning and Intermediate Algebra

Question

Solve each problem. The surface, or rim, of a canyon is at altitude \(0 .\) On a hike down into the canyon, a party of hikers stops for a rest at \(130 \mathrm{~m}\) below the surface. The hikers then descend another \(54 \mathrm{~m}\). Write the new altitude as a signed number.

Step-by-Step Solution

Answer

-184 meters

Step 1 - Identify the initial altitude

The initial altitude is given as the surface of the canyon, which is at altitude 0 meters.

Step 2 - Determine the altitude after the first descent

The hikers first stop at 130 meters below the surface. Represent this as -130 meters.

Step 3 - Calculate the new altitude after the second descent

After resting, the hikers descend another 54 meters from their current altitude. Represent this descent as an additional 54 meters below -130 meters: Initial altitude: -130 Descent: -54 New altitude: -130 + (-54) = -184 meters

Step 4 - Write the new altitude as a signed number

The new altitude after both descents is -184 meters. This signed number indicates the hikers are 184 meters below the canyon's surface.

Key Concepts

Signed Numbers

Signed numbers are numbers that have a '+' or '-' sign in front of them to indicate whether they are positive or negative.
This concept is crucial in various mathematical operations, including algebra and real-life scenarios.
A number without a sign is assumed to be positive by default.
Common examples include:

• +5 for positive five
• -3 for negative three

Signed numbers are essential to represent gains and losses, elevations, and temperatures. For instance, in our exercise, the altitude below the canyon's rim is expressed using negative numbers since it is below a reference point of zero meters.
In mathematical operations:

• Positive numbers increase value (e.g., gaining height above the surface)
• Negative numbers decrease value (e.g., descending below the surface)

Adding Integers

Adding integers involves combining positive and negative values to reach a sum.
When adding integers, it's essential to understand how positive and negative signs interact. Here are some guidelines:

• When adding two positive integers, the sum remains positive (e.g., 5 + 3 = 8)
• When adding two negative integers, the sum also remains negative (e.g., -4 + (-6) = -10)
• When adding a positive integer and a negative integer, the sum depends on their absolute values. For instance, adding a larger positive integer and a smaller negative integer will result in a positive sum, and vice versa (e.g., 7 + (-2) = 5, and -7 + 2 = -5)

In the exercise, the hikers' descent involves adding negative integers:
The initial altitude was -130 meters (first descent), and descending further 54 meters adds another negative number: -130 + (-54) = -184.

This results in the hikers being 184 meters below the surface.

Altitude Calculation

Altitude calculation refers to determining the height or depth of a point relative to sea level or another reference point, such as the surface of a canyon.
In our example, the surface (or rim) of the canyon is our reference point at 0 meters.
To compute altitude changes correctly, we must pay attention to signed numbers as they indicate direction:

• Positive altitudes indicate heights above the reference point.
• Negative altitudes indicate depths below the reference point.

For the exercise, the hikers' descent involves two steps:

• First, they descend 130 meters below the surface, represented as -130 meters.
• Then, they descend another 54 meters from that position, represented as -54 meters.

Combining these changes gives us:
Initial altitude: -130 meters
Further descent: -54 meters
New altitude = -130 + (-54) = -184 meters.
This means the hikers are now 184 meters below the canyon's surface, which we express as a signed number: -184 meters.