Problem 77 - Algebra A Combined Function

Question

Solve. Mauna Kea in Hawaii has an elevation of 13,796 feet above sea level. The Mid- America Trench in the Pacific Ocean has an elevation of 21,857 feet below sea level. Find the difference in elevation between those two points. (Source: National Geographic Society and Defense Mapping Agency)

Step-by-Step Solution

Answer

The difference in elevation is 35,653 feet.

Step 1: Identify Elevations

First, let's identify the given elevations from the problem. Mauna Kea has an elevation of 13,796 feet above sea level, and the Mid-America Trench has an elevation of 21,857 feet below sea level.

Step 2: Represent Elevations Mathematically

Convert the elevations into mathematical expressions. The elevation of Mauna Kea is "+13,796" feet because it is above sea level. The elevation of the Mid-America Trench is "-21,857" feet because it is below sea level.

Step 3: Calculate the Difference

To find the difference in elevation, subtract the elevation of the Mid-America Trench from the elevation of Mauna Kea. The calculation is as follows: \[ 13,796 - (-21,857) \]

Step 4: Apply the Double Negative Rule

Applying the rule of subtracting a negative number, which is the same as adding the absolute value, we get: \[ 13,796 + 21,857 \]

Step 5: Perform the Addition

Calculate the sum: \[ 13,796 + 21,857 = 35,653 \]

Step 6: State the Result

The difference in elevation between Mauna Kea and the Mid-America Trench is 35,653 feet.

Key Concepts

Understanding Elevation

Elevation refers to the height or depth of a point in relation to sea level. When we talk about something being "above sea level," we mean it is higher, like a mountain peak. Conversely, "below sea level" indicates a depression, such as an oceanic trench. For example, Mauna Kea in Hawaii is 13,796 feet above sea level. Meanwhile, the Mid-America Trench is 21,857 feet below sea level. In everyday terms, elevation gives us a way to describe how high or low a location is by using sea level as a reference point. This measurement is crucial in geography, meteorology, and various scientific fields to understand the Earth's surface features.

The Role of Subtraction in Calculating Differences

Subtraction is the mathematical operation used to find the difference between two numbers. In our example, we want to know how much higher Mauna Kea is than the Mid-America Trench. To find this difference, we subtract the elevation of the trench from that of the mountain. Since the Trench's elevation is negative, we perform the operation \[ 13,796 - (-21,857) \]. Here, subtraction becomes a tool to measure the gap between different elevations, whether involving positive or negative heights. Subtraction of negative numbers can often seem tricky, but it actually simplifies the result by converting the subtraction into an addition operation.

The Importance of Absolute Value

Absolute value plays a key role when dealing with negative numbers. It represents the distance a number is from zero on a number line, without considering whether the number is positive or negative. When you subtract a negative number, such as in our problem, the act is the same as adding its absolute value. Therefore, \[ 13,796 - (-21,857) \] turns into \[ 13,796 + 21,857 \]. By using absolute value, we ensure that the distance between the two elevations is correctly calculated. Since it essentially makes all numbers positive, absolute value provides a straightforward way to assess such differences accurately. Understanding this concept helps demystify why subtracting negatives results in addition.