Problem 67 - Elementary and Intermediate Algebra

Question

The tallest mountain in the world, as measured from base to peak, is Mauna Kea in Hawaii. From a base \(19,684\) ft below sea level, it rises \(33,480\) ft. What is the elevation of its peak? Source: The Guinness Book of Records

Step-by-Step Solution

Answer

The elevation of Mauna Kea's peak is 13,796 ft.

Step 1 - Identify the given values

Mauna Kea rises from a base that is located 19,684 ft below sea level and extends upwards by 33,480 ft. These values are given in the problem.

Step 2 - Understanding the elevation

Elevation is measured from sea level. This means we need to account for the entire height starting from below sea level and adding the height above sea level.

Step 3 - Calculate the elevation

The elevation of the peak can be found by adding the height below sea level to the height it rises. Therefore, the elevation is: \[ 33,480 \text{ ft} - 19,684 \text{ ft} \]

Step 4 - Perform the addition

Adding these values gives: \[ 33,480 \text{ ft} - 19,684 \text{ ft} = 13,796 \text{ ft} \]

Key Concepts

height measurement

Height measurement is a fundamental concept in various fields, including geography and construction. It refers to the process of determining how tall an object is, from its base to its peak. The height of an object can be measured from different reference points, such as below sea level, at sea level, or above sea level. In the context of Mauna Kea, its height is measured from its base, which is located below sea level, all the way to its peak. This comprehensive measurement gives us an accurate sense of the mountain's total vertical extent.

base to peak

When measuring the height of a mountain, we often refer to 'base to peak' height. This indicates the total vertical extension of the mountain from its very bottom, or base, to its highest point, or peak. For Mauna Kea, the base is located underwater, 19,684 feet below sea level. The peak, on the other hand, rises to 33,480 feet above the base. Understanding this concept helps us appreciate the true size of natural formations. By using 'base to peak' measurements, we recognize both the visible part of the mountain and the portion that lies below the sea surface.

subtraction

Subtraction is a basic arithmetic operation that we use to calculate differences between numbers. In this case, to determine the elevation of Mauna Kea’s peak, we subtract the depth of the base from the height it rises. The calculation is straightforward:

• Start with the height Mauna Kea rises above its base:
  • 33,480 feet
• Subtract the depth of the underwater base:
  • 19,684 feet
• The resulting elevation is:
  • 33,480 feet - 19,684 feet = 13,796 feet

This means the elevation of Mauna Kea's peak, calculated from sea level, is 13,796 feet. Subtraction allows us to determine differences, making it a critical skill in math and science.