Problem 96 - Introductory and Intermediate Algebra for College Students 4th

Question

The peak of Mount Whitney is \(14,494\) feet above sea level. Mount Whitney can be seen directly above Death Valley, which is 282 feet below sea level. What is the difference in elevation between these geographic locations?

Step-by-Step Solution

Answer

The difference in elevation is \(14,776\) feet.

Step 1: Define the Heights

Mount Whitney is \(14,494\) feet above sea level and Death Valley is \(282\) feet below sea level. These can be defined as \(+14,494\) and \(-282\) respectively.

Step 2: Calculate the Difference

To find the difference in elevation, these two amounts need to be added together because one is described as a negative value. So, \(+14,494\) + \(-282\) = \(14,776\) feet.

Key Concepts

Elevation Difference

When trying to determine the elevation difference between two locations, you have to consider their heights relative to sea level: one might be high above, while the other could be below it.

This was the case in the exercise, where the peak of Mount Whitney stands at an impressive 14,494 feet above sea level, while Death Valley takes a dip below at 282 feet below sea level.

The difference in elevation is a straightforward concept. It's essentially about measuring how much higher or lower one point is compared to another.

To find this elevation difference, you use a combination of addition and subtraction operations, depending on whether you're dealing with heights above or below sea level. The key is consistency in defining your zero point, which is the sea level.

So, if you want to calculate the difference, imagine your two points on a vertical number line, with sea level being zero.

• Mount Whitney: +14,494 feet
• Death Valley: -282 feet

Adding these values gives you a total elevation difference of 14,776 feet.

Integers

Integers are the set of whole numbers that include positive numbers, negative numbers, and zero. They do not include fractions, decimals, or mixed numbers.

This makes integers perfectly suited for representing simple concepts like elevation, especially when distinguishing between heights above and below sea level.

In the exercise we tackled, Mount Whitney was described as +14,494 feet, which is a positive integer, reflecting its elevation above sea level. Meanwhile, Death Valley was represented by a negative integer, -282 feet, because it is below sea level.

Working with integers allows us to easily visualize and calculate differences in these elevations with straightforward mathematical operations. These integer representations give clarity and precision needed to determine aspects of geography, like elevation difference.

Moreover, understanding how to manipulate integers lays a solid foundation for more advanced topics in algebra and mathematics as a whole.

Addition of Integers

Adding integers, whether positive or negative, is a fundamental concept in mathematics, and it comes in handy when measuring elevation differences, among other things.

To find the elevation difference between Mount Whitney and Death Valley, we consider one value as positive and one as negative.

So, you take the two integer values:

• Mount Whitney's elevation: +14,494
• Death Valley's elevation: -282

To calculate the difference, add these two integers: \[+14,494 + (-282)\]

This involves simple addition rules: if you add a positive and a negative integer, you essentially subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value.

Thus, the result is \[14,494 - 282 = 14,776\]

This simple addition operation helps us find that the difference in elevation between these two geographical features is 14,776 feet.