Question

A unit of area, often used in expressing areas of land, is the hectare, defined as \(10^{4} \mathrm{~m}^{2}\). An open-pit coal mine consumes 77 hectares of land, down to a depth of \(26 \mathrm{~m}\), each year. What volume of earth, in cubic kilometers, is removed in this time?

Short answer

The volume of earth removed each year is 0.02002 cubic kilometers

Step-by-step solution

Convert area from hectares to square meters

Given that 1 hectare equals \(10^{4} m^{2}\), the total area is \(77 hectares * 10^{4} m^{2}/hectare = 770000 m^{2}\)

Calculate volume in cubic meters

The volume in cubic meters can be calculated by multiplying the area by the depth. Volume = area * depth = \(770000 m^{2} * 26 m = 20020000 m^{3}\).

Convert volume from cubic meters to cubic kilometers

Since 1 cubic kilometer equals \(10^{9} m^{3}\), we divided the volume in cubic meters by \(10^{9} m^{3}/km^{3}\) to get the volume in cubic kilometers. Volume = \(20020000 m^{3}/10^{9} m^{3}/km^{3} = 0.02002 km^{3}\)

Key concepts

Area Conversion

When you hear about converting area measurements, it's usually about changing units from one system to another, like hectares to square meters. This is a crucial step when you need detailed measurements for calculations.
For instance, 1 hectare is a popular unit of area that's used quite often in agriculture and land measurement. It's equivalent to 10,000 square meters. When you're told that your plot of land is "x" hectares, you are essentially saying it's "x times 10,000" in square meters.
This conversion helps simplify more complex operations, like calculating volume, which depends on the area of the surface you're working with. So, if you have 77 hectares, as in our exercise, you'll multiply by 10,000 to find out what that is in square meters, which is particularly useful when you're calculating the volume of earth displaced during mining or construction.

Unit Conversion

Unit conversion is like translating a sentence into a different language; only the language is numbers. It allows us to comprehend information in units we are familiar with or that are more practical for our purposes, such as converting volume from cubic meters to cubic kilometers.
This task can seem complicated, but it often involves straightforward multiplication or division. For our exercise, converting the volume from cubic meters to cubic kilometers involves dividing by 1,000,000,000 since 1 cubic kilometer contains 1 billion cubic meters. This helps you understand the magnitude of the number in a more comprehensible scale.
Being adept at unit conversion is especially critical in large-scale operations, like mining, where dealing with a multitude of units could lead to errors if not handled carefully. Practicing with smaller scale conversions builds the skill sets necessary for interpreting and manipulating data accurately.

Volume of Earth

Calculating the volume of earth removed, as we explored in the given exercise, involves understanding area and depth. Volume calculation becomes an essential task here to gauge the scale of operations like mining or excavation.
We calculate the volume by multiplying the converted area (in square meters) by depth (in meters). This provides the volume in cubic meters. Since operations over large areas, like the open-pit mine example, result in enormous volumes, these are typically represented in cubic kilometers for convenience. For instance, 20,020,000 cubic meters, once converted, equals about 0.02002 cubic kilometers.
Understanding these large numbers is crucial, as they can influence decisions about logistics, environmental impact assessments, and resource management strategies in industries that manipulate large volumes of earth.