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Chapter 1: Problem 46

A unit of area often used in measuring land areas is the hectare, defined as \(10^{4} \mathrm{~m}^{2}\). An open-pit coal mine consumes 75 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

Expert verified
0.0195 cubic kilometers of earth is removed each year.

Step by step solution

01

Define Variables and Units

First, let's define the units and variables we will use. We know 1 hectare equals \(10,000 \ m^2\). We need to calculate the volume of the earth in cubic kilometers. Given that 75 hectares are used, and the depth is 26 meters per year, we need to find the total volume in \(km^3\).
02

Calculate Total Area in Square Meters

To find the total area in square meters, multiply the number of hectares by the area of one hectare. This is: \(75 \text{ hectares} \times 10^4 \ m^2/\text{hectare} = 750,000 \ m^2\).
03

Calculate the Volume in Cubic Meters

Now, compute the volume of the earth removed by multiplying the total area obtained in Step 2 by the depth of the mine. So, \(750,000 \ m^2 \times 26 \ m = 19,500,000 \ m^3\).
04

Convert Cubic Meters to Cubic Kilometers

Finally, to convert the volume from cubic meters to cubic kilometers, recall that \(1 km^3 = 10^9 m^3\). Thus, divide the volume in cubic meters by \(10^9\): \(19,500,000 \ m^3 \div 10^9 = 0.0195 \ km^3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion
Understanding unit conversion is crucial when dealing with measurements in different scales. In this exercise, we need to convert various units to calculate the volume of earth removed in the coal mine.
To start, we convert hectares to square meters since the measurement in hectares must be translated to a compatible unit for further calculations.
  • 1 hectare is equivalent to \(10,000 \, m^2\).
  • Therefore, 75 hectares translate to \(75 \times 10,000 = 750,000 \, m^2\).
After calculating the total area in square meters, we compute the volume in cubic meters and subsequently convert it to cubic kilometers because our final answer needs to be in \(km^3\). Remember,
  • 1 cubic kilometer equals \(10^9 \, m^3\).
  • Thus, dividing the volume in cubic meters by \(10^9\) allows us to express the volume in kilometers cubed.
These multiple conversions ensure our calculations are consistent and accurate to provide results in the desired measurement units.
Area Measurement
The concept of area measurement is foundational in determining how expansive a space is within a two-dimensional plane, like a surface. Here, we began with an area measured in hectares, where converting to square meters was essential for further calculations.
To understand area measurement, let's dive into the essentials:
  • 1 hectare is defined as \(10,000 \, m^2\), often representing a size for large pieces of land, such as farms or mines.
  • When you know the measurement of an area in hectares, you multiply by \(10,000 \, m^2\) to assess its size in square meters.
These measurements allow significant calculations like finding how much land is utilized or affected in large operations.
Cubic Measurement
Once we have the area, cubic measurement comes into play when determining the volume. Volume measurement combines area with depth, giving us a three-dimensional space figure. Here, after calculating the area, the depth of the pit lets us find the total volume in cubic meters before converting to cubic kilometers.
Here is how we calculate this:
  • The total area in square meters (\(750,000 \, m^2\)) is multiplied by the depth (26 meters), resulting in volume in cubic meters.
  • This yields \(19,500,000 \, m^3\), the total volume excavated each year.
Ensuing this, we convert to cubic kilometers, a necessary conversion for comprehending quantities on a larger geographical scale. Volume in cubic kilometers offers a clearer picture of earth materials removed or consumed, which is particularly relevant in industries such as mining or construction.

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Most popular questions from this chapter

Suppose that, while lying on a beach near the equator watching the Sun set over a calm ocean, you start a stopwatch just as the top of the Sun disappears. You then stand, elevating your eyes by a height \(H=1.70 \mathrm{~m}\), and stop the watch when the top of the Sun again disappears. If the elapsed time is \(t=11.1 \mathrm{~s}\), what is the radius \(r\) of Earth?

A standard interior staircase has steps each with a rise (height) of \(19 \mathrm{~cm}\) and a run (horizontal depth) of \(23 \mathrm{~cm}\). Research suggests that the stairs would be safer for descent if the run were, instead, \(28 \mathrm{~cm}\). For a particular staircase of total height \(4.57 \mathrm{~m}\), how much farther into the room would the staircase extend if this change in run were made?

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