Question

Antarctica is roughly semicircular in shape with a radius of \(2000 \mathrm{~km}\). The average thickness of the ice cover is \(3000 \mathrm{~m}\). How many cubic centimeters of ice does Antarctica contain? (Ignore the curvature of the Earth.)

Short answer

The volume of ice that Antarctica contains is \( V = \frac{2}{3} * \pi * (300000)^3 \) cubic centimeters. Calculate the expression to get the final numerical value.

Step-by-step solution

Data Conversion

The radius r is given in kilometers and needs to be converted into centimeters. We know that 1 km = 100000 cm. So, \( r = 2000 * 100000 = 200000000 \) cm. Similarly, convert the thickness from meters to centimeters. Therefore, \( thickness = 3000 * 100 = 300000 \) cm. To find the volume of ice, the thickness will be used as the radius.

Calculation of Volume

Now that we have the radius, we can use the volume formula to calculate the volume of the hemisphere. Using the volume formula \( V = \frac{2}{3} * \pi * r^3 \), where r = 300000 cm, we get \( V = \frac{2}{3} * \pi * (300000)^3 \).

Simplifying the Result

Solving the expression will give us the volume of the ice in cubic centimeters.

Key concepts

Antarctica ice cover

Antarctica, the southernmost continent, is known for its vast ice cover. This massive ice sheet is crucial for our planet's climate regulation and sea level stability. The ice cover's thickness and area make it a significant focus in environmental studies. In calculations like these, we approximate the shape of Antarctica to be semicircular, and for this exercise, the radius is given as 2000 kilometers. This simplification helps in mathematical modeling and visualization. In the real world, the actual shape and distribution of ice are more complex due to geographical features and environmental influences.

Cubic centimeters

Cubic centimeters (cm³) are a standard unit of volume used in the metric system. They are particularly useful for calculating volumes involving solid objects, offering a great deal of precision. Understanding this unit is crucial for geographical and scientific computations, especially when dealing with massive volumes like that of Antarctica's ice. Since the Earth's dimensions and the components under study are large, the conversion from kilometers or meters to centimeters allows us to appreciate the significance in more tangible terms. For comparative ease, think of cubic centimeters like the volume of a small cube with each side measuring 1 centimeter.

Data conversion

Data conversion is the process of changing measurements from one unit to another, necessary for accurate calculation and modeling. In the case of Antarctica's ice, both the radius and thickness were converted from kilometers and meters, respectively, to centimeters before using them in the volume calculation. The conversion factors, 1 km = 100,000 cm and 1 m = 100 cm, are fundamental to remember for accurate results. Performing these conversions ensures the measurements are in consistent units, which is crucial when applying mathematical formulas. This step avoids common mistakes that arise from unit discrepancies and establishes a solid foundation for any scientific analysis.

Hemisphere volume formula

The hemisphere volume formula we use here is an adaptation of the full sphere volume formula to suit a hemispherical shape, given by \[ V = \frac{2}{3} \pi r^3 \]. Here, \( r \) is the radius of the hemisphere, or in the context of the exercise, the thickness of the ice in centimeters. Applying this formula provides a means to calculate the volume of the semicircular ice sheet of Antarctica. It factors in only one half of a hypothetical sphere. Using this approach simplifies calculations when dealing with hemispherical shapes and ensures a more accurate representation of real-world phenomena where perfect spheres are rarely encountered. Recognizing this formula and how to apply it is fundamental for any volume calculations involving hemispheres.