Questions: Problem 3: Antarctica is roughly semicircular, with a radius of r:=2000 km. The average thickness of its ice cover is hi:=3000 m. How many cubic centimeters of ice does Antarctica contain? (Ignore the curvature of Earth.) Think of Antarctica as a half slice of a flat cylinder, where the slice is along the radius of the cylinder.

Problem 3: Antarctica is roughly semicircular, with a radius of r:=2000 km. The average thickness of its ice cover is hi:=3000 m. How many cubic centimeters of ice does Antarctica contain? (Ignore the curvature of Earth.) Think of Antarctica as a half slice of a flat cylinder, where the slice is along the radius of the cylinder.

Transcript text: Problem 3: Antarctica is roughly semicircular, with a radius of $r:=2000 \mathrm{~km}$. The average thickness of its ice cover is $h_{i}:=3000 \cdot m$. How many cubic centimeters of ice does Antarctica contain? (Ignore the curvature of Earth.) Think of antarctica as a half slice of a flat cylinder. Where the slice is along the radius of the cylinder.

Solution

Solution Steps

To find the volume of ice in Antarctica, we can model it as a semicircular cylinder. First, calculate the area of the semicircular base using the formula for the area of a circle and then divide by two. Multiply this area by the thickness of the ice to get the volume in cubic kilometers. Finally, convert the volume from cubic kilometers to cubic centimeters.

Step 1: Calculate the Area of the Semicircular Base

The area of a full circle is given by the formula $ A = \pi r^2 $. Since Antarctica is modeled as a semicircle, the area of the base is half of that: \[ A_{\text{semicircle}} = \frac{\pi \times (2000 \, \text{km})^2}{2} = 6,283,185.3072 \, \text{km}^2 \]

Step 2: Calculate the Volume of the Semicircular Cylinder

The volume of a cylinder is given by the formula $ V = A \times h $, where $ A $ is the area of the base and $ h $ is the height (or thickness in this case). The thickness of the ice is 3000 meters, which is equivalent to 3 kilometers. Therefore, the volume is: \[ V = 6,283,185.3072 \, \text{km}^2 \times 3 \, \text{km} = 18,849,555.9215 \, \text{km}^3 \]

Step 3: Convert Volume to Cubic Centimeters

To convert the volume from cubic kilometers to cubic centimeters, use the conversion factor $ 1 \, \text{km}^3 = 10^{15} \, \text{cm}^3 $: \[ V = 18,849,555.9215 \, \text{km}^3 \times 10^{15} \, \text{cm}^3/\text{km}^3 = 1.8849555922 \times 10^{22} \, \text{cm}^3 \]