Use the chart below to answer the questions:

Question

Country Population
(1997 estimate) Area
(square km)
U.S. 272,648,000 9,364,520
Romania 22,609,000 237,391
Ghana 18,339,000 237,533

Which of these countries has the greatest population density? Which has the least? Provide an explanation and proof for your answer to receive full credit.

Question 4 (Essay Worth 10 points)
(06.04 HC)

A figure is located at (2, 0), (2, −2), (6, −2), and (6, 0) on a coordinate plane. What kind of 3-D shape would be created if the figure was rotated around the x-axis? Provide an explanation and proof of your answer to receive full credit. Include the dimensions of the 3-D shape in your explanation.

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FDK451.12

Answer 1

To determine which country has the greatest and least population density, we first need to calculate the population density for each country. Population density is calculated using the formula:

\[ \text{Population Density} = \frac{\text{Population}}{\text{Area}} \]

Let's calculate the population density for each country based on the provided data:

1. United States:

   • Population: 272,648,000
   • Area: 9,364,520 square km
   • Population Density: \[ \frac{272,648,000}{9,364,520} \approx 29.1 \text{ people per square km} \]

2. Romania:

   • Population: 22,609,000
   • Area: 237,391 square km
   • Population Density: \[ \frac{22,609,000}{237,391} \approx 95.5 \text{ people per square km} \]

3. Ghana:

   • Population: 18,339,000
   • Area: 237,533 square km
   • Population Density: \[ \frac{18,339,000}{237,533} \approx 77.1 \text{ people per square km} \]

Now, let’s summarize the population densities:

• United States: ~29.1 people/sq km
• Romania: ~95.5 people/sq km
• Ghana: ~77.1 people/sq km

From our calculations, we can conclude:

• Greatest Population Density: Romania, with approximately 95.5 people per square kilometer.
• Least Population Density: United States, with approximately 29.1 people per square kilometer.

Explanation and Proof:

1. Romania has the highest ratio of population to area, meaning it has a far greater concentration of people in a smaller area compared to the other countries.
2. The U.S. has the largest land area and the highest population number but has the lowest density because its vast area results in a smaller number of people per square kilometer compared to Romania and Ghana.

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Question 4:

To determine the 3-D shape created by rotating the figure defined by the points (2, 0), (2, −2), (6, −2), and (6, 0) around the x-axis, we first visualize the shape.

The points describe a rectangle:

• The rectangle extends in the y-direction from (2, 0) down to (2, -2) and horizontally from (2, -2) to (6, -2) and back up to (6, 0).
• The dimensions of the rectangle are:
  • Height (y-direction) = 2 units (from y = 0 to y = -2).
  • Width (x-direction) = 4 units (from x = 2 to x = 6).

When this rectangle is rotated around the x-axis, it forms a cylindrical shape.

Explanation and Proof of the 3-D Shape:

1. Shape: Cylinder
2. Dimensions:
   • Height of the cylinder = width of the rectangle = distance on the x-axis (from 2 to 6) = 4 units.
   • Radius of the cylinder = distance from the center of the rectangle to the x-axis = 2 units (the furthest y-value from the x-axis is -2, thus the radius is 2).

Thus, the resulting 3D shape is a cylinder with:

• Radius = 2 units
• Height = 4 units

This cylinder will have circular bases on the top (at y = 0) and bottom (at y = -2).