Problem 54 - Algebra for College Students

Question

Alaska has an area of approximately \((6.15)\left(10^{5}\right)\) square miles. In 1999 the state had a population of approximately 619,000 people. Compute the population density to the nearest hundredth. Population density is the number of people per square mile. Express the result in decimal form rounded to the nearest hundredth.

Step-by-Step Solution

Answer

The population density is approximately 1.01 people per square mile.

Step 1: Understand the Formula for Population Density

Population density is calculated as the number of people divided by the area (in square miles) of the location. The formula is:\[\text{Population Density} = \frac{\text{Population}}{\text{Area}}\]

Step 2: Identify Given Values

From the problem, you have been given the population of Alaska in 1999 as 619,000 people and the area as \((6.15) \times (10^{5})\) square miles.

Step 3: Plug Values into the Formula

Substitute the given population and area into the population density formula:\[\text{Population Density} = \frac{619,000}{6.15 \times 10^{5}}\]

Step 4: Calculate the Result

Perform the division to find the population density:\[\text{Population Density} = \frac{619,000}{615,000} \approx 1.0065\]

Step 5: Round to the Nearest Hundredth

Round 1.0065 to the nearest hundredth to express the population density:\[1.0065 \approx 1.01\]

Key Concepts

Population Density

Population density is a measure that gives us an idea of how crowded or sparse a particular area is. To compute population density, you divide the total number of people living in an area by the size of that area. In mathematical terms, this can be represented by the formula:

• Population Density = Number of People / Area

For instance, if a region has a large area but a small number of people, the population density will be low, indicating sparsity. Conversely, a small area with many people will have a high population density, suggesting crowding.
When dealing with population density, it’s important to use consistent units to ensure an accurate calculation. In most cases, such as in this example with Alaska, the area is measured in square miles and the population in individuals.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. This is particularly useful in fields like science and engineering where such extremes are frequently encountered. In scientific notation, a number is expressed as a product of two factors:

• A coefficient, typically between 1 and 10
• A power of 10

For example, Alaska's area was given as \((6.15) \times (10^{5})\) square miles. Here, 6.15 is the coefficient and \(10^{5}\) indicates that we multiply 6.15 by 100,000.
This notation makes calculations simpler and helps you comprehend and write down big or small numbers efficiently without losing precision. To interpret scientific notation, multiply the coefficient by 10 raised to the specified power.

Rounding Numbers

Rounding is the process of adjusting a number to make it simpler and easier to use in calculations, while still remaining close to the original value. This is crucial when numbers have more decimal places than necessary for a given purpose.
In the context of our calculation, the population density of Alaska was approximately 1.0065. To round to the nearest hundredth, focus on the thousandths place:

• If it’s 5 or more, round the hundredths place up
• If it’s 4 or less, keep the hundredths place the same

Therefore, 1.0065 rounds to 1.01 because the digit in the thousandths place is 6, which is more than 5. Rounding is essential because it ensures results are presented in an easily interpretable manner without unnecessary precision.

Division in Algebra

Division in algebra involves splitting a number into equal parts or determining how many times one number is contained within another. In the context of calculating population density, we need to divide the population by the area. This can be expressed as a fraction and computed using division:

• Begin with the fraction for population density: \(\frac{619,000}{6.15 \times 10^{5}}\)
• Next, calculate the denominator: \(6.15 \times 10^{5}\) becomes 615,000
• Then perform the division: \(\frac{619,000}{615,000} \approx 1.0065\)

Understanding how to handle division with large numbers, especially when in scientific notation, is crucial. It ensures accuracy and allows for solving real-world problems efficiently. Like in this case, achieving the correct population density result required proper division handling.