Problem 159 - General Chemistry

Question

The total length of all the DNA molecules contained in a human body is \(1 \times 10^{10}\) miles. The population of the United States is about 300 million. What is the total length of all the DNA of the U.S. population in lightyears? (A lightyear is the distance that light travels in a year and is \(9.46 \times 10^{15} \mathrm{~m} .\)

Step-by-Step Solution

Answer

The total DNA length for the U.S. population is approximately 51,000 lightyears.

Step 1: Define the Problem

We need to find the total length of the DNA for the entire U.S. population. We know the DNA length for one person and want to extend that to the entire U.S. population. Then, we will convert this measurement to lightyears.

Step 2: Calculate the Total DNA Length for the U.S.

The total length of DNA for one person is given as \(1 \times 10^{10}\) miles. The U.S. population is about 300 million people. The total DNA length for the entire U.S. population is the product of these two values: \(300,000,000 \times 1 \times 10^{10}\) miles.

Step 3: Perform the Multiplication

Multiply the number of people by the length of DNA per person: \(300,000,000 \times 1 \times 10^{10} = 3 \times 10^{17}\) miles of DNA for the entire U.S. population.

Step 4: Convert Miles to Lightyears

First, convert miles to meters, knowing that 1 mile = 1609 meters. Then, divide by the number of meters in a lightyear, \(9.46 \times 10^{15}\) meters:\[\text{Total meters of DNA} = 3 \times 10^{17} \times 1609\]\[\text{Total lightyears of DNA} = \frac{3 \times 10^{17} \times 1609}{9.46 \times 10^{15}}\]

Step 5: Calculation and Interpretation

Calculate the total distance in lightyears:\[\text{Total lightyears of DNA} = \frac{3 \times 1609}{9.46} \approx 5.1 \times 10^{4} \text{ lightyears}\] Therefore, the DNA from the entire U.S. population stretches approximately \(5.1 \times 10^{4}\) lightyears.

Key Concepts

Unit Conversion

Unit conversion is the process of converting a measurement from one unit to another, which in this problem involves converting miles to meters, and then meters to lightyears. Remember, unit conversion is essential when the initial data is given in one form, but the objective is to express the results in another.

Here's how you can perform unit conversions effectively:

• Identify the conversion factor, which is a ratio that lets you convert from one unit to another. For example, 1 mile is equal to 1609 meters.
• Multiply the given value by the conversion factor to switch units. In this problem, you convert miles to meters by multiplying the total miles of DNA by 1609 to get meters.

Once the miles are converted to meters, you can then convert meters to lightyears using:\[\text{Lightyears} = \frac{\text{Total meters}}{\text{Meters in a lightyear}}\]Keep in mind these steps whenever you need to convert between units.

Population Scaling

Population scaling is the method of scaling a measurement from a single instance to a larger group, like a population. In this exercise, you need to scale the DNA length from one individual to the entire U.S. population.

Here's the approach:

• First, recognize the measurement for a single instance. For one person, the DNA length is given as \(1 \times 10^{10}\) miles.
• Multiply this by the total number of instances or individuals in the population. The U.S. population is estimated at 300 million people, so the multiplication is:

\[300,000,000 \times 1 \times 10^{10} = 3 \times 10^{17}\text{ miles}\]This gives you the total length of DNA for the whole population. Scaling helps to extend single-case data to larger groups easily.

Lightyear

A lightyear is a unit of distance that measures how far light can travel in one year. It's often used in astronomy to express distances between celestial objects.
In this exercise, we need to convert total DNA length in meters into lightyears. The essential information is that one lightyear is\[9.46 \times 10^{15} \text{ meters}\]To find out how many lightyears the U.S. population's DNA length stretches, you first convert the length to meters and then divide by the number of meters per lightyear.

The formula is:\[\text{Total lightyears} = \frac{\text{Total meters}}{9.46 \times 10^{15}}\]Understanding the scale of a lightyear can help us grasp the enormity of distances even within the universe and this exercise.

Multiplication in Scientific Notation

Scientific notation is a helpful way to express very large or very small numbers by using powers of ten. In this problem, you'll see calculations like \(1 \times 10^{10}\) miles or \(3 \times 10^{17}\) miles.

Here’s how to efficiently multiply numbers in scientific notation:

• First, multiply the coefficients (the numbers in front). For instance, in this problem, you multiply \(300,000,000\times 1 = 3.0\).
• Next, add the exponents of ten together. For example, \(10^8\times 10^{10} = 10^{18}\).

Combining both results, we have:\[3.0 \times 10^{18}\]Being comfortable with multiplying in scientific notation can drastically simplify working with large numbers, especially in scientific calculations.