Question

The rate of continental drift is on the order of \(10.0 \mathrm{~mm} / \mathrm{yr}\). Approximately how long did it take North America and Europe to reach their current separation of about \(3000 \mathrm{mi} ?\)

Short answer

Answer: It took approximately 4.8 x 10^11 years for North America and Europe to reach their current separation.

Step-by-step solution

Convert miles to millimeters.

We need to convert 3000 miles to millimeters. We can use the following conversion factors: 1 mile = 1.60934 km 1 km = 1000 m 1 m = 1000 mm So, 3000 mi = 3000 * 1.60934 * 1000 * 1000 mm #Step 2: Calculate the time taken#

Calculate the time taken for the continents to drift apart.

Now that we have the distance in millimeters, we can calculate the time taken using the formula time = distance / rate. So we have: time = (3000 * 1.60934 * 1000 * 1000 mm) / (10.0 mm/yr) #Step 3: Simplify the expression#

Simplify the expression to get the time in years.

When we simplify the expression, we get: time = 3000 * 1.60934 * 1000 * 1000 / 10.0 years time ≈ 4.8 x 10^11 years So, it took approximately 4.8 x 10^11 years for North America and Europe to reach their current separation.

Key concepts

Converting Miles to Millimeters

Understanding unit conversion is critical in solving many scientific and mathematical problems, including calculating continental drift. The exercise at hand requires us to convert miles to millimeters to facilitate computation of the drift rate over time. Let's put this into perspective: one mile equates to 1.60934 kilometers, a kilometer equals to 1000 meters, and a meter corresponds to 1000 millimeters.

To convert from a larger unit to a smaller one, we multiply (since one unit of the larger measure contains many units of the smaller measure). So, for 3000 miles, we need to multiply each conversion step by the next:

• 1 mile = 1.60934 kilometers
• 1 kilometer = 1000 meters
• 1 meter = 1000 millimeters

Thus, we have:
\( 3000 \text{ miles} \times 1.60934 \times 1000 \times 1000 = 3000 \times 1.60934 \times 10^6 \text{ millimeters} \).

Understand that knowing how to convert units is not just about memorizing conversion factors; it's about understanding the relationships between different scales of measurements and being able to shift seamlessly between them to solve real-world problems.

Calculating Time Using Distance and Rate

After converting the distance between continents from miles to millimeters, the next step is to calculate the time it took for the continents to drift apart using the given rate of drift. The formula for time based on distance and rate is:

\( \text{Time} = \frac{\text{Distance}}{\text{Rate}} \)

In our scenario, the distance represents the converted separation between North America and Europe, and the rate is the speed of continental drift per year. By substituting the distance in millimeters and the drift rate (10 mm/year), we can find out how many years it would take at this steady rate to cover the converted distance between the continents.

The simplification of this formula, taking into account the scale of millimeters and years, allows for a precise calculation, providing an estimate on how long the natural process of continental movement has been in progress. Understanding this concept is foundational for grasping rates of change in natural phenomena and their impact on Earth's geography over vast timespans.

Simplifying Mathematical Expressions

Simplifying mathematical expressions means reducing them to their simplest form, making them easier to interpret or compute. In the context of our problem, we've reached a complex numerical expression that represents the time taken for continental drift:

\( \text{time} = 3000 \times 1.60934 \times 10^6 / 10.0 \text{ years} \)

This expression can be intimidating due to the large numbers and units involved. To simplify, we start by performing multiplications and divisions step by step. Dividing by 10 is straightforward, effectively just moving the decimal place one position to the left in each number to be divided.

When dealing with scientific notation, recognize that it's a convenient way to express large or small numbers. In our case, \(10^6\) means a one followed by six zeros. Incorporating this into our calculations and simplifying allows us to clearly see the magnitude of time being discussed. Essential skills in algebra and number theory come to play here, ensuring that students can complete calculations efficiently and accurately.