Problem 40 - Astronomy: At Play in the Cosmos

Question

How many years would 1,500 generations represent, if each generation was 25 years? Give your answer in scientific notation.

Step-by-Step Solution

Answer

1,500 generations represent approximately \(3.75 \times 10^4\) years in scientific notation.

Step 1: Determine Number of Years for One Generation

We are given that each generation lasts for 25 years. So, we start by noting this fact: 1 generation = 25 years.

Step 2: Calculate Total Years for Multiple Generations

To find the total number of years for 1,500 generations, we need to multiply the number of generations by the number of years per generation. This gives us: \(1,500 \text{ generations} \times 25 \text{ years/generation} = 37,500 \text{ years}\).

Step 3: Convert Total Years to Scientific Notation

To express 37,500 years in scientific notation, we rewrite it as a number between 1 and 10 multiplied by a power of 10. We shift the decimal point four places to the left: \(3.75 \times 10^{4}\).

Key Concepts

Generational Time Calculation

Generational time calculation involves determining the total number of years spanned by multiple generations given the average length of one generation. In this context, we are given a situation where each generation lasts 25 years. Calculating the total time for 1,500 generations involves simple multiplication. We need to multiply the average generational length by the number of generations. Here’s how you do it:

• Determine the duration of one generation, which is 25 years.
• Multiply this by the total number of generations, which is 1,500.
• This gives us a total of 37,500 years for 1,500 generations.

With this generational time calculation, you can easily convert from number of generations to total years for any given generational length.

Mathematical Conversion

Mathematical conversion is the process of changing a number or value from one form to another, such as converting a linear measurement into scientific notation. Scientific notation is particularly useful for expressing large numbers concisely.
In our example, we needed to convert 37,500 years into scientific notation. Here’s the step-by-step process:

• First, identify the most significant digit in the number when expressed without leading zeroes, which is 3.75 in this case.
• Next, count the number of places the decimal point was moved to get this number: four places to the left.
• Finally, express this in notation format as: \(3.75 \times 10^{4}\).

This conversion allows numbers of vast or minute scale to be presented more simply, making calculations and comparisons easier.

Multiplication in Scientific Calculations

Multiplication in scientific calculations frequently involves working with values in scientific notation. The use of scientific notation simplifies multiplication by reducing the need for long-hand arithmetic, especially with large numbers.
When multiplying a number of generations by the average years per generation, as in our problem, we carry out basic multiplication:

• Calculate 1,500 generations times 25 years per generation
• This gives a product of 37,500 years

This result can then be more conveniently expressed in scientific notation. Understanding such basic multiplication is crucial, as it provides a foundational skill required for more intricate scientific calculations. This form of multiplication emphasizes the significance of scientific notation in handling large quantities quickly and efficiently.