Problem 46 - Elementary Algebra

Question

Convert the numbers used in the following problems to scientific notation. Manganese-53 has a half-life of 59,918,000,000,000 seconds \((1,900,000\) years \()\).

Step-by-Step Solution

Answer

Question: Express the half-life of Manganese-53 in scientific notation for both seconds and years. Answer: The half-life of Manganese-53 can be expressed as approximately \(5.9918 × 10^{13}\) seconds and \(1.9 × 10^{6}\) years in scientific notation.

Step 1: Understand the format of scientific notation.

Scientific notation involves writing a number as the product of a number between 1 and 10 and a power of 10. The general format is: \(a × 10^b\) where \(1 \leq a < 10\) and \(b\) is an integer.

Step 2: Convert the half-life of Manganese-53 in seconds to scientific notation.

To convert the given half-life of Manganese-53, which is 59,918,000,000,000 seconds, into scientific notation, follow these steps: 1. Move the decimal point 13 places to the left to make the number 5.9918: \(59,918,000,000,000 \rightarrow 5.9918\) 2. Multiply the resulting number by \(10^{13}\) to get the number back to its original value: \(5.9918 × 10^{13}\) So, the half-life of Manganese-53 in seconds is approximately \(5.9918 × 10^{13}\) seconds in scientific notation.

Step 3: Convert the half-life of Manganese-53 in years to scientific notation.

To convert the given half-life of Manganese-53, which is 1,900,000 years, into scientific notation, follow these steps: 1. Move the decimal point 6 places to the left to make the number 1.9: \(1,900,000 \rightarrow 1.9\) 2. Multiply the resulting number by \(10^{6}\) to get the number back to its original value: \(1.9 × 10^{6}\) So, the half-life of Manganese-53 in years is approximately \(1.9 × 10^{6}\) years in scientific notation. In conclusion, the half-life of Manganese-53 can be expressed as: - Seconds: \(5.9918 × 10^{13}\) - Years: \(1.9 × 10^{6}\)

Key Concepts

Half-Life

The concept of half-life is frequently used in fields such as physics, chemistry, and geology. **Half-life** refers to the time required for a quantity to reduce to half its initial value. It's commonly applied to radioactive substances, where it describes how long it takes for half of the radioactive atoms in a sample to decay. For example, with Manganese-53, the half-life is given as 59,918,000,000,000 seconds or 1,900,000 years. This means that after 1,900,000 years, half of the Manganese-53 atoms would have decayed into another element. When examining half-life, consider these features:

• It's a constant value and does not depend on the initial amount of substance or the specific amount left.
• Half-life can be represented in different units such as seconds or years.

Knowing how to express half-life in various forms, notably scientific notation, can make complex numbers easier to understand and work with in calculations.

Mathematical Conversion

Mathematical conversion is a process used to change a number or quantity from one form to another. This is particularly useful when dealing with large or small numbers, like those seen in scientific contexts. **Scientific notation**, as used in the example with Manganese-53, is a type of mathematical conversion. To convert a regular number into scientific notation involves the following steps:

• Identify where the decimal point would be placed to create a number between 1 and 10.
• Count how many places the decimal moves, which becomes the power of ten.

Converting Manganese-53's half-life from 59,918,000,000,000 seconds to 5.9918 x 10^{13} seconds involves moving the decimal 13 places to the left. Similarly, converting 1,900,000 years to 1.9 x 10^{6} years means moving the decimal 6 places left. This method helps simplify large numbers for calculations and comparisons, making them more manageable and understandable.

Power of Ten

A **power of ten** is an expression that represents a number as ten raised to an integer exponent. In scientific notation, large and small numbers are often expressed as a product of a coefficient and a power of ten. Understanding powers of ten involves:

• A positive exponent (e.g., 10^{13}) indicates the number 10 is multiplied by itself a certain number of times.
• A negative exponent signifies a division process, representing fractional quantities.

In the context of Manganese-53, the power of 10 is 13 for seconds and 6 for years. This means that the decimals move 13 and 6 places respectively to transform the original numbers into a more compact and usable form. Learning how to work with powers of ten is fundamental in mathematical conversion tasks, enabling the simplification of numbers, especially within scientific calculations. It provides a more readable and consistent way of expressing very large or tiny quantities.