Problem 67 - Algebra and Trigonometry

Question

Write the number indicated in each statement in scientific notation. (a) A light-year, the distance that light travels in one year, is about \(5,900,000,000,000\) mi. (b) The diameter of an electron is about 0.00000000000004 \(\mathrm{cm} .\) (c) A drop of water contains more than 33 billion billion molecules.

Step-by-Step Solution

Answer

(a) \(5.9 \times 10^{12}\), (b) \(4 \times 10^{-14}\), (c) \(3.3 \times 10^{19}\).

Step 1: Understand Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10. The format is: \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

Step 2: Convert Light-Year Distance to Scientific Notation

Identify the significant figure in 5,900,000,000,000, which is 5.9. Count the number of spaces from the decimal point to the end of the number to find the power of ten. There are 12 spaces, so the number in scientific notation is \( 5.9 \times 10^{12} \).

Step 3: Convert Electron Diameter to Scientific Notation

Identify the significant figure in 0.00000000000004, which is 4. Count how many times you need to move the decimal point to the right to get a number between 1 and 10. You move it 14 spaces, so the number in scientific notation is \( 4 \times 10^{-14} \).

Step 4: Convert Water Molecules Quantity to Scientific Notation

Identify the significant figures in 33 billion billion, which is \( 33,000,000,000,000,000,000 \). Count the spaces to achieve a number between 1 and 10. Moving the decimal 19 places to the left, the notation is \( 3.3 \times 10^{19} \).

Key Concepts

Light-Year Distance

The term 'light-year' might sound like it relates to time, but it's actually a measure of distance in astronomy. A light-year is how far light travels in one year.
Light is incredibly fast, traveling about 186,282 miles per second. Students of science use this measurement to express vast distances in space, as numbers can become unwieldy to handle in their conventional form.
The distance light covers in one year is approximately 5,900,000,000,000 miles.

• Using scientific notation to express this number, we identify 5.9 as the leading significant figure.
• The digits following zeros take us to 12 places to the right of 5.9.
• Thus, the light-year distance can be written in scientific notation as \(5.9 \times 10^{12}\).

This notation simplifies handling extreme values, making calculations and comparisons much easier.

Electron Diameter

The diameter of an electron is unbelievably small, making it challenging to comprehend. In physical science, electrons are subatomic particles, part of an atom's structure, and are extremely tiny in scope.
When we express the electron's diameter as 0.00000000000004 cm, it naturally becomes cumbersome to use in calculations.

• In scientific notation, it's vital to find a significant figure between 1 and 10. For this measure, 4 is the significant figure.
• Counting the decimal places is crucial, as we need to explain how many positions the decimal moves to achieve a suitable scientific notation.
• Here, moving 14 places results in the scientific expression: \(4 \times 10^{-14}\).

Scientific notation streamlines usage and comprehension of remarkably small numbers like that of an electron.

Water Molecules Quantity

Water molecules in a single drop of water are surprisingly plentiful, far beyond everyday counting.
A typical drop of water contains more than 33 billion billion molecules, an amount that's tricky to write out fully.

• Understanding in scientific notation becomes essential when faced with such large numbers.
• The significant figure in this context is 3.3, accompanied by 19 additional digits to the right.
• This transforms the count to a scientific notation of \(3.3 \times 10^{19}\).

Such representation not only eases the numerical problem but also aids in comparing molecular quantities across different substances.