Verified Answer :
- Represent the mass of one feather as \(8.2 \times 10^{-3} \text{ g}\).
- Set up the division of the total weight \(W = 4.1 \times 10^3 \text{ g}\) by the individual weight \(w = 8.2 \times 10^{-3} \text{ g}\).
- Calculate the quotient to get \(N = 0.5 \times 10^6\).
- Adjust to proper scientific notation to obtain \(5 \times 10^5\), so the total number of feathers is \(\boxed{5 \times 10^5}\).
Explanation
- Analysis of the Problem
The objective is to determine the number of feathers, \(N\), required to reach a total mass of \(W = 4.1 \times 10^3\) g, given that a single feather has a mass of \(w = 0.0082\) g. The relationship between total mass, individual mass, and quantity is \(W = N \times w\). Therefore, we need to calculate \(N = \frac{W}{w}\).
- Converting to Scientific Notation
First, convert the mass of a single feather \(w = 0.0082\) g into scientific notation to facilitate division. Moving the decimal point 3 places to the right gives:
\[w = 8.2 \times 10^{-3} \text{ g}.\]
The total target mass is already in scientific notation:
\[W = 4.1 \times 10^3 \text{ g}.\]
- Calculating the Number of Feathers
Now, calculate the number of feathers \(N\) by dividing the total weight \(W\) by the individual weight \(w\):
\[N = \frac{4.1 \times 10^3}{8.2 \times 10^{-3}}.\]
Dividing the coefficients \(4.1 \div 8.2 = 0.5\), and applying the rule for exponents by subtracting them \(10^3 \div 10^{-3} = 10^{3 - (-3)} = 10^6\):
\[N = 0.5 \times 10^6.\]
- Final Answer Formatting
To express the result \(N = 0.5 \times 10^6\) in standard scientific notation, where the coefficient must be at least 1 and less than 10, rewrite \(0.5\) as \(5 \times 10^{-1}\):
\[N = (5 \times 10^{-1}) \times 10^6 = 5 \times 10^{6-1} = 5 \times 10^5.\]
Thus, the total number of feathers required is \(5 \times 10^5\).
Examples
Scientific notation is frequently used in fields like biology or chemistry when dealing with microscopic or macroscopic quantities. For example, if you are studying the mass of microscopic dust particles in an industrial cleanroom, you might have a filter that collects billions of particles, each weighing a fraction of a gram. Expressing these quantities in scientific notation, such as \(5 \times 10^5\) particles, allows for easier calculations, comparisons, and communication of vast data sets without dealing with long strings of zeros.