To find out how many feathers you would need to make a weight of \( 4.1 \times 10^3 \) g, you can use the following formula:
\[ \text{Number of feathers} = \frac{\text{Total weight}}{\text{Weight of one feather}} \]
In this case:
- Total weight = \( 4.1 \times 10^3 \) g
- Weight of one feather = \( 0.0082 \) g
First, we convert \( 0.0082 \) g into scientific notation:
\[ 0.0082 = 8.2 \times 10^{-3} \text{ g} \]
Now, we can set up the equation:
\[ \text{Number of feathers} = \frac{4.1 \times 10^3 \text{ g}}{8.2 \times 10^{-3} \text{ g}} \]
To divide in scientific notation, you divide the coefficients and subtract the exponents:
-
Divide the coefficients: \[ \frac{4.1}{8.2} = 0.5 \]
-
Subtract the exponents: \[ 10^{3 - (-3)} = 10^{3 + 3} = 10^6 \]
Putting it together:
\[ \text{Number of feathers} = 0.5 \times 10^6 \]
To express \( 0.5 \) in scientific notation, we rewrite it as \( 5.0 \times 10^{-1} \):
\[ 0.5 \times 10^6 = 5.0 \times 10^{-1} \times 10^6 = 5.0 \times 10^{5} \]
Therefore, the number of feathers needed is:
\[ \boxed{5.0 \times 10^5} \]