To find out how many feathers you need to make a weight of \( 4.1 \times 10^3 \) grams, you can use the following formula:
\[ \text{Number of Feathers} = \frac{\text{Total Weight}}{\text{Weight of One Feather}} \]
Given:
- Total Weight = \( 4.1 \times 10^3 \) g
- Weight of One Feather = \( 0.0082 \) g
Now, plug in the values:
\[ \text{Number of Feathers} = \frac{4.1 \times 10^3 , \text{g}}{0.0082 , \text{g}} \]
To perform the division, we first convert \( 0.0082 \) g into scientific notation:
\[ 0.0082 = 8.2 \times 10^{-3} , \text{g} \]
Now the equation becomes:
\[ \text{Number of Feathers} = \frac{4.1 \times 10^3}{8.2 \times 10^{-3}} \]
When dividing, divide the coefficients and subtract the exponents:
\[ \text{Number of Feathers} = \frac{4.1}{8.2} \times 10^{3 - (-3)} = \frac{4.1}{8.2} \times 10^{3 + 3} \]
Now, calculate \( \frac{4.1}{8.2} \):
\[ \frac{4.1}{8.2} = 0.5 \]
Now, substituting this value back into the equation:
\[ \text{Number of Feathers} = 0.5 \times 10^{6} \]
To express \( 0.5 \) in scientific notation:
\[ 0.5 = 5.0 \times 10^{-1} \]
Thus:
\[ \text{Number of Feathers} = 5.0 \times 10^{-1} \times 10^{6} = 5.0 \times 10^{5} \]
Therefore, the number of feathers needed is:
\[ \boxed{5.0 \times 10^{5}} \]
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