The formula for the volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cone. We know the volume \( V = 261.8 , \text{cm}^3 \) and the height \( h = 10 , \text{cm} \). We need to solve for the radius \( r \).
Substituting the known values into the volume formula, we have:
\[ 261.8 = \frac{1}{3} \pi r^2 (10) \]
This simplifies to:
\[ 261.8 = \frac{10}{3} \pi r^2 \]
Now, multiply both sides by \( 3 \) to eliminate the fraction:
\[ 3 \times 261.8 = 10 \pi r^2 \]
This results in:
\[ 785.4 = 10 \pi r^2 \]
Next, divide both sides by \( 10 \pi \):
\[ \frac{785.4}{10 \pi} = r^2 \]
Calculating \( \frac{785.4}{10} = 78.54 \), we have:
\[ r^2 = \frac{78.54}{\pi} \]
Now, using \( \pi \approx 3.14 \):
\[ r^2 = \frac{78.54}{3.14} \approx 25 \]
Taking the square root of both sides gives:
\[ r \approx \sqrt{25} \approx 5 \]
Thus, the approximate radius of the cone is:
\[ \boxed{5} , \text{cm} \]
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