To find the slant height of a cone, we can use the formula for the total surface area of a cone:
\[ A = \pi r (r + l) \]
where \( A \) is the total surface area, \( r \) is the radius, and \( l \) is the slant height.
Given:
- Total surface area \( A = 625 , \text{in}^2 \)
- Radius \( r = 5 , \text{in} \)
Substituting the values into the formula:
\[ 625 = \pi (5) (5 + l) \]
To simplify, divide both sides by \( \pi \):
\[ \frac{625}{\pi} = 5 (5 + l) \]
Calculating \( \frac{625}{\pi} \):
\[ \approx \frac{625}{3.14159} \approx 199.1 \]
Thus, the equation can be rewritten as:
\[ 199.1 = 5 (5 + l) \]
Now, divide both sides by \( 5 \):
\[ \approx 39.82 = 5 + l \]
Now, solve for \( l \):
\[ l = 39.82 - 5 \approx 34.82 \]
Rounding to the nearest whole number, we find:
\[ l \approx 35 , \text{in} \]
Thus, the slant height of the cone is approximately \( \boxed{35} , \text{in} \).
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