To find the slant height of a square pyramid, you can use the Pythagorean theorem. The slant height \( l \) is related to the height \( h \) and half of the base length \( b \).
Given:
- The length of the pyramid \( h = 13 \) (this is the vertical height)
- The base length \( b = 16 \)
First, we find half of the base length:
\[ \frac{b}{2} = \frac{16}{2} = 8 \]
Now, we can apply the Pythagorean theorem:
\[ l = \sqrt{h^2 + \left(\frac{b}{2}\right)^2} \]
Substituting the known values:
\[ l = \sqrt{13^2 + 8^2} \] \[ l = \sqrt{169 + 64} \] \[ l = \sqrt{233} \]
Thus, the slant height is:
\[ l \approx 15.23 \]
Since 15.23 is not one of the options given (14, 8, 289, 17), we conclude that none of those values represent the slant height based on the calculations.
If you require a specific reasonable approximation based on those options, the closest whole number would be 17, although it's not the exact calculation.