To find the slant height of a square pyramid, we can use the Pythagorean theorem.
In this case, we know:
- The height (h) of the pyramid from the base to the apex (middle) is 15 units, and
- The base of the pyramid is a square with side length 16 units.
The base's half diagonal can be calculated using:
- The side length of the base is 16 units, so the distance from the center of the base to a vertex of the base (which forms a right triangle with the slant height) can be calculated as half the side length:
\[ \text{Half base length} = \frac{16}{2} = 8 \text{ units} \]
Now, we can use the height (15 units) and the half base length (8 units) to find the slant height (l) of the pyramid using the Pythagorean theorem.
\[ l^2 = h^2 + \left(\text{Half base length}\right)^2 \]
Substituting the values:
\[ l^2 = 15^2 + 8^2 \] \[ l^2 = 225 + 64 \] \[ l^2 = 289 \]
Now, take the square root of both sides to solve for l:
\[ l = \sqrt{289} = 17 \text{ units} \]
Thus, the slant height of the square pyramid is 17.
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