Bot GPT 4-o mini To find the slant height of the square pyramid, we first need to identify the necessary dimensions from the information given.
- The base edge of the pyramid (which is a square) is given as 16 feet.
- The height from the vertex of the pyramid to the center of the base is given as 15 feet.
The slant height (\(l\)) of the pyramid can be found using the Pythagorean theorem. The slant height is the length of the line segment from the vertex of the pyramid to the midpoint of one of the base edges.
Step 1: Calculate Half the Length of the Base Edge
Since the base is a square with a length of 16 feet, half of this length is: \[ \frac{16}{2} = 8 \text{ feet} \]
Step 2: Use the Pythagorean Theorem
You can visualize a right triangle formed by:
- The height of the pyramid (15 feet)
- Half the base edge (8 feet)
- The slant height as the hypotenuse (\(l\))
According to the Pythagorean theorem: \[ l^2 = height^2 + \left(\frac{base , edge}{2}\right)^2 \] Substituting in the values: \[ l^2 = 15^2 + 8^2 \] \[ l^2 = 225 + 64 \] \[ l^2 = 289 \] Taking the square root of both sides gives: \[ l = \sqrt{289} = 17 \]
Conclusion
The slant height of the square pyramid is 17 feet.
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