Final answer:
The volume of a square pyramid with base edges of 48 cm and a slant height of 26 cm can be found using the Pythagorean theorem to first determine the pyramid's height, and then applying the volume formula for a pyramid. The calculated volume of the pyramid is 7680 cm³.
Step-by-step explanation:
To find the volume of a square pyramid, you need to know the base area and the height of the pyramid. However, the question provides the slant height, not the height of the pyramid. To proceed, we first need to find the height using the Pythagorean theorem. Assuming the base edges are all equal, since it's a square base, we can work as follows:
First, calculate the area of the base (which is a square):
Area of base = side × side = 48 cm × 48 cm = 2304 cm²
Now, to find the height (h) of the pyramid we use the slant height (l) and half of the base's length (b/2) where b is the base's side. This forms a right-angled triangle:
(l)² = (h)² + (b/2)²
Then,
(26 cm)² = (h)² + (24 cm)²
(h)² = (676 cm²) - (576 cm²) = 100 cm²
h = √(100 cm²) = 10 cm
Now that we have the height, we can find the volume of the pyramid:
Volume = (1/3) × base area × height
Volume = (1/3) × 2304 cm² × 10 cm = 7680 cm³
Therefore, the volume of the square pyramid is 7680 cm³.