1. **State the problem:** We need to find the volume of the frustum of a rectangular-based pyramid. The frustum is formed by cutting the original pyramid horizontally. 2. **Given dimensions:** - Small top pyramid base sides: $3$ cm and $6$ cm - Small top pyramid height: $9$ cm - Frustum height: $15$ cm - Large base of frustum sides: $8$ cm and $16$ cm 3. **Find the height of the original large pyramid:** The frustum is the lower part of the original pyramid after cutting off the smaller top pyramid. Since the bases are similar rectangles, the scale factor for the sides from the small pyramid to the large base is: $$\text{scale} = \frac{8}{3} = \frac{16}{6} = \frac{8}{3} \approx 2.6667$$ The height scales the same way, so the height of the original pyramid is: $$h_{original} = 9 \times \frac{8}{3} = 24 \text{ cm}$$ 4. **Volume formula for a pyramid:** $$V = \frac{1}{3} \times \text{base area} \times \text{height}$$ 5. **Calculate volumes:** - Volume of original large pyramid: $$V_{original} = \frac{1}{3} \times (8 \times 16) \times 24 = \frac{1}{3} \times 128 \times 24 = \frac{1}{3} \times 3072 = 1024 \text{ cm}^3$$ - Volume of small top pyramid: $$V_{small} = \frac{1}{3} \times (3 \times 6) \times 9 = \frac{1}{3} \times 18 \times 9 = \frac{1}{3} \times 162 = 54 \text{ cm}^3$$ 6. **Volume of the frustum:** The frustum volume is the original pyramid volume minus the small top pyramid volume: $$V_{frustum} = V_{original} - V_{small} = 1024 - 54 = 970 \text{ cm}^3$$ **Final answer:** $$\boxed{970 \text{ cm}^3}$$