1. **State the problem:** Find the volume of a composite solid consisting of a rectangular prism with a square base of side 2 ft and height 2 ft, topped by a pyramid with a square base of side 2 ft and a slant edge of 3 ft. 2. **Volume of the rectangular prism:** The formula for the volume of a rectangular prism is $$V = Bh$$ where $B$ is the area of the base and $h$ is the height. Since the base is a square with side length 2 ft, $$B = 2 \times 2 = 4 \text{ ft}^2$$ and the height is 2 ft. So, $$V_{prism} = 4 \times 2 = 8 \text{ ft}^3$$ 3. **Volume of the pyramid:** The formula for the volume of a pyramid is $$V = \frac{1}{3}Bh$$ where $B$ is the area of the base and $h$ is the height of the pyramid. The base area is the same as the prism's top face, $$B = 4 \text{ ft}^2$$. We need to find the height $h$ of the pyramid. Given the slant edge (3 ft) and the base side (2 ft), we can find the height using the Pythagorean theorem. The slant edge forms the hypotenuse of a right triangle with half the base side as one leg and the height as the other leg: $$\text{half base} = \frac{2}{2} = 1 \text{ ft}$$ Using Pythagoras: $$3^2 = h^2 + 1^2$$ $$9 = h^2 + 1$$ $$h^2 = 8$$ $$h = \sqrt{8} = 2\sqrt{2} \text{ ft}$$ 4. **Calculate the volume of the pyramid:** $$V_{pyramid} = \frac{1}{3} \times 4 \times 2\sqrt{2} = \frac{4}{3} \times 2\sqrt{2} = \frac{8\sqrt{2}}{3} \text{ ft}^3$$ 5. **Total volume of the composite figure:** $$V_{total} = V_{prism} + V_{pyramid} = 8 + \frac{8\sqrt{2}}{3} = \frac{24}{3} + \frac{8\sqrt{2}}{3} = \frac{24 + 8\sqrt{2}}{3} \text{ ft}^3$$ **Final answer:** $$\boxed{\frac{24 + 8\sqrt{2}}{3} \text{ ft}^3}$$