1. **State the problem:** We have a rectangular prism birdcage with base side lengths 6 m and 8 m, and height 7 m. A branch of length $b$ stretches from one corner of the prism to the opposite corner (space diagonal). We need to find: (a) The length $a$ of the diagonal on the base (bottom face). (b) The length $b$ of the branch (space diagonal). 2. **Formula and rules:** - The diagonal $a$ on the base is found using the Pythagorean theorem in 2D: $$a = \sqrt{6^2 + 8^2}$$ - The space diagonal $b$ in 3D is found using the Pythagorean theorem extended to three dimensions: $$b = \sqrt{a^2 + 7^2}$$ 3. **Calculate $a$:** $$a = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$$ 4. **Calculate $b$ using $a=10$:** $$b = \sqrt{10^2 + 7^2} = \sqrt{100 + 49} = \sqrt{149}$$ 5. **Approximate $b$ to the nearest tenth:** $$b \approx 12.2$$ **Final answers:** - (a) $a = 10$ m - (b) $b \approx 12.2$ m