1. **State the problem:** We have a right triangular prism with a base triangle having legs 5 in and 9 in, and height 12 in. We need to find: (a) The length $a$, which is the hypotenuse of the right triangle base. (b) The length $b$, the diagonal tape length from one bottom corner to the opposite top corner of the prism. 2. **Formula and rules:** - To find the hypotenuse $a$ of a right triangle with legs $5$ and $9$, use the Pythagorean theorem: $$a = \sqrt{5^2 + 9^2}$$ - To find the diagonal $b$ across the prism, treat $a$ and the height $12$ as legs of another right triangle: $$b = \sqrt{a^2 + 12^2}$$ 3. **Calculate $a$:** $$a = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106}$$ 4. **Calculate $b$ using $a$:** $$b = \sqrt{a^2 + 12^2} = \sqrt{106 + 144} = \sqrt{250}$$ 5. **Simplify and approximate:** $$a = \sqrt{106} \approx 10.2956 \text{ in}$$ $$b = \sqrt{250} = \sqrt{25 \times 10} = 5\sqrt{10} \approx 15.8114 \text{ in}$$ 6. **Final answers:** (a) $a \approx 10.3$ in (rounded to one decimal place) (b) $b \approx 15.8$ in (rounded to one decimal place)