1. **State the problem:** We have an oblique triangular prism and a right cylinder with the same base area and volume. The prism's base is a triangle with an altitude of 8 and two known sides: 10.6 and 10, and an unknown base edge perpendicular to the altitude. The cylinder has height 8 and radius 4. 2. **Formulas and rules:** - Volume of prism: $$V_{prism} = \text{Base Area} \times \text{Length}$$ - Volume of cylinder: $$V_{cyl} = \pi r^2 h$$ - Base area of prism (triangle): $$A = \frac{1}{2} \times \text{base} \times \text{altitude}$$ Since volumes and base areas are equal, we can set up equations to find the unknown base edge. 3. **Calculate cylinder volume:** $$V_{cyl} = \pi \times 4^2 \times 8 = 128\pi$$ 4. **Calculate prism volume:** Let the unknown base edge be $x$. The base area of the prism triangle is: $$A = \frac{1}{2} \times x \times 8 = 4x$$ The prism volume is: $$V_{prism} = A \times 10.6 = 4x \times 10.6 = 42.4x$$ 5. **Set volumes equal:** $$42.4x = 128\pi$$ 6. **Solve for $x$:** $$x = \frac{128\pi}{42.4}$$ 7. **Simplify fraction with cancellation:** $$x = \frac{\cancel{128}\times \pi}{\cancel{42.4}} = \frac{128\pi}{42.4}$$ (Note: 42.4 is 10.6 times 4, so no common integer factor to cancel exactly, keep as is.) 8. **Calculate numerical value:** $$x \approx \frac{128 \times 3.1416}{42.4} \approx \frac{402.12}{42.4} \approx 9.48$$ **Final answer:** The length of the base edge perpendicular to the altitude is approximately **9.48** units.