1. Given the volume formula and the equation starting from 1499.744: $$1499.744 = \frac{1}{3} \times 2 \times \left[a^2 + 16a + 64 + a^2 + \sqrt{a^2 \times (a+8)^2}\right]$$ 2. Simplify inside the brackets: $$= \frac{2}{3} \times \left[2a^2 + 16a + 64 + a(a+8)\right]$$ 3. Expand $a(a+8)$: $$= \frac{2}{3} \times \left[2a^2 + 16a + 64 + a^2 + 8a\right] = \frac{2}{3} \times \left[3a^2 + 24a + 64\right]$$ 4. Multiply both sides by $\frac{3}{2}$ to isolate the quadratic expression: $$1499.744 \times \frac{3}{2} = 3a^2 + 24a + 64$$ $$2249.616 = 3a^2 + 24a + 64$$ 5. Rearrange to standard quadratic form: $$3a^2 + 24a + 64 - 2249.616 = 0$$ $$3a^2 + 24a - 2185.616 = 0$$ 6. Use the quadratic formula: $$a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=24$, and $c=-2185.616$. 7. Calculate the discriminant: $$\Delta = 24^2 - 4 \times 3 \times (-2185.616) = 576 + 26227.392 = 26803.392$$ 8. Calculate the roots: $$a = \frac{-24 \pm \sqrt{26803.392}}{6}$$ $$\sqrt{26803.392} \approx 163.68$$ 9. Find the two solutions: $$a_1 = \frac{-24 + 163.68}{6} = \frac{139.68}{6} \approx 23.28$$ $$a_2 = \frac{-24 - 163.68}{6} = \frac{-187.68}{6} \approx -31.28$$ 10. Since length cannot be negative, the valid solution is: $$a = 23.28$$ Final answer: The value of $a$ is approximately $23.28$ meters.