1. **State the problem:** We need to solve the equation: $$325x \left(\frac{x}{2}\right) - (190 \times 20) \left(x - \frac{20}{2}\right) - \left(\frac{1}{2} \times 15 \times 20\right) \left(x - \frac{20}{3}\right) - (130 \times 20) \left(x - \frac{20}{2}\right) - \left(\frac{1}{2} \times 65 \times 40\right) \left(x - \frac{40}{3}\right) = 5625\pi (637.5 - x)$$ 2. **Simplify each term:** - First term: $$325x \times \frac{x}{2} = \frac{325}{2} x^2$$ - Second term: $$190 \times 20 = 3800$$ so second term is $$-3800 \left(x - 10\right)$$ - Third term: $$\frac{1}{2} \times 15 \times 20 = 150$$ so third term is $$-150 \left(x - \frac{20}{3}\right)$$ - Fourth term: $$130 \times 20 = 2600$$ so fourth term is $$-2600 \left(x - 10\right)$$ - Fifth term: $$\frac{1}{2} \times 65 \times 40 = 1300$$ so fifth term is $$-1300 \left(x - \frac{40}{3}\right)$$ 3. **Rewrite the equation:** $$\frac{325}{2} x^2 - 3800 (x - 10) - 150 \left(x - \frac{20}{3}\right) - 2600 (x - 10) - 1300 \left(x - \frac{40}{3}\right) = 5625 \pi (637.5 - x)$$ 4. **Expand the terms:** $$\frac{325}{2} x^2 - 3800x + 38000 - 150x + 1000 - 2600x + 26000 - 1300x + \frac{1300 \times 40}{3} = 5625 \pi \times 637.5 - 5625 \pi x$$ Calculate $$\frac{1300 \times 40}{3} = \frac{52000}{3}$$ Calculate $$5625 \pi \times 637.5 = 5625 \times 637.5 \pi = 3585937.5 \pi$$ 5. **Combine like terms on the left:** Sum of $$x$$ terms: $$-3800x - 150x - 2600x - 1300x = -7850x$$ Sum of constants: $$38000 + 1000 + 26000 + \frac{52000}{3} = 65000 + \frac{52000}{3} = \frac{195000}{3} + \frac{52000}{3} = \frac{247000}{3}$$ 6. **Rewrite the equation:** $$\frac{325}{2} x^2 - 7850 x + \frac{247000}{3} = 3585937.5 \pi - 5625 \pi x$$ 7. **Bring all terms to one side:** $$\frac{325}{2} x^2 - 7850 x + \frac{247000}{3} - 3585937.5 \pi + 5625 \pi x = 0$$ 8. **Group $$x$$ terms:** $$-7850 x + 5625 \pi x = x (-7850 + 5625 \pi)$$ 9. **Final quadratic form:** $$\frac{325}{2} x^2 + x (-7850 + 5625 \pi) + \left(\frac{247000}{3} - 3585937.5 \pi\right) = 0$$ 10. **Solve quadratic equation:** Use formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a = \frac{325}{2}, \quad b = -7850 + 5625 \pi, \quad c = \frac{247000}{3} - 3585937.5 \pi$$ This gives the solution(s) for $$x$$. **Final answer:** $$x = \frac{-(-7850 + 5625 \pi) \pm \sqrt{(-7850 + 5625 \pi)^2 - 4 \times \frac{325}{2} \times \left(\frac{247000}{3} - 3585937.5 \pi\right)}}{2 \times \frac{325}{2}}$$