1. **State the problem:** We need to simplify and 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 each term:** - $$-3800 (x - 10) = -3800x + 38000$$ - $$-150 \left(x - \frac{20}{3}\right) = -150x + 1000$$ - $$-2600 (x - 10) = -2600x + 26000$$ - $$-1300 \left(x - \frac{40}{3}\right) = -1300x + \frac{52000}{3}$$ 5. **Combine all terms on the left:** $$\frac{325}{2} x^2 - 3800x + 38000 - 150x + 1000 - 2600x + 26000 - 1300x + \frac{52000}{3} = 5625\pi (637.5 - x)$$ 6. **Combine like terms:** - Combine $x$ terms: $$-3800x - 150x - 2600x - 1300x = -7850x$$ - Combine constants: $$38000 + 1000 + 26000 + \frac{52000}{3} = 65000 + \frac{52000}{3} = \frac{195000}{3} + \frac{52000}{3} = \frac{247000}{3}$$ 7. **Rewrite left side:** $$\frac{325}{2} x^2 - 7850x + \frac{247000}{3} = 5625\pi (637.5 - x)$$ 8. **Expand right side:** $$5625\pi \times 637.5 - 5625\pi x = 3585937.5\pi - 5625\pi x$$ 9. **Bring all terms to one side:** $$\frac{325}{2} x^2 - 7850x + \frac{247000}{3} - 3585937.5\pi + 5625\pi x = 0$$ 10. **Group $x$ terms:** $$\frac{325}{2} x^2 + (-7850 + 5625\pi) x + \left(\frac{247000}{3} - 3585937.5\pi\right) = 0$$ This is a quadratic equation in standard form: $$ax^2 + bx + c = 0$$ where $$a = \frac{325}{2}, \quad b = -7850 + 5625\pi, \quad c = \frac{247000}{3} - 3585937.5\pi$$ 11. **Use quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ 12. **Final answer:** The solutions for $x$ are given by the quadratic formula above with the coefficients calculated. **Note:** Numerical approximation can be done if needed, but the exact form is preferred for clarity.