1. **Minimize area of a closed cylinder with volume 250\pi ml** Given volume $V = 250\pi$, volume formula: $$V = \pi r^2 h = 250\pi \implies r^2 h = 250$$ Surface area $A$ (including top and bottom) is: $$A = 2\pi r^2 + 2\pi r h$$ Express $h$: $$h = \frac{250}{r^2}$$ Rewrite $A$: $$A = 2\pi r^2 + 2\pi r \cdot \frac{250}{r^2} = 2\pi r^2 + \frac{500\pi}{r}$$ 2. **Minimize $A$ by differentiation** $$\frac{dA}{dr} = 4\pi r - \frac{500\pi}{r^2} = 0$$ Multiply both sides by $r^2$: $$4\pi r^3 = 500\pi$$ Divide both sides by $4\pi$: $$r^3 = \frac{500}{4} = 125$$ 3. **Solve for radius $r$** $$r = \sqrt[3]{125} = 5$$ Answer: The radius should be $5$ ml. 4. **Solve the system:** $\frac{x+y}{3} - \frac{x-y}{4} = \frac{2}{3}$ Multiply by 12 (common denominator): $$4(x+y) - 3(x-y) = 8$$ $$4x + 4y - 3x + 3y = 8$$ $$x + 7y = 8$$ (1) $\frac{2x-3}{3} - \frac{2y+3}{4} = \frac{19}{2}$ Multiply by 12: $$4(2x-3) - 3(2y+3) = 114$$ $$8x - 12 - 6y - 9 = 114$$ $$8x - 6y - 21 = 114$$ $$8x - 6y = 135$$ (2) 5. **Solve system (1) and (2):** From (1): $$x = 8 - 7y$$ Substitute into (2): $$8(8 - 7y) - 6y = 135$$ $$64 - 56y - 6y = 135$$ $$64 - 62y = 135$$ $$-62y = 71$$ $$y = -\frac{71}{62}$$ Calculate $x$: $$x = 8 - 7 \times \left(-\frac{71}{62}\right) = 8 + \frac{497}{62} = \frac{496}{62} + \frac{497}{62} = \frac{993}{62}$$ Answer: $$x = \frac{993}{62}, y = -\frac{71}{62}$$ 6. **Club members contribution problem:** Let number of members be $n$. Original contribution each: $$\frac{480000}{n}$$ Four members withdrew; remaining members = $n - 4$. Each remaining member pays $$\frac{480000}{n} + 20000$$ Equation: $$(n - 4) \left( \frac{480000}{n} + 20000 \right) = 480000$$ Expand: $$(n - 4) \frac{480000}{n} + 20000(n - 4) = 480000$$ $$480000 - \frac{1920000}{n} + 20000n - 80000 = 480000$$ Simplify: $$400000 + 20000n - \frac{1920000}{n} = 480000$$ Subtract 480000 both sides: $$20000n - \frac{1920000}{n} = 80000$$ Multiply by $n$: $$20000 n^2 - 1920000 = 80000 n$$ Rearranged: $$20000 n^2 - 80000 n - 1920000 = 0$$ Divide by 4000: $$5 n^2 - 20 n - 480 = 0$$ 7. **Solve quadratic:** $$n^2 - 4 n - 96 = 0$$ Use quadratic formula: $$n = \frac{4 \pm \sqrt{16 + 384}}{2} = \frac{4 \pm \sqrt{400}}{2} = \frac{4 \pm 20}{2}$$ Possible values: $$n=12\ (positive)$$ or $$n = -8\ (discard)$$ Original contribution: $$\frac{480000}{12} = 40000$$ Answer: Each member originally contributed 40000. 8. **Calculate log 50 given log 3, log 5, and log 2:** $$\log 50 = \log (5 \times 10) = \log 5 + \log 10$$ $$\log 10 = 1$$ So: $$\log 50 = 0.6990 + 1 = 1.6990$$ Final answers: - Radius of tin: $5$ - $x = \frac{993}{62}$, $y = -\frac{71}{62}$ - Original contribution: $40000$ - $\log 50 = 1.6990$