1. Solve equation: $$\frac{m}{2} + \frac{m}{3} + 3 = 2 + \frac{m}{6}$$ Multiply both sides by 6 (LCM of denominators) to clear fractions: $$6\times\left(\frac{m}{2} + \frac{m}{3} + 3\right) = 6\times\left(2 + \frac{m}{6}\right)$$ Which gives: $$3m + 2m + 18 = 12 + m$$ Simplify: $$5m + 18 = 12 + m$$ Bring terms with m to one side: $$5m - m = 12 - 18$$ $$4m = -6$$ Divide both sides by 4: $$m = -\frac{6}{4} = -\frac{3}{2} = -1.5$$ 2. Find the value: $$\sqrt{\frac{(15.62)^2}{29.21 \times \sqrt{10.52}}}$$ Calculate inside the square root stepwise: $$15.62^2 = 243.9844$$ Calculate $$\sqrt{10.52} \approx 3.244$$ Calculate denominator: $$29.21 \times 3.244 \approx 94.77$$ Calculate fraction: $$\frac{243.9844}{94.77} \approx 2.575$$ Finally, square root: $$\sqrt{2.575} \approx 1.605$$ 3. Solve system: $$y = x + 2$$ $$x^2 + y^2 = 28$$ Substitute $$y$$: $$x^2 + (x + 2)^2 = 28$$ Expand: $$x^2 + x^2 + 4x + 4 = 28$$ Simplify: $$2x^2 + 4x + 4 = 28$$ $$2x^2 + 4x + 4 - 28 = 0$$ $$2x^2 + 4x - 24 = 0$$ Divide by 2: $$x^2 + 2x - 12 = 0$$ Factor or use quadratic formula: $$x = \frac{-2 \pm \sqrt{4 + 48}}{2} = \frac{-2 \pm \sqrt{52}}{2} = \frac{-2 \pm 2\sqrt{13}}{2}$$ Simplify: $$x = -1 \pm \sqrt{13}$$ Find $$y$$ values: $$y = x + 2$$ So, $$y = -1 + \sqrt{13} + 2 = 1 + \sqrt{13}$$ or $$y = -1 - \sqrt{13} + 2 = 1 - \sqrt{13}$$ 4. Solve for $$y$$: $$\sqrt{\frac{y+2}{3-y}} = -15 + 10$$ Simplify right side: $$-15 + 10 = -5$$ Note the square root function always non-negative, so no solution since left side $$\geq 0$$, right side $$< 0$$. Therefore, no real solutions. 5. Find force $$F$$ given: $$F = G^{\frac{m_1 m_2}{d^2}}$$ Where $$G = 6.67 \times 10^{-11}$$, $$m_1 = 7.36$$, $$m_2 = 15.5$$, $$d = 22.6$$ Calculate exponent: $$\frac{m_1 m_2}{d^2} = \frac{7.36 \times 15.5}{(22.6)^2} = \frac{114.08}{510.76} \approx 0.2233$$ Calculate $$F$$: $$F = (6.67 \times 10^{-11})^{0.2233} = e^{0.2233 \ln(6.67 \times 10^{-11})}$$ Calculate natural log: $$\ln(6.67 \times 10^{-11}) = \ln(6.67) + \ln(10^{-11}) \approx 1.897 - 11 \times 2.303 = 1.897 - 25.333 = -23.436$$ Calculate exponent: $$0.2233 \times (-23.436) = -5.237$$ Calculate $$F$$: $$F = e^{-5.237} \approx 0.00531$$ Express in standard form: $$F \approx 5.310 \times 10^{-3}$$ 6. Triangle area: Given $$a = 3.60$$, $$b = 4.00$$, $$c = 5.20$$ Calculate semi-perimeter: $$s = \frac{a+b+c}{2} = \frac{3.60 + 4.00 + 5.20}{2} = \frac{12.8}{2} = 6.4$$ Calculate area: $$A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{6.4(6.4 - 3.6)(6.4 - 4.0)(6.4 - 5.2)}$$ Calculate: $$= \sqrt{6.4 \times 2.8 \times 2.4 \times 1.2} = \sqrt{51.9936} \approx 7.212$$ 7. Find constants $$a$$ and $$b$$ in $$F = aL + b$$: Two data points: When $$F=5.6, L=8.0$$ When $$F=4.4, L=2.0$$ Set up equations: $$5.6 = 8a + b$$ $$4.4 = 2a + b$$ Subtract second from first: $$5.6 - 4.4 = 8a - 2a + b - b$$ $$1.2 = 6a$$ $$a = 0.2$$ Substitute $$a$$ back: $$4.4 = 2(0.2) + b \Rightarrow b = 4.4 - 0.4 = 4.0$$ Find $$F$$ when $$L=6.5$$: $$F = 0.2 \times 6.5 + 4.0 = 1.3 + 4.0 = 5.3$$