1. Solve the equation: $$\frac{m}{2} + \frac{m}{3} + 3 = 2 + \frac{m}{6}$$ Step 1: Find a common denominator for the fractions on the left side. The denominators are 2, 3, and 6. The least common denominator (LCD) is 6. Step 2: Rewrite each fraction with denominator 6: $$\frac{m}{2} = \frac{3m}{6}, \quad \frac{m}{3} = \frac{2m}{6}$$ Step 3: Substitute back into the equation: $$\frac{3m}{6} + \frac{2m}{6} + 3 = 2 + \frac{m}{6}$$ Step 4: Combine like terms: $$\frac{5m}{6} + 3 = 2 + \frac{m}{6}$$ Step 5: Subtract $$\frac{m}{6}$$ from both sides: $$\frac{5m}{6} - \frac{m}{6} + 3 = 2$$ Simplify: $$\frac{4m}{6} + 3 = 2$$ Step 6: Simplify fraction: $$\frac{2m}{3} + 3 = 2$$ Step 7: Subtract 3 from both sides: $$\frac{2m}{3} = -1$$ Step 8: Multiply both sides by $$\frac{3}{2}$$: $$m = -1 \times \frac{3}{2} = -\frac{3}{2}$$ Answer: $$m = -\frac{3}{2}$$ or $$-1.5$$ --- 2. Find the value of $$\sqrt{\frac{(15.03)^2}{29.21 \times 10.52}}$$ Step 1: Calculate the numerator: $$(15.03)^2 = 15.03 \times 15.03 = 225.9009$$ Step 2: Calculate the denominator: $$29.21 \times 10.52 = 307.3492$$ Step 3: Compute the fraction: $$\frac{225.9009}{307.3492} \approx 0.7353$$ Step 4: Take the square root: $$\sqrt{0.7353} \approx 0.8576$$ Answer: Approximately $$0.858$$ --- 3. Solve the system: $$y = x + 2$$ $$x^2 + y^2 = 28$$ Step 1: Substitute $$y = x + 2$$ into the second equation: $$x^2 + (x+2)^2 = 28$$ Step 2: Expand: $$x^2 + (x^2 + 4x + 4) = 28$$ Step 3: Simplify: $$2x^2 + 4x + 4 = 28$$ Step 4: Subtract 28 from both sides: $$2x^2 + 4x + 4 - 28 = 0$$ Simplify: $$2x^2 + 4x - 24 = 0$$ Step 5: Divide entire equation by 2: $$x^2 + 2x - 12 = 0$$ Step 6: Factor or use quadratic formula. Factors of -12 that sum to 2 are 4 and -3, so: $$(x + 4)(x - 3) = 0$$ Step 7: Solve for $$x$$: $$x = -4 \quad \text{or} \quad x = 3$$ Step 8: Find corresponding $$y$$: For $$x = -4$$: $$y = -4 + 2 = -2$$ For $$x = 3$$: $$y = 3 + 2 = 5$$ Answer: Solutions are $$(x, y) = (-4, -2)$$ and $$(3, 5)$$ --- 4. Determine value of $$y$$: $$\sqrt{\frac{y+2}{3-y}} = -15 + 10$$ Step 1: Simplify the right side: $$-15 + 10 = -5$$ Step 2: Note that the square root function outputs non-negative values. Since right side is negative (-5), equation has no real solution. Answer: No real solutions. --- 5. Find force $$F$$ given: $$F = G \frac{m_1 m_2}{d^2}$$ $$G = 6.67 \times 10^{-11}$$ $$m_1 = 7.36$$ $$m_2 = 15.5$$ $$d = 22.6$$ Step 1: Calculate $$d^2$$: $$22.6^2 = 510.76$$ Step 2: Calculate numerator: $$G \times m_1 \times m_2 = 6.67 \times 10^{-11} \times 7.36 \times 15.5$$ Calculate constant part: $$7.36 \times 15.5 = 114.08$$ Step 3: Multiply by $$G$$: $$6.67 \times 10^{-11} \times 114.08 = 7.6067 \times 10^{-9}$$ Step 4: Divide by $$d^2$$: $$F = \frac{7.6067 \times 10^{-9}}{510.76} = 1.49 \times 10^{-11}$$ Step 5: Round to 3 decimal places in standard form: $$F = 1.489 \times 10^{-11}$$ Answer: $$F = 1.489 \times 10^{-11}$$ Newtons --- 6. Triangle area given: $$a=3.60, b=4.00, c=5.20$$ Formula: $$s = \frac{a+b+c}{2}$$ $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ Step 1: Calculate $$s$$: $$s = \frac{3.60 + 4.00 + 5.20}{2} = \frac{12.80}{2} = 6.40$$ Step 2: Compute each term: $$s - a = 6.40 - 3.60 = 2.80$$ $$s - b = 6.40 - 4.00 = 2.40$$ $$s - c = 6.40 - 5.20 = 1.20$$ Step 3: Calculate area: $$A = \sqrt{6.40 \times 2.80 \times 2.40 \times 1.20}$$ Calculate product: $$6.40 \times 2.80 = 17.92$$ $$17.92 \times 2.40 = 43.008$$ $$43.008 \times 1.20 = 51.6096$$ Step 4: Square root: $$A = \sqrt{51.6096} \approx 7.186$$ Answer: Area $$\approx 7.186 \text{ cm}^2$$ --- 7. Find constants $$a$$ and $$b$$ in: $$F = aL + b$$ Given: When $$F=5.6$$, $$L=8.0$$ When $$F=4.4$$, $$L=2.0$$ Step 1: Write equations: $$5.6 = 8a + b$$ $$4.4 = 2a + b$$ Step 2: Subtract second equation from first: $$(5.6 - 4.4) = (8a - 2a) + (b - b)$$ $$1.2 = 6a$$ Step 3: Solve for $$a$$: $$a = \frac{1.2}{6} = 0.2$$ Step 4: Substitute $$a$$ into second equation: $$4.4 = 2(0.2) + b = 0.4 + b$$ Step 5: Solve for $$b$$: $$b = 4.4 - 0.4 = 4.0$$ Step 6: Find $$F$$ when $$L=6.5$$: $$F = 0.2 \times 6.5 + 4.0 = 1.3 + 4.0 = 5.3$$ Answer: $$a = 0.2,$$ $$b = 4.0,$$ $$F = 5.3 \text{ when } L=6.5$$