answer:Given that x divided by (9p) is an odd prime number and the least value of x is 54, we can set up the equation: x / (9p) = odd prime number Since x is given to be 54, we can substitute that into the equation: 54 / (9p) = odd prime number Now, we can simplify the left side of the equation by dividing 54 by 9: 6 / p = odd prime number Since p is a prime number and the result must be an odd prime number, the smallest odd prime number is 3. Therefore, we can set the equation to: 6 / p = 3 Now, we can solve for p: p = 6 / 3 p = 2 However, p must be a prime number, and 2 is the only even prime number. Since the result must be an odd prime number, p cannot be 2. The next smallest odd prime number is 3. Let's check if p can be 3: 6 / p = 3 6 / 3 = 3 This works because 3 is an odd prime number. Therefore, the least value of p is 3. Now, we can find the least value of x using the original equation with p = 3: x / (9p) = odd prime number x / (9 * 3) = 3 x / 27 = 3 Now, we can solve for x: x = 3 * 27 x = 81 Therefore, the least value of x is boxed{81} .

answer:First, we need to convert the speed of the train and the man from km/h to m/s to make the units consistent. To convert km/h to m/s, we use the conversion factor: 1 km/h = 1000 m / 3600 s. Speed of the train in m/s = 60 km/h * (1000 m / 3600 s) = 16.67 m/s Speed of the man in m/s = 6 km/h * (1000 m / 3600 s) = 1.67 m/s Since the train and the man are moving in opposite directions, their relative speed is the sum of their individual speeds. Relative speed = Speed of the train + Speed of the man Relative speed = 16.67 m/s + 1.67 m/s = 18.34 m/s Now, we need to calculate the time it takes for the train to pass the man. The time can be calculated using the formula: Time = Distance / Relative speed The distance to be covered is the length of the train, which is 550 meters. Time = 550 m / 18.34 m/s ≈ 29.99 seconds Therefore, it will take approximately boxed{30} seconds for the train to pass the man.

answer:Solution: (Ⅰ) Given the right focus is F(sqrt{3}, 0), we have c = sqrt{3}. Since the distance from point F to one endpoint of the minor axis is equal to the focal distance, we have a = 2c, which means a = 2sqrt{3}. Then b^2 = a^2 - c^2 = 9. Therefore, the equation of ellipse C is frac{x^2}{12} + frac{y^2}{9} = 1; (Ⅱ) Let the point A(x_0, y_0) (x_0 > 0, y_0 > 0), then y_0 = kx_0, Let AB intersect the x-axis at point D. By symmetry, we know: S_{triangle OAB} = 2S_{triangle OAD} = 2 times frac{1}{2}x_0y_0 = kx_0^2, From begin{cases} y_0 = kx_0 frac{x_0^2}{12} + frac{y_0^2}{9} = 1 end{cases}, we get x_0^2 = frac{36}{3 + 4k^2}, Thus, S_{triangle OAB} = k frac{36}{3 + 4k^2} = frac{36}{frac{3}{k} + 4k} leq frac{36}{2sqrt{frac{3}{k} cdot 4k}} = 3sqrt{3}, Equality holds if and only if frac{3}{k} = 4k, k = frac{sqrt{3}}{2}, Therefore, the maximum value of the area of triangle OAB is boxed{3sqrt{3}}.

answer:1. **Percentage Change Calculations**: - A 20% increase in population means the population becomes 120% of its original, expressed as a multiplication by frac{6}{5}. - A 30% decrease means the population becomes 70% of the previous amount, expressed as a multiplication by frac{7}{10}. 2. **Cumulative Change Calculation**: - Apply the changes over four years: [ text{Final Population} = text{Initial Population} times frac{6}{5} times frac{7}{10} times frac{6}{5} times frac{7}{10} ] - Simplifying, we find: [ frac{6}{5} times frac{7}{10} = frac{42}{50} = frac{21}{25} ] [ frac{6}{5} times frac{7}{10} = frac{42}{50} = frac{21}{25} ] [ frac{21}{25} times frac{21}{25} = frac{441}{625} ] 3. **Deriving the Net Percentage Change**: - Determine the net change relative to the initial population: [ text{Net Change} = left(frac{441}{625} - 1right) times 100% ] [ = left(frac{441 - 625}{625}right) times 100% ] [ = frac{-184}{625} times 100% ] [ = -29.44% ] Since the result is not easily rounded to a simple integer, we calculate it to the nearest percent as approximately -29%. Conclusion: - The final population after these changes is frac{441}{625} times the original population, resulting in a net change of approximately -29.44%, and when rounded, -29%. The final answer is boxed{textbf{(A)} -29}