answer:First, we need to convert the speed from kilometers per hour (km/hr) to meters per second (m/s) because the length of the train is given in meters. The conversion factor is: 1 km/hr = 1000 m / 3600 s So, 36 km/hr = 36 * (1000 m / 3600 s) = 10 m/s Now, we have the speed of the train in meters per second. Next, we need to calculate the time it takes for the train to pass a tree. Since the length of the train is 280 meters and it's moving at a speed of 10 meters per second, we can use the formula: Time = Distance / Speed Time = 280 m / 10 m/s = 28 seconds Therefore, it will take the train boxed{28} seconds to pass the tree.

answer:: 1. **Identifying the Number of Lines**: There are 31 horizontal and 31 vertical lines on the grid. 2. **Choosing Horizontal Sides of the Rectangles**: - The number of ways to choose 2 out of the 31 horizontal lines is given by: [ binom{31}{2} = frac{31 times 30}{2} = 465 ] 3. **Choosing Vertical Sides of the Rectangles**: - Similarly, the number of ways to choose 2 out of the 31 vertical lines is also: [ binom{31}{2} = frac{31 times 30}{2} = 465 ] 4. **Total Number of Rectangles**: - The total number of rectangles formed by choosing 2 horizontal and 2 vertical lines is: [ 465 times 465 = 216225 ] 5. **Counting the Squares**: - Squares are special cases of rectangles where the distance between the chosen horizontal lines is equal to the distance between the chosen vertical lines. 6. **Defining Distance Between Lines**: - Let the smallest distance between adjacent lines be defined as 1 unit. - Then the distances must occur in multiples of 1 and 2 units due to the 1:2 ratio provided. 7. **Possible Sizes of Squares**: - The minimum side length of a square is 1 unit, and the maximum side length is formed by the distance between the furthest parallel lines, which is 30 units. 8. **Calculations of Squares Based on Side Length**: - For sides of length 3n: The count is (31 - 2n) possible positions horizontally, and the same count vertically. So, the total count is: [ sum_{n=1}^{15} (31 - 2n)^2 ] - For sides of length 3n-1: The count is (16 - n) horizontally and vertically. So, the total count is: [ sum_{n=1}^{15} (16 - n)^2 ] - For sides of length 3n+1: The count is (15 - n) horizontally and vertically. So, total count is: [ sum_{n=1}^{15} (15 - n)^2 ] 9. **Total Calculations**: - Adding the calculated totals leads to: [ A = sum_{n=1,3,5,ldots,29} n^2 ] [ B = sum_{n=0,2,4,ldots,14} (15-n)^2 ] - Finding these sums using the sum of squares formula: [ S_n = frac{n(n+1)(2n+1)}{6} ] - Thus, the total number of squares is: [ N = S_{30} - 2 cdot S_{15} ] 10. **Final Calculation**: [ N = frac{30 times 31 times 61}{6} - 2 times frac{15 times 16 times 31}{6} = 9455 - 2 times 1240 = 6975 ] 11. **Conclusion**: [ boxed{N = 6975} ]

answer:Starting with the equation: [ 2 + 3iw = 4 - 2iw ] 1. Bring all terms involving w to one side and constant terms to the other side: [ 3iw + 2iw = 4 - 2 ] [ 5iw = 2 ] 2. Solve for w: [ w = frac{2}{5i} ] 3. Simplify w by multiplying the numerator and denominator by -i: [ w = frac{2}{5i} cdot frac{-i}{-i} = frac{-2i}{5} ] [ w = boxed{frac{-2i}{5}} ]

answer:In the Cartesian coordinate system, the coordinates of a point are given as (x, y). For a point P(2,1), its coordinates with respect to the origin are determined by reflecting it across the origin. This reflection changes the sign of both the x and y coordinates. Starting with the point P(2,1), we reflect it across the origin to find its symmetric point. The reflection process involves changing the sign of both coordinates: - The x coordinate changes from 2 to -2. - The y coordinate changes from 1 to -1. Therefore, the coordinates of the point symmetric to P(2,1) with respect to the origin are (-2,-1). So, the final answer is boxed{(-2,-1)}.