answer:To find the speed of the train, we first need to determine the total distance the train travels while crossing the bridge. This distance is the length of the train plus the length of the bridge. Total distance = Length of the train + Length of the bridge Total distance = 120 meters + 255 meters Total distance = 375 meters The train covers this distance in 30 seconds. To find the speed in meters per second (m/s), we use the formula: Speed = Distance / Time Speed = 375 meters / 30 seconds Speed = 12.5 m/s To convert this speed to kilometers per hour (km/hr), we use the conversion factor that 1 m/s is equivalent to 3.6 km/hr. Speed in km/hr = Speed in m/s * 3.6 Speed in km/hr = 12.5 m/s * 3.6 Speed in km/hr = 45 km/hr Therefore, the speed of the train is boxed{45} km/hr.

answer:Given: [ f(x) = x^2 - 12x + 36 ] and the set ( A = { a mid 1 leq a leq 10, a in mathbf{N} } ). First, expand the given function: [ f(x) = x^2 - 12x + 36 = (x-6)^2 ] Thus, rewrite the function: [ f(x) = (x-6)^2 ] Define set ( B ): [ B = { b mid b = f(a), a in A } = { b mid b = (a-6)^2, a in A } ] Evaluate ( f(a) ) for each ( a in A ): - For ( a = 1 ): [ f(1) = (1-6)^2 = (-5)^2 = 25 ] - For ( a = 2 ): [ f(2) = (2-6)^2 = (-4)^2 = 16 ] - For ( a = 3 ): [ f(3) = (3-6)^2 = (-3)^2 = 9 ] - For ( a = 4 ): [ f(4) = (4-6)^2 = (-2)^2 = 4 ] - For ( a = 5 ): [ f(5) = (5-6)^2 = (-1)^2 = 1 ] - For ( a = 6 ): [ f(6) = (6-6)^2 = 0^2 = 0 ] - For ( a = 7 ): [ f(7) = (7-6)^2 = 1^2 = 1 ] - For ( a = 8 ): [ f(8) = (8-6)^2 = 2^2 = 4 ] - For ( a = 9 ): [ f(9) = (9-6)^2 = 3^2 = 9 ] - For ( a = 10 ): [ f(10) = (10-6)^2 = 4^2 = 16 ] Thus, we form the set ( B ): [ B = { 25, 16, 9, 4, 1, 0 } ] Next, find the intersection of ( A ) and ( B ): [ A = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } ] [ B = { 25, 16, 9, 4, 1, 0 } ] Intersection ( A cap B ) results in: [ A cap B = { 1, 4, 9 } ] Conclusion: [ boxed{{1, 4, 9}} ]

answer:1. **Identifying Points and Order**: The points given are (0,0), (5,0), (5,5), (0,5), (0,3), (3,3), (3,0), and back to (0,0). These points form a complex polygon. 2. **Breaking into Simpler Shapes**: The polygon can be divided into a rectangle (0,0) to (5,5) and a cut-out smaller rectangle (0,3) to (3,5). 3. **Calculating Areas**: - Area of large rectangle = length × width = 5 units × 5 units = 25 square units. - Area of cut-out rectangle = length × width = 3 units × 2 units = 6 square units. - Total enclosed area = Area of large rectangle - Area of cut-out rectangle = 25 square units - 6 square units = 19 square units. The enclosed area of the polygon is boxed{19} square units.

answer:Sofia asked a total of 185 students who suggested mashed potatoes and 125 students who suggested bacon. To find the total number of students she asked, we simply add these two numbers together: 185 (students who suggested mashed potatoes) + 125 (students who suggested bacon) = 310 students So, Sofia asked boxed{310} students for suggestions.