answer:To calculate how much money Selena will be left with after her indulgence at a 5-star hotel, we need to follow these steps: 1. Calculate the total cost for the steak meals: - The cost per plate of steak is 24, and she ordered 2 plates. - Therefore, the total cost for the steak meals is 24 times 2 = 48. 2. Calculate the total cost for the burgers: - The cost per burger is 3.5, and she ordered 2 burgers. - Therefore, the total cost for the burgers is 3.5 times 2 = 7. 3. Calculate the total cost for the cups of ice cream: - The cost per cup of ice cream is 2, and she ordered 3 cups. - Therefore, the total cost for the ice cream is 2 times 3 = 6. 4. Calculate the total amount spent at the hotel: - The total amount spent is the sum of the costs for the steak meals, burgers, and ice cream. - Therefore, the total amount spent is 48 + 7 + 6 = 61. 5. Calculate the amount Selena is left with: - Selena started with a tip of 99. - After spending 61 at the hotel, the amount she is left with is 99 - 61. Therefore, Selena is left with boxed{38}.

answer:Solution: (1) The inequality f(x) leqslant 0 is equivalent to |x-2| leqslant |2x+1|, which is equivalent to 3x^2 + 8x - 3 geqslant 0. Solving this, we get x geqslant frac{1}{3} or x leqslant -3, Therefore, the solution set of the inequality f(x) leqslant 0 is left{x | x geqslant frac{1}{3} text{ or } x leqslant -3right}, (2) The function f(x) = |x-2| - |2x+1| can be expressed as: f(x) = begin{cases} x + 3, & x < frac{1}{2} -3x + 1, & -frac{1}{2} leqslant x leqslant 2 -x - 3, & x > 2 end{cases} Therefore, the maximum value of f(x) is fleft(-frac{1}{2}right) = frac{5}{2}. Since for all x in mathbb{R}, f(x) - 2m^2 leqslant 4m always holds, it follows that 2m^2 + 4m geqslant frac{5}{2}, which is equivalent to 4m^2 + 8m - 5 geqslant 0, Solving this, we get m geqslant frac{1}{2} or m leqslant -frac{5}{2}, Therefore, the range of m is boxed{(-infty, -frac{5}{2}] cup [frac{1}{2}, +infty)}.

answer:Solution: (1) Let the equation of the ellipse C be frac{x^{2}}{a^{2}} + frac{y^{2}}{b^{2}} = 1 (a > b > 0), Since e = frac{c}{a} = frac{1}{2}, and it passes through the point Mleft(1, frac{3}{2}right), Therefore, frac{1}{4c^{2}} + frac{3}{4c^{2}} = 1, Solving this, we get c^{2} = 1, a^{2} = 4, b^{2} = 3, Thus, the equation of the ellipse C is boxed{frac{x^{2}}{4} + frac{y^{2}}{3} = 1}. (2) If there exists a line l that meets the condition, assuming the line l has an equation y = k(x-2) + 1, From begin{cases} frac{x^{2}}{4} + frac{y^{2}}{3} = 1 y = k(x-2) + 1 end{cases}, We get (3+4k^{2})x^{2} - 8k(2k-1)x + 16k^{2} - 16k - 8 = 0. Since line l intersects the ellipse C at two distinct points A, B, Let the coordinates of points A, B be (x_{1},y_{1}), (x_{2},y_{2}), Therefore, Delta = [-8k(2k-1)]^{2} - 4 cdot (3+4k^{2}) cdot (16k^{2} - 16k - 8) > 0. After simplification, we get 32(6k+3) > 0. Solving this, we find k > -frac{1}{2}, Also, x_{1} + x_{2} = frac{8k(2k-1)}{3+4k^{2}}, x_{1}x_{2} = frac{16k^{2} - 16k - 8}{3+4k^{2}}, Since overrightarrow{PA} cdot overrightarrow{PB} = overrightarrow{PM}^{2}, That is, ((x_{1}-2)(x_{2}-2) + (y_{1}-1)(y_{2}-1) = frac{5}{4}, Therefore, ((x_{1}-2)(x_{2}-2))(1+k^{2}) = |overrightarrow{PM}|^{2} = frac{5}{4}, That is, left[ x_{1}x_{2} - 2(x_{1} + x_{2}) + 4 right](1+k^{2}) = frac{5}{4}, Therefore, left[ frac{16k^{2} - 16k - 8}{3+4k^{2}} - 2 cdot frac{8k(2k-1)}{3+4k^{2}} + 4 right](1+k^{2}) = frac{4+4k^{2}}{3+4k^{2}} = frac{5}{4}, Solving this, we get k = pm frac{1}{2}. Since A, B are distinct points, Therefore, k = frac{1}{2}, Thus, there exists a line l that meets the condition, and its equation is boxed{y = frac{1}{2}x}.

answer:To find out how long it takes for the slower train to cross the faster train, we need to calculate the relative speed at which they are approaching each other and then determine the time it takes for them to cover the combined length of both trains. Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds. So, the relative speed is: Relative speed = Speed of Train 1 + Speed of Train 2 Relative speed = 110 km/hr + 90 km/hr Relative speed = 200 km/hr Now, we need to convert this speed into meters per second to be consistent with the lengths of the trains, which are given in kilometers. We know that 1 km/hr is equal to (1 km/hr) * (1000 m/km) / (3600 s/hr) = 5/18 m/s. So, the relative speed in meters per second is: Relative speed (m/s) = 200 km/hr * (5/18 m/s per km/hr) Relative speed (m/s) = (200 * 5/18) m/s Relative speed (m/s) = (1000/18) m/s Relative speed (m/s) ≈ 55.56 m/s The combined length of both trains is: Combined length = Length of Train 1 + Length of Train 2 Combined length = 1.10 km + 0.9 km Combined length = 2.00 km We need to convert this combined length into meters: Combined length (m) = 2.00 km * 1000 m/km Combined length (m) = 2000 m Now, we can calculate the time it takes for the slower train to cross the faster train using the formula: Time = Distance / Speed Time (s) = Combined length (m) / Relative speed (m/s) Time (s) = 2000 m / 55.56 m/s Time (s) ≈ 36 seconds Therefore, it takes approximately boxed{36} seconds for the slower train to cross the faster train.