Appearance
question:At a certain crosswalk, the pedestrian signal alternates between red and green lights, with the red light lasting for 40 seconds. If a pedestrian arrives at the crosswalk and encounters a red light, the probability that they need to wait at least 15 seconds for the green light to appear is ______.
answer:Since the red light lasts for 40 seconds, and one needs to wait at least 15 seconds for the green light, it means that if a pedestrian arrives during the first 25 seconds of the red light, they will encounter it. Therefore, the probability of having to wait at least 15 seconds for the green light is dfrac{25}{40} = dfrac{5}{8}. Hence, the answer is boxed{dfrac{5}{8}}. By determining that a pedestrian arriving in the first 25 seconds encounters the red light, we can find the probability of having to wait at least 15 seconds for the green light. This question tests the calculation of probability, geometric probability models, and basic computational skills.
question:Marco, his father, and his sister went strawberry picking. Marco picked 1 pound of strawberries, then an additional 3 kilograms. His sister picked one and a half times the amount Marco did. Their father picked twice as much as his daughter. If 1 kilogram equals about 2.2 pounds, how much did each of them pick in pounds?
answer:First, let's convert the 3 kilograms that Marco picked into pounds. Since 1 kilogram equals 2.2 pounds: 3 kilograms * 2.2 pounds/kilogram = 6.6 pounds Now, let's add the 1 pound of strawberries Marco initially picked: 1 pound + 6.6 pounds = 7.6 pounds Marco picked a total of 7.6 pounds of strawberries. Next, let's calculate how much Marco's sister picked. She picked one and a half times the amount Marco did: 1.5 * 7.6 pounds = 11.4 pounds Marco's sister picked 11.4 pounds of strawberries. Finally, let's calculate how much their father picked. He picked twice as much as his daughter: 2 * 11.4 pounds = 22.8 pounds Their father picked 22.8 pounds of strawberries. In summary: - Marco picked 7.6 pounds of strawberries. - His sister picked 11.4 pounds of strawberries. - Their father picked boxed{22.8} pounds of strawberries.
question:Given the hyperbola x^{2} frac{y^{2}}{3}=1 with left and right foci F_{1} and F_{2}, respectively, and eccentricity e. If a point P on the hyperbola satisfies frac{sin angle PF_{2}F_{1}}{sin angle PF_{1}F_{2}}=e, find the value of overrightarrow{F_{2}P}cdot overrightarrow{F_{2}F_{1}}. A: 3 B: 2 C: 3 D: 2
answer:The hyperbola has a=1, b=sqrt{3}, and c=sqrt{1+3}=2. Thus, e=frac{c}{a}=2. The foci are F_{1}(-2,0) and F_{2}(2,0), and point P is on the right branch of the hyperbola. By the Sine Law, |PF_{1}|=2|PF_{2}|. According to the definition of a hyperbola, |PF_{1}|-|PF_{2}|=2a=2. Solving for |PF_{1}| and |PF_{2}|, we get |PF_{1}|=4 and |PF_{2}|=2. In triangle PF_{2}F_{1}, using the Cosine Law, we have cos angle PF_{2}F_{1}=frac{2^{2}+4^{2}-4^{2}}{2times 2times 4}=frac{1}{4}. Consequently, overrightarrow{F_{2}P}cdot overrightarrow{F_{2}F_{1}}=|overrightarrow{F_{2}P}|cdot |overrightarrow{F_{2}F_{1}}|cdot cos angle PF_{2}F_{1}=2times 4times frac{1}{4}=boxed{2}. Therefore, the answer is B.
question:Frank went to a shop to buy some food for his breakfast. He bought 10 buns for 0.1 each, some bottles of milk, and a carton of eggs, which was three times more expensive than one bottle of milk. Frank paid 11 for his breakfast shopping. How much did each bottle of milk cost?
answer:Let's denote the cost of one bottle of milk as ( x ) dollars. Frank bought 10 buns for 0.1 each, so the total cost for the buns is ( 10 times 0.1 = 1 ) dollar. The carton of eggs was three times more expensive than one bottle of milk, so the cost of the carton of eggs is ( 3x ) dollars. The total cost for Frank's breakfast shopping is the sum of the cost of the buns, the milk, and the eggs, which is 11. So we can write the equation: [ 1 + x + 3x = 11 ] Combining like terms, we get: [ 4x + 1 = 11 ] Subtracting 1 from both sides, we get: [ 4x = 10 ] Dividing both sides by 4, we get: [ x = frac{10}{4} ] [ x = 2.5 ] Therefore, each bottle of milk cost boxed{2.5} .