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question:The graph of the function y=sin left( 2xfrac{pi }{4} right) is translated frac{pi }{6} units to the left to obtain the graph of the function fleft( x right). Find the value of fleft( frac{pi }{12} right)=( ) A: frac{sqrt{2}+sqrt{6}}{4} B: frac{sqrt{3}+sqrt{6}}{4} C: frac{sqrt{3}}{2} D: frac{sqrt{2}}{2}
answer:**Analysis** This problem tests the understanding of transformations of trigonometric function graphs and evaluation of these functions. It is of medium difficulty. **Step-by-step Solution** 1. Identify the transformation: The function f(x) is obtained by shifting the graph of y=sin left( 2x-frac{pi }{4} right) to the left by frac{pi }{6} units. 2. Express f(x) mathematically: This left translation results in the function f(x) = sin left( 2left( x + frac{pi}{6} right) - frac{pi}{4} right) = sin left( 2x + frac{pi}{12} right). 3. Evaluate fleft( frac{pi }{12} right): Substitute x = frac{pi }{12} into the function obtained in step 2. fleft( frac{pi }{12} right) = sin left( 2left( frac{pi }{12} right) + frac{pi}{12} right) = sin left( frac{pi }{4} right) = boxed{frac{sqrt{2}}{2}}.
question:On a doubletrack railway, there is a passenger train running at 72 kilometers per hour. The driver of the passenger train notices an oncoming freight train running at 90 kilometers per hour. It takes 8 seconds for the freight train to pass by him. What is the length of the freight train?
answer:When the two trains pass each other, the distance they cover is the length of the freight train. The relative speed is the sum of the speeds of the two trains. To find the length of the freight train, we multiply the sum of the speeds of the two trains by the time it takes for them to pass each other, which is 8 seconds. The key to solving this problem is to understand that the relative speed of the two trains is the sum of their speeds. Let's calculate the length of the freight train: - First, convert the speeds of the trains into meters per second. The passenger train runs at 72 times frac{1000}{3600} = 20 m/s, and the freight train runs at 90 times frac{1000}{3600} = 25 m/s. - The relative speed of the two trains is 20 + 25 = 45 m/s. - The length of the freight train, which is the distance it travels relative to the passenger train in 8 seconds, is 45 times 8 = 360 meters. Therefore, the length of the freight train is boxed{360 text{ meters}}.
question:Pizzas are sized by diameter. Calculate the percent increase in area if a pizza increases from an 8inch pizza to a 14inch pizza.
answer:1. **Calculate the area of the 8-inch pizza**: The radius of the 8-inch pizza is 4 inches. Hence, its area is: [ A = pi r^2 = pi (4^2) = 16pi text{ square inches}. ] 2. **Calculate the area of the 14-inch pizza**: The radius of the 14-inch pizza is 7 inches. Hence, its area is: [ A = pi r^2 = pi (7^2) = 49pi text{ square inches}. ] 3. **Determine the increase in area**: [ text{Increase in area} = 49pi - 16pi = 33pi text{ square inches}. ] 4. **Calculate the percent increase**: [ text{Percent increase} = left(frac{33pi}{16pi}right) times 100 = left(frac{33}{16}right) times 100 approx 206.25%. ] Thus, the percent increase in area is (boxed{206.25%}).
question:Store A and Store B sell the same product and earn the same profit. The selling price of Store A is 10 yuan lower than the maximum market price, and the profit is 10% of the selling price. The selling price of Store B is 20 yuan lower than the maximum price, and the profit is 20% of the selling price. What is the maximum price of the product? A: 30 yuan B: 40 yuan C: 70 yuan D: 100 yu
answer:Let the cost price of the product be x yuan. According to the problem, we can set up the equation: (x - 10) times 10% = (x - 20) times 20%, Solving this equation, we get: x = 30. Therefore, the maximum price of the product is 30 yuan. Hence, the answer is boxed{A}.