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question:Given the function f(x) = sin(2x + frac{π}{3}), g(x) = sinx, to obtain the graph of the function y = g(x), you only need to take all the points on the graph of the function y = f(x) and: A: Shrink the xcoordinate to frac{1}{2} of its original length, then shift it to the right by frac{π}{6} units. B: Shrink the xcoordinate to frac{1}{2} of its original length, then shift it to the right by frac{π}{3} units. C: Stretch the xcoordinate to twice its original length, then shift it to the right by frac{π}{6} units. D: Stretch the xcoordinate to twice its original length, then shift it to the right by frac{π}{3} units.
answer:The function is f(x) = sin(2x + frac{π}{3}). First, stretch the x-coordinate to twice its original length to get: y = sin(x + frac{π}{3}). Then, shift it to the right by frac{π}{3} units to get g(x) = sinx. Therefore, the answer is boxed{text{D}}. This solution uses the identity transformation and the translation and scaling transformations of trigonometric function relationships. This problem tests the student's computational and transformational abilities and is a basic problem type.
question:Noa scored 30 points to win a bowl, and Phillip scores a certain multiple of Noa's points. The total number of points Noa and Phillip scored to win the bowl is 90. What is the ratio of Phillip's points to Noa's points?
answer:Let's denote the multiple by which Phillip's score exceeds Noa's as ( x ). So Phillip's score is ( 30x ). The total number of points scored by Noa and Phillip is given as 90. Therefore, we can write the equation: [ Noa's points + Phillip's points = Total points ] [ 30 + 30x = 90 ] Now, we solve for ( x ): [ 30x = 90 - 30 ] [ 30x = 60 ] [ x = frac{60}{30} ] [ x = 2 ] So Phillip scored twice as many points as Noa. Therefore, the ratio of Phillip's points to Noa's points is ( boxed{2:1} ).
question:a sevenstory building, house spirits live. The elevator travels between the first and last floors, stopping at each floor. On each floor, starting from the first, one house spirit entered the elevator, but no one exited. When the thousandth house spirit entered the elevator, it stopped. On which floor did this happen? Explain your answer.
answer:1. Let's begin by understanding the problem: The elevator makes stops at each floor of a seven-story building to let in one house spirit at each floor, but no one leaves. Since the elevator moves between the first and seventh floor, stopping at all floors, we will analyze how house spirits enter over multiple trips. 2. First, establish the number of house spirits that enter the elevator in a complete up-and-down trip. During an upward trip from the 1st to the 7th floor, one house spirit gets on at each of the 7 floors (since spirits enter the elevator on every floor). On the return trip from the 7th back to the 1st floor, house spirits again enter the elevator at every floor. However, recall that no spirits exit the elevator. Specifically: - On the 1st floor, only one house spirit enters. - From the 2nd to the 6th floors, two spirits enter: one during the upward trip and one during the downward trip. - On the 7th floor, only one house spirit enters. Therefore, the total number of house spirits entering the elevator in one complete cycle is calculated as: [ 1 + 2 times 5 + 1 = 1 + 10 + 1 = 12 text{ spirits} ] 3. Determine how many complete cycles the elevator makes to approach the total of 1000 house spirits. To calculate the number of complete cycles: [ leftlfloor frac{1000}{12} rightrfloor = leftlfloor 83.33ldots rightrfloor = 83 text{ complete cycles} ] 4. Calculate the total number of house spirits after 83 complete cycles: [ 83 times 12 = 996 text{ spirits} ] 5. Since the elevator must accommodate 1000 house spirits, the remaining spirits to reach 1000 is: [ 1000 - 996 = 4 text{ spirits} ] 6. Identify the floors where the next 4 house spirits will get on in the subsequent trip: - One spirit enters on the first floor. - One spirit enters on the second floor. - One spirit enters on the third floor. - One spirit enters on the fourth floor. The elevator then has its 1000th spirit when it reaches the fourth floor of its 84th ascent. Conclusion: The 1000th house spirit enters the elevator on the **fourth floor**. [ boxed{4} ]
question:Tommy is making steaks for his family. There are 5 of them in total. If each member wants one pound and the steaks are a certain weight each, Tommy needs to buy 4 steaks. How much does each steak weigh?
answer:If Tommy needs to buy 4 steaks to feed 5 family members, and each family member wants one pound of steak, then the total amount of steak needed is 5 pounds. To find out how much each steak weighs, we divide the total amount of steak needed by the number of steaks Tommy needs to buy: 5 pounds / 4 steaks = 1.25 pounds per steak Each steak weighs boxed{1.25} pounds.