Appearance
question:Suppose that ( n = 50 ), and that the terms of the initial sequence ( r_1, r_2, dots, r_{50} ) are distinct from one another and are in random order. Determine the probability that the number that begins as ( r_{10} ) will end up, after one bubble pass, in the ( 40^{mbox{th}} ) place. Express your answer as ( p/q ) in lowest terms and find ( p + q ).
answer:For ( r_{10} ) to end up in the 40th place after one bubble pass: - ( r_{10} ) must be the largest among ( r_1, ldots, r_{10} ) to ensure it starts moving forward. - ( r_{10} ) must be larger than ( r_{11}, r_{12}, ldots, r_{40} ) but smaller than ( r_{41} ) to stop at the 40th place. We need to calculate the probability that in a sequence of 41 distinct numbers (from ( r_1 ) to ( r_{41} )): - The largest number is in position 41 (to ensure ( r_{10} ) does not move beyond 40th position). - The second largest is in position 10 (so ( r_{10} ) moves up to the 40th position). There are ( 41! ) ways to arrange 41 numbers. The number of favorable arrangements: - Fix the largest number in the 41st position. - Fix the second largest number in the 10th position. - The remaining 39 numbers can be in any order: ( 39! ) ways. Thus, the favorable probability is ( frac{39!}{41!} = frac{1}{41 times 40} = frac{1}{1640} ). Therefore, ( p/q = 1/1640 ) and ( p + q = 1 + 1640 = boxed{1641} ).
question:There are 11 baskets numbered from 1 to 11 and filled with apples. 10 children are asked to pick apples one by one from each basket such that the number of apples picked by each child from each basket is equal to the number marked on the basket. The baskets were filled in such a way that none of the basket became empty in the process. In the end, there were 340 apples left. How many apples were there in total before the children started picking?
answer:Each child picks a number of apples equal to the number marked on the basket from each basket. Since there are 10 children, the total number of apples picked from each basket will be 10 times the number marked on the basket. Let's calculate the total number of apples picked from all the baskets: For basket 1: 10 children pick 1 apple each, so 10 * 1 = 10 apples are picked. For basket 2: 10 children pick 2 apples each, so 10 * 2 = 20 apples are picked. For basket 3: 10 children pick 3 apples each, so 10 * 3 = 30 apples are picked. For basket 4: 10 children pick 4 apples each, so 10 * 4 = 40 apples are picked. For basket 5: 10 children pick 5 apples each, so 10 * 5 = 50 apples are picked. For basket 6: 10 children pick 6 apples each, so 10 * 6 = 60 apples are picked. For basket 7: 10 children pick 7 apples each, so 10 * 7 = 70 apples are picked. For basket 8: 10 children pick 8 apples each, so 10 * 8 = 80 apples are picked. For basket 9: 10 children pick 9 apples each, so 10 * 9 = 90 apples are picked. For basket 10: 10 children pick 10 apples each, so 10 * 10 = 100 apples are picked. For basket 11: 10 children pick 11 apples each, so 10 * 11 = 110 apples are picked. Now, let's add up all the apples picked from all the baskets: 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100 + 110 = 660 apples picked in total. After the children finished picking, there were 340 apples left. To find the total number of apples before the children started picking, we add the apples picked to the apples left: 660 apples picked + 340 apples left = boxed{1000} apples in total before the children started picking.
question:Given a line l passing through point P(1, 2), and the cosine of its inclination angle is frac{sqrt{2}}{2}. (1) Find the general form equation of line l; (2) Calculate the volume of the geometric body formed by rotating the triangle enclosed by line l and the coordinate axes around the yaxis in space.
answer:(1) Given that cosalpha = frac{sqrt{2}}{2}, it follows that k = tanalpha = 1, Since line l passes through point P(-1, 2), the equation of line l is y - 2 = x + 1, which simplifies to x - y - 3 = 0. (2) The line l and the coordinate axes form an isosceles right triangle. The geometric body formed by rotating this triangle around the y-axis in space is a cone with a base radius of 3 and a height of 3. The volume V of the cone is V = frac{1}{3}pi cdot 3^{2} cdot 3 = 9pi. Therefore, the answers are: (1) The general form equation of line l is boxed{x - y - 3 = 0}. (2) The volume of the geometric body formed is boxed{9pi}.
question:abel has 204 in her piggy bank. She spent 40% of the amount on a toy for herself. Then, with the remaining money, she spent 50% on a book for her brother and saved 20% for a charity donation. After that, she spent 30% of the remaining amount on a gift for her friend and decided to save 10% of what was left for her future expenses. How much money, in dollars, was left over in her piggy bank after all transactions?
answer:First, let's calculate how much Isabel spent on the toy: 40% of 204 = 0.40 * 204 = 81.60 Now, let's subtract that amount from her total to find out how much she had left: 204 - 81.60 = 122.40 Next, she spent 50% on a book for her brother and saved 20% for charity. Let's calculate those amounts: 50% of 122.40 = 0.50 * 122.40 = 61.20 (spent on the book) 20% of 122.40 = 0.20 * 122.40 = 24.48 (saved for charity) Now, let's subtract the amounts spent on the book and saved for charity from the remaining money: 122.40 - 61.20 - 24.48 = 36.72 Then, she spent 30% of the remaining amount on a gift for her friend: 30% of 36.72 = 0.30 * 36.72 = 11.016 Subtract the amount spent on the gift from the remaining money: 36.72 - 11.016 = 25.704 Finally, she decided to save 10% of what was left for her future expenses: 10% of 25.704 = 0.10 * 25.704 = 2.5704 Now, let's subtract the amount saved for future expenses from the remaining money: 25.704 - 2.5704 = 23.1336 Therefore, Isabel had 23.1336 left over in her piggy bank after all transactions. However, since we're dealing with money, we should round to the nearest cent, which gives us: boxed{23.13} left over in her piggy bank.