stanford

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question:f sqrt{5+n}=7, then what is the value of n?

answer:To solve for n, I need to isolate it on one side of the equation. First, I can square both sides of the equation to get rid of the square root. Squaring both sides gives me 5+n=49. Next, I can subtract 5 from both sides to get n=44. This is the value of n that satisfies the equation.

question:The students in Mrs. Reed's English class are reading the same 760page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. Chandra and Bob, who each have a copy of the book, decide that they can save time by `team reading' the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel?

answer:I need to find the page number, say p, such that the time Chandra spends reading from page 1 to page p is equal to the time Bob spends reading from page p + 1 to page 760. To compare the times, I need to convert them to the same units. Since the problem gives the rate of reading in seconds per page, I will use seconds as the unit of time. If Chandra reads a page in 30 seconds, then she reads p pages in 30p seconds. Similarly, Bob reads 760 - p pages in 45(760 - p) seconds. So I want to solve the equation 30p = 45(760 - p), which means that the two times are equal. To solve for p, I can expand the right-hand side and collect the terms with p on one side: 30p = 34200 - 45p implies 75p = 34200 implies p = 456. This means that Chandra should read up to page 456 and Bob should read from page 457 to page 760.

question:Of the numbers 1, 2, 3, ..., 15, which number has the greatest number of divisors (the dots mean that we are including all the integers between 1 and 15)?

answer:I want to find the number that has the most factors among 1, 2, 3, ..., 15. One way to do this is to list all the factors of each number and count them. For example, 1 has only one factor, 1 itself. 2 has two factors, 1 and 2. 3 has two factors, 1 and 3. 4 has three factors, 1, 2, and 4. 5 has two factors, 1 and 5. 6 has four factors, 1, 2, 3, and 6. 7 has two factors, 1 and 7. 8 has four factors, 1, 2, 4, and 8. 9 has three factors, 1, 3, and 9. 10 has four factors, 1, 2, 5, and 10. 11 has two factors, 1 and 11. 12 has six factors, 1, 2, 3, 4, 6, and 12. 13 has two factors, 1 and 13. 14 has four factors, 1, 2, 7, and 14. 15 has four factors, 1, 3, 5, and 15. I can see that 12 has the most factors, six in total. Therefore, 12 is the number that has the greatest number of divisors among 1, 2, 3, ..., 15.

question:w many distinct arrangements of the letters in the word basic'' are there?

answer:I need to find the number of ways to order five letters, where order matters. This is a permutation problem, since the order of the letters matters. The formula for permutations of n objects is n!, which means n times (n-1) times (n-2) and so on down to 1. In this case, n = 5, so the number of permutations is 5!, which is 5 times 4 times 3 times 2 times 1. I can multiply these numbers together to get the final answer.