answer:1. The function f(x) is defined as the dot product of overrightarrow{m} and overrightarrow{n}: f(x) = overrightarrow{m} cdot overrightarrow{n} = cos 2x + left(frac{sqrt{3}}{2}sin x - frac{1}{2}cos xright)^2. Expanding the squared term, we get f(x) = cos 2x + frac{3}{4}sin^2 x + frac{1}{4}cos^2 x - frac{sqrt{3}}{2}sin xcos x. Using the double-angle formulae, we can rewrite f(x) as f(x) = frac{3}{4}cos 2x - frac{sqrt{3}}{4}sin 2x + frac{1}{2}. Now, we can express f(x) as a single cosine function with a phase shift: f(x) = frac{sqrt{3}}{2}cosleft(2x + frac{pi}{6}right) + frac{1}{2}. The maximum value of f(x) is attained when cosleft(2x + frac{pi}{6}right) = 1, which occurs when 2x + frac{pi}{6} = 2kpi for some integer k. Solving for x, we get x = kpi - frac{pi}{12}. Thus, the set of values of x for which f(x) attains its maximum value is boxed{{x mid x = kpi - frac{pi}{12}, k in mathbb{Z}}}. 2. Since A, B, and C are the internal angles of an acute triangle ABC, we have cos B = frac{3}{5}. Therefore, sin B = sqrt{1 - cos^2 B} = frac{4}{5}. Given that f(C) = -frac{1}{4}, we can write frac{sqrt{3}}{2}cosleft(2C + frac{pi}{6}right) + frac{1}{2} = -frac{1}{4}. Solving for cosleft(2C + frac{pi}{6}right), we obtain cosleft(2C + frac{pi}{6}right) = -frac{sqrt{3}}{2}. This implies 2C + frac{pi}{6} = frac{5pi}{6}, so C = frac{pi}{3}. Finally, to find sin A, we use the sine angle sum identity: sin A = sinleft(frac{2pi}{3} - Bright) = frac{sqrt{3}}{2}cos B + frac{1}{2}sin B. Substituting the values of cos B and sin B, we get sin A = frac{sqrt{3}}{2} cdot frac{3}{5} + frac{1}{2} cdot frac{4}{5} = boxed{frac{4 + 3sqrt{3}}{10}}.

answer:Let's denote the number of tables made this month for each style as x_1, x_2, ..., x_10, where each x_i represents the number of tables made for style i. According to the problem, the sum of all x values for this month is 100: x_1 + x_2 + ... + x_10 = 100 For last month, the number of tables made for each style would be x_1 - 3, x_2 - 3, ..., x_10 - 3. The total number of tables made last month would be: (x_1 - 3) + (x_2 - 3) + ... + (x_10 - 3) = (x_1 + x_2 + ... + x_10) - 3*10 = 100 - 30 = 70 The total number of tables made in both months would be the sum of the tables made this month and last month: Total tables = (x_1 + x_2 + ... + x_10) + ((x_1 - 3) + (x_2 - 3) + ... + (x_10 - 3)) = 100 + 70 = 170 tables However, we have the constraint that the carpenter worked a fixed number of hours each month and the total area of tables made this month is 200 square feet. This implies that the carpenter can only make a certain number of tables based on the time it takes to produce each style and the area each table occupies. Let's denote the area of each table style as A_1, A_2, ..., A_10. The total area for the tables made this month would be: A_1*x_1 + A_2*x_2 + ... + A_10*x_10 = 200 square feet Since each table style has a unique size and takes a different amount of time to produce, we cannot determine the exact number of tables made without additional information about the area and production time for each table style. However, we can say that the carpenter made boxed{170} tables in total over the two months, given the constraints of fixed work hours and the total area of tables made this month. To find the exact number of tables for each style, we would need more specific information about the area and production time for each table style.

answer:- As before, n/(n+1) is in simplest form since n and n+1 are consecutive integers, and their greatest common divisor is 1. - The fraction frac{n}{n+1} has a repeating decimal if the denominator (n+1) is not solely composed of the primes 2 and 5. - For 1 le n le 150, n+1 ranges from 2 to 151. Calculate how many of these numbers are not solely composed of the factors 2 and 5. First, list the numbers between 2 to 151 that consist only of the primes 2 and/or 5: {2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128}. There are 16 such numbers. Since there are 150 values of n (from n=1 to n=150), and 16 of these result in n/(n+1) having terminating decimals, the remaining numbers will have repeating decimals. Thus, there are 150 - 16 = boxed{134} integers n such that frac{n}{n+1} is a repeating decimal.

answer:Let's analyze the situation in detail to understand where the calculations went wrong and what really happened with the money. 1. **Initial Plan**: Karl, the cobbler, instructed his son Hans to sell a pair of boots for 25 talers at the market. 2. **Market Situation**: At the market, Hans met two handicapped men who each needed only one boot. Hans agreed to sell one boot to each man for 12.5 talers. 3. **Total Money Collected**: Hans sold the two boots and collected a total of: [ 12.5 , text{talers} + 12.5 , text{talers} = 25 , text{talers} ] 4. **Father's Reaction**: When Hans reported this to Karl, Karl decided that each handicapped man should have paid 10 talers instead of 12.5 talers per boot. Therefore, he instructed Hans to return: [ 12.5 , text{talers} - 10 , text{talers} = 2.5 , text{talers} text{ to each man} ] 5. **Total to Return**: Karl gave Hans 5 talers to return to the two customers in total: [ 2.5 , text{talers} times 2 = 5 , text{talers} ] 6. **Discrepancy by Purchases**: While looking for the two men to return the money, Hans spent 3 talers on candy. 7. **Money Returned to Customers**: Hans found the two men and, because he had spent 3 talers, he could return only: [ 5 , text{talers} - 3 , text{talers} = 2 , text{talers} ] He gave 1 taler to each man: [ frac{2 , text{talers}}{2} = 1 , text{taler} text{ per man} ] 8. **Calculation of Effective Payments**: Each man effectively paid: [ 12.5 , text{talers} - 1 , text{taler} = 11.5 , text{talers} ] 9. **Total Money for Boots**: The total for the two boots was: [ 2 times 11.5 , text{talers} = 23 , text{talers} ] 10. **Understanding the Complete Transactions**: - Hans collected 25 talers initially by selling the boots. - Hans was given 5 talers to return. - He used 3 talers for candy. So, effectively he returned 2 talers back. 11. **Summing Collected Owing to Misinterpretation**: Hans added the 23 talers (boots) and 3 talers (candy), summing incorrectly to 26 talers instead of correctly understanding Karl received the net proceeds correctly. When looking at the final money, it actually does not create a sensible conclusion by adding spent into earnings. # Proper Financial Impact: When considering only the sale - Receipts: 23 talers (after correction of giving back 2 talers) - Expenditure: Candy (3 talers - waste ) - Karl's final correct income: originally intended 25 but effectively - 2 reps (correction returning) Therefore confirming: Hans miscalculated it including spending, instead actual clear logical difference of adjustments [ 25 presentable correct No mystery of extra taler Karl ultimately net sum - proper clarity is in recognizing not spending addition but considerate right receivable and payable track tracing correct Hassan undue lashings injustly for misinterpretation solving enigma cleaner. Conclusion: [ boxed{clear} text{ distribution Karl inline original sums outcome tracing verified }.