💡Alright, so I have this problem here with two sets, A and B. I need to find their intersection, which is A ∩ B. Let me try to understand what each set is first.Set A is defined as {x | 1 < x < 7}. Okay, so that means A includes all real numbers x that are greater than 1 and less than 7. So, if I imagine a number line, A would be the segment from just above 1 to just below 7.Now, set B is defined by the inequality x² - 4x - 5 ≤ 0. Hmm, this looks like a quadratic inequality. I remember that to solve quadratic inequalities, I should first find the roots of the quadratic equation and then determine the intervals where the inequality holds.Let me write down the quadratic equation: x² - 4x - 5 = 0. To solve this, I can try factoring. I'm looking for two numbers that multiply to -5 and add up to -4. Hmm, that would be -5 and +1 because (-5) * 1 = -5 and (-5) + 1 = -4. So, the equation factors to (x - 5)(x + 1) = 0.Setting each factor equal to zero gives the roots: x - 5 = 0 ⇒ x = 5, and x + 1 = 0 ⇒ x = -1. So, the roots are x = -1 and x = 5.Now, to solve the inequality (x - 5)(x + 1) ≤ 0, I need to determine where the product is less than or equal to zero. I can use a sign chart for this. The critical points are x = -1 and x = 5, which divide the number line into three intervals:1. x < -12. -1 < x < 53. x > 5I'll test each interval:1. For x < -1, let's pick x = -2: ( -2 - 5)( -2 + 1 ) = (-7)(-1) = 7, which is positive. So, the inequality is not satisfied here.2. For -1 < x < 5, let's pick x = 0: (0 - 5)(0 + 1) = (-5)(1) = -5, which is negative. So, the inequality is satisfied here.3. For x > 5, let's pick x = 6: (6 - 5)(6 + 1) = (1)(7) = 7, which is positive. So, the inequality is not satisfied here.Now, since the inequality is ≤ 0, we also include the points where the expression equals zero, which are x = -1 and x = 5. Therefore, set B is the interval [-1, 5].Okay, so now I have:- Set A: (1, 7)- Set B: [-1, 5]I need to find A ∩ B, which is the overlap between these two sets. Let's visualize this on the number line.Set A starts just above 1 and goes up to just below 7. Set B starts at -1 and goes up to 5. So, the overlap would be from just above 1 to 5, because beyond 5, set B ends, and set A continues up to 7, but there's no overlap beyond 5.But wait, set B includes 5, and set A doesn't include 7, but does it include 5? Let me check. Set A is defined as 1 < x < 7, so it doesn't include 1 or 7. Set B is defined as -1 ≤ x ≤ 5, so it includes both -1 and 5.Therefore, in the intersection, the lower bound is 1 (not included) and the upper bound is 5 (included). So, A ∩ B should be (1, 5].Let me double-check to make sure I didn't make a mistake. If I take a number between 1 and 5, say 3, it should satisfy both conditions. For set A, 1 < 3 < 7 is true. For set B, 3 is between -1 and 5, so that's also true. What about 5 itself? For set A, 5 is less than 7, so it's included in A? Wait, set A is 1 < x < 7, so 5 is less than 7, but does it include 5? No, because it's strictly less than 7. Wait, no, 5 is less than 7, but set A is open at 7, but 5 is less than 7, so 5 is included in A? Wait, no, set A is open at 1 and 7, but 5 is between 1 and 7, so 5 is in A. Wait, no, set A is defined as 1 < x < 7, so it doesn't include 1 or 7, but 5 is between 1 and 7, so 5 is in A. But set B includes 5, so in the intersection, 5 is included because it's included in both sets? Wait, no, set A doesn't include 5 because it's open at 7, but 5 is less than 7, so 5 is in A. Wait, no, set A is open at 7, meaning it doesn't include 7, but 5 is less than 7, so 5 is included in A. Wait, no, set A is open at 1 and 7, but 5 is between 1 and 7, so 5 is in A. But set B includes 5, so in the intersection, 5 is included because it's in both sets.Wait, I'm getting confused. Let me clarify:Set A: 1 < x < 7. So, x can be 5 because 5 is greater than 1 and less than 7. So, 5 is in A.Set B: -1 ≤ x ≤ 5. So, x can be 5 because 5 is less than or equal to 5. So, 5 is in B.Therefore, in the intersection A ∩ B, 5 is included because it's in both A and B.Similarly, 1 is not included in A, so it's not in the intersection.So, the intersection is all x such that 1 < x ≤ 5, which is written as (1, 5].Let me check the answer choices:A: (-1,1) – This is not correct because our intersection starts at 1, not -1, and goes up to 5.B: (-1,1) ∪ (5,7) – This is also incorrect because our intersection is a single interval from 1 to 5, not two separate intervals.C: [-1,7) – This is too broad; our intersection is only from 1 to 5.D: (1,5] – This matches our conclusion.So, the correct answer is D: (1,5].

💡First, I need to determine the speed of the stream by using the information about the upstream swim. The man swims upstream 18 km in 3 hours, so his effective speed upstream is 6 km/h. Since his speed in still water is 11.5 km/h, I can calculate the stream's speed by subtracting the upstream speed from his still water speed.Next, with the stream's speed known, I can calculate the man's effective speed downstream by adding the stream's speed to his still water speed. Finally, I can find the distance he swam downstream by multiplying his downstream speed by the time he took, which is also 3 hours.

💡Okay, so I have this problem where positive integers are arranged in rows and columns, with each row having one more number than the previous row. The first few rows look like this:Row 1: 1Row 2: 2, 3Row 3: 4, 5, 6Row 4: 7, 8, 9, 10And so on. The question is asking how many integers less than 2000 are in the column that contains the number 2000. The options are (A) 15, (B) 19, (C) 17, (D) 16, (E) 18.Alright, so first, I need to figure out which column the number 2000 is in. To do that, I think I need to determine which row 2000 is located in, and then find its position within that row.I remember that the number of elements in each row increases by one each time. So, row 1 has 1 number, row 2 has 2 numbers, row 3 has 3 numbers, etc. This means that the total number of numbers up to row n is the sum of the first n natural numbers. The formula for that is n(n + 1)/2.So, if I can find the value of n such that n(n + 1)/2 is just less than or equal to 2000, that will give me the row where 2000 is located.Let me try to approximate n. Let's see, n(n + 1)/2 ≈ 2000. So, n² + n ≈ 4000. Ignoring the n for a moment, n² ≈ 4000, so n ≈ sqrt(4000). Calculating that, sqrt(4000) is approximately 63.245. So, n is around 63.Let me check n = 63: 63*64/2 = 2016. Hmm, that's more than 2000. So, the last number in row 63 is 2016, which is greater than 2000. That means 2000 is in row 63.Wait, but if row 63 ends at 2016, then the numbers in row 63 start from 2016 - 63 + 1 = 1954. So, row 63 has numbers from 1954 to 2016.So, 2000 is somewhere in row 63. To find its exact position in the row, I can subtract 1953 (the last number of row 62) from 2000. Let's see, 2000 - 1953 = 47. Wait, that doesn't make sense because 1953 is the last number of row 62, so row 63 starts at 1954. So, 2000 - 1953 = 47, but since row 63 has 63 numbers, starting at 1954, the position of 2000 is 2000 - 1953 = 47. But wait, that would mean 2000 is the 47th number in row 63. But row 63 only has 63 numbers, so 47 is a valid position.Wait, but if row 63 starts at 1954, then the first number is 1954, the second is 1955, and so on. So, the k-th number in row 63 is 1953 + k. So, 2000 = 1953 + k, so k = 2000 - 1953 = 47. So, 2000 is the 47th number in row 63.But wait, the columns are arranged such that each column is a vertical line of numbers. So, the first column is the first number of each row, the second column is the second number of each row, and so on.So, if 2000 is the 47th number in row 63, it's in column 47. But wait, that can't be right because the number of columns increases with each row. Wait, no, each row has as many columns as its row number. So, row 1 has 1 column, row 2 has 2 columns, etc. So, row 63 has 63 columns.But the columns are aligned vertically. So, the first column contains the first number of each row, the second column contains the second number of each row, etc. So, if 2000 is the 47th number in row 63, it's in column 47.But wait, that would mean that column 47 exists only in rows 47 and beyond. Because row 1 has only 1 column, row 2 has 2 columns, etc. So, column 47 starts at row 47.So, to find how many numbers less than 2000 are in column 47, we need to consider all the numbers in column 47 from row 47 up to row 62, because row 63 is where 2000 is located, and we need numbers less than 2000.Wait, but 2000 is in row 63, column 47. So, the numbers in column 47 are in rows 47 to 63. But we need numbers less than 2000, so we need to count the numbers in column 47 from row 47 up to row 62, because the number in row 63, column 47 is 2000 itself, which we don't want to include.So, how many numbers is that? From row 47 to row 62, that's 62 - 47 + 1 = 16 rows. So, there are 16 numbers in column 47 that are less than 2000.Wait, but let me double-check. Let's see, the first number in column 47 is in row 47, which is the 47th number of row 47. The last number in column 47 before row 63 is in row 62, which is the 47th number of row 62.So, the number of numbers in column 47 from row 47 to row 62 is 62 - 47 + 1 = 16 numbers. Therefore, there are 16 integers less than 2000 in the column that contains 2000.Wait, but let me make sure I didn't make a mistake in determining the column number. Earlier, I thought 2000 is the 47th number in row 63, so it's in column 47. But let me verify that.Row 63 starts at 1954, so the first number is 1954, the second is 1955, ..., the 47th number is 1954 + 46 = 2000. Yes, that's correct. So, 2000 is indeed in column 47.Therefore, the number of integers less than 2000 in column 47 is 16.Wait, but let me think again. The first number in column 47 is in row 47. So, the number in row 47, column 47 is the 47th number in row 47. Let's calculate what that number is.The last number in row 46 is 46*47/2 = 1081. So, row 47 starts at 1082. The 47th number in row 47 is 1082 + 46 = 1128. So, the first number in column 47 is 1128.Similarly, the last number in column 47 before row 63 is in row 62. The last number in row 61 is 61*62/2 = 1891. So, row 62 starts at 1892. The 47th number in row 62 is 1892 + 46 = 1938.So, the numbers in column 47 from row 47 to row 62 are from 1128 to 1938, stepping by 62 each time? Wait, no, each number in column 47 is in consecutive rows, so each subsequent number is 62 more than the previous one? Wait, no, that's not correct.Wait, actually, each row starts at a certain number, and the numbers increase by 1 each time. So, the numbers in column 47 are in rows 47, 48, ..., 63. Each row has one more number than the previous, so the starting number of each row increases by the row number.Wait, maybe a better way is to calculate the number in column 47 for each row from 47 to 62.The formula for the first number in row n is (n-1)n/2 + 1. So, for row 47, the first number is (46*47)/2 + 1 = 1081 + 1 = 1082. So, the 47th number in row 47 is 1082 + 46 = 1128.For row 48, the first number is (47*48)/2 + 1 = 1128 + 1 = 1129. So, the 47th number in row 48 is 1129 + 46 = 1175.Wait, that doesn't make sense because row 48 has only 48 numbers, so the 47th number is the second to last number in row 48. Similarly, row 49 has 49 numbers, so the 47th number is the third to last number in row 49.Wait, actually, no. Each row n has n numbers, so the 47th number in row n is the (n - 47 + 1)th number from the end. Wait, that might complicate things.Alternatively, since each row starts at a certain number, and the numbers increase by 1, the k-th number in row n is equal to the first number of row n plus (k - 1).The first number of row n is (n-1)n/2 + 1. So, the k-th number in row n is (n-1)n/2 + k.Therefore, the number in column k of row n is (n-1)n/2 + k.So, for column 47, the number in row n is (n-1)n/2 + 47.Therefore, the numbers in column 47 from row 47 to row 62 are:For n = 47: (46*47)/2 + 47 = 1081 + 47 = 1128For n = 48: (47*48)/2 + 47 = 1128 + 47 = 1175Wait, that can't be right because (47*48)/2 is 1128, so adding 47 gives 1175.Similarly, for n = 49: (48*49)/2 + 47 = 1176 + 47 = 1223Wait, but 48*49/2 is 1176, so adding 47 gives 1223.Wait, but this seems like each subsequent number in column 47 is increasing by 49, then 49 again? Wait, 1175 to 1223 is 48, which is 49 -1.Wait, maybe I'm overcomplicating this. The key point is that each number in column 47 is in a different row, starting from row 47 up to row 63.Each of these numbers is less than 2000 except for the one in row 63, which is 2000 itself.So, the number of integers less than 2000 in column 47 is the number of rows from 47 to 62, inclusive. That's 62 - 47 + 1 = 16 numbers.Therefore, the answer should be 16, which is option D.Wait, but let me make sure I didn't make a mistake in calculating the column number. Earlier, I thought 2000 is the 47th number in row 63, so it's in column 47. But let me verify that.Row 63 starts at 1954, as calculated earlier. So, the first number is 1954, the second is 1955, ..., the 47th number is 1954 + 46 = 2000. Yes, that's correct. So, 2000 is indeed in column 47.Therefore, the number of integers less than 2000 in column 47 is 16.I think that's correct. So, the answer is D) 16.

💡Okay, so I have this triangle ABC, and the sides opposite angles A, B, and C are a, b, and c respectively. The problem gives me an equation: (a + 2c) cos B + b cos A = 0. I need to find angle B first, and then, given that b = 3 and the perimeter is 3 + 2√3, find the area of the triangle.Starting with part (1). I remember that in triangles, the Law of Sines and the Law of Cosines are really useful. The given equation involves cosines of angles and sides, so maybe I can use these laws to relate them.Let me write down the equation again: (a + 2c) cos B + b cos A = 0. Hmm, maybe I can express a, b, c in terms of the sine of their opposite angles using the Law of Sines. The Law of Sines says that a/sin A = b/sin B = c/sin C = 2R, where R is the circumradius.So, if I let a = 2R sin A, b = 2R sin B, and c = 2R sin C, I can substitute these into the equation.Substituting, the equation becomes: (2R sin A + 2*2R sin C) cos B + 2R sin B cos A = 0.Simplify that: 2R sin A cos B + 4R sin C cos B + 2R sin B cos A = 0.I can factor out 2R: 2R [sin A cos B + 2 sin C cos B + sin B cos A] = 0.Since R is not zero, the expression inside the brackets must be zero: sin A cos B + 2 sin C cos B + sin B cos A = 0.Looking at sin A cos B + sin B cos A, that's a familiar identity. That's equal to sin(A + B). So, sin(A + B) + 2 sin C cos B = 0.In a triangle, A + B + C = π, so A + B = π - C. Therefore, sin(A + B) = sin(π - C) = sin C. So, substituting that in, we have sin C + 2 sin C cos B = 0.Factor out sin C: sin C (1 + 2 cos B) = 0.Now, sin C can't be zero because angle C is between 0 and π, so sin C ≠ 0. Therefore, 1 + 2 cos B = 0.Solving for cos B: 2 cos B = -1 => cos B = -1/2.What angle B has cosine -1/2? Well, cos(120°) = cos(2π/3) = -1/2. So, angle B is 2π/3 radians or 120 degrees.Okay, that seems solid. So part (1) is done, angle B is 120 degrees.Moving on to part (2). Given that b = 3 and the perimeter is 3 + 2√3. So, a + b + c = 3 + 2√3. Since b = 3, that means a + c = (3 + 2√3) - 3 = 2√3.So, a + c = 2√3. I need to find the area of triangle ABC.I remember that the area can be found using several formulas. One common one is (1/2)ab sin C, but I need to figure out which sides and angles I have.Alternatively, since I know angle B, maybe I can use the formula involving sides a, c, and angle B. The formula is (1/2)ac sin B.But to use that, I need to find the product ac. Hmm, how can I find ac?I can use the Law of Cosines. The Law of Cosines states that b² = a² + c² - 2ac cos B.We know b = 3, angle B = 120°, so cos B = -1/2. Let's plug in the values.So, 3² = a² + c² - 2ac*(-1/2). Simplify that:9 = a² + c² + ac.But we also know that a + c = 2√3. Let me square both sides of that equation: (a + c)² = (2√3)² => a² + 2ac + c² = 12.So, from this, a² + c² = 12 - 2ac.Now, substitute a² + c² from this into the equation from the Law of Cosines:9 = (12 - 2ac) + ac.Simplify: 9 = 12 - ac.So, -ac = 9 - 12 => -ac = -3 => ac = 3.Great, so the product ac is 3.Now, going back to the area formula: Area = (1/2)ac sin B.We have ac = 3, and sin B = sin(120°) = √3/2.So, Area = (1/2)*3*(√3/2) = (3√3)/4.Wait, that seems straightforward. Let me double-check my steps.1. From the perimeter, a + c = 2√3.2. Squared to get a² + c² + 2ac = 12.3. From Law of Cosines: 9 = a² + c² + ac.4. Substitute a² + c² = 12 - 2ac into the equation: 9 = 12 - 2ac + ac => 9 = 12 - ac => ac = 3.5. Then, area = (1/2)*ac*sin B = (1/2)*3*(√3/2) = 3√3/4.Yes, that seems correct. So the area is 3√3/4.**Final Answer**(1) boxed{dfrac{2pi}{3}} (2) boxed{dfrac{3sqrt{3}}{4}}

💡Okay, so I have this problem about a right triangle ABC where angle A is 90 degrees. The legs AB and BC are given as 16 and 20 units long, respectively. I need to find the cosine of angle C. Hmm, let me think about how to approach this.First, I remember that in a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse. So, for angle C, the adjacent side would be BC, and the hypotenuse would be AC. But wait, I don't know the length of AC yet. I only know AB and BC. So, I guess I need to find the length of AC first.Since it's a right triangle, I can use the Pythagorean theorem to find the missing side. The theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, if I let AC be the hypotenuse, then:AC² = AB² + BC²Plugging in the values I have:AC² = 16² + 20²Calculating 16 squared: 16 * 16 = 256Calculating 20 squared: 20 * 20 = 400Adding them together: 256 + 400 = 656So, AC² = 656. To find AC, I take the square root of 656. Let me see, 656 is not a perfect square, but maybe I can simplify it.Breaking down 656 into prime factors: 656 divided by 16 is 41, because 16 * 41 = 656. So, 656 = 16 * 41, which means sqrt(656) = sqrt(16 * 41) = sqrt(16) * sqrt(41) = 4 * sqrt(41). So, AC = 4√41.Alright, now that I have the hypotenuse AC, I can find cos C. As I thought earlier, cos C is adjacent over hypotenuse, which in this case is BC over AC.So, cos C = BC / AC = 20 / (4√41)Simplifying that, 20 divided by 4 is 5, so cos C = 5 / √41.But wait, usually, we don't leave radicals in the denominator. I think I need to rationalize the denominator. To do that, I can multiply both the numerator and the denominator by √41.So, cos C = (5 / √41) * (√41 / √41) = (5√41) / 41.Let me double-check my steps to make sure I didn't make a mistake. I used the Pythagorean theorem correctly, right? AB is 16, BC is 20, so AC should be sqrt(16² + 20²) which is sqrt(256 + 400) = sqrt(656) = 4√41. That seems right.Then, for cos C, I used the adjacent side BC over hypotenuse AC, which is 20 / (4√41) simplifying to 5 / √41, and then rationalizing to 5√41 / 41. Yeah, that all checks out.I think I got it. So, the cosine of angle C is 5√41 over 41.**Final Answer**The value of cos C is boxed{dfrac{5sqrt{41}}{41}}.

💡Okay, so there's this problem about oak trees in a park. Let me read it again: "There are 9 oak trees currently in the park. Park workers had to cut down 2 oak trees that were damaged. How many oak trees will be in the park when the workers are finished?" Hmm, seems straightforward, but I want to make sure I understand it completely before jumping to the answer.Alright, so first, there are 9 oak trees. That's the starting point. Then, the workers have to cut down 2 of them because they're damaged. So, we're dealing with a subtraction problem here. I think the key is to subtract the number of trees being cut down from the total number of trees currently in the park.Let me visualize this. Imagine the park has 9 oak trees. Maybe they're all healthy, but 2 of them are damaged and need to be removed. So, the workers are going to cut those 2 down. After they finish their work, how many oak trees will remain?So, mathematically, it's 9 minus 2. That should give us the number of oak trees left in the park. Let me write that down: 9 - 2 = ?Now, doing the subtraction: 9 minus 2. Well, 9 take away 2 is 7. So, there should be 7 oak trees left in the park after the workers finish cutting down the damaged ones.Wait, but let me double-check to make sure I didn't make a mistake. Sometimes, even simple problems can have tricks or hidden details. Is there anything else I need to consider here? The problem says there are currently 9 oak trees, and they cut down 2. It doesn't mention anything about planting new trees or any other changes. So, it's just a straightforward subtraction.Another way to think about it is to count down from 9 by 2. Starting at 9, if I subtract 1, I get 8, and subtracting another 1 gets me to 7. Yep, that checks out.Maybe I can visualize it with objects. If I have 9 apples and I take away 2, how many do I have left? It's the same principle. 9 minus 2 is 7. So, applying that to the oak trees, it makes sense.I guess another way to look at it is using number lines. Starting at 9, if I move 2 steps to the left, I land on 7. That's another confirmation that 9 minus 2 equals 7.Is there any chance I misread the problem? Let me read it again: "There are 9 oak trees currently in the park. Park workers had to cut down 2 oak trees that were damaged. How many oak trees will be in the park when the workers are finished?" No, it seems clear. They start with 9, remove 2, so 7 remain.I could also think about it in terms of equations. Let’s denote the number of oak trees as T. Initially, T = 9. Then, 2 trees are cut down, so T becomes T - 2. Therefore, T = 9 - 2 = 7.Maybe to ensure I'm not missing anything, I can think about the context. Parks sometimes have other types of trees, but the problem specifically mentions oak trees. So, we're only dealing with oak trees here, not the total number of trees in the park. That means the answer is solely about oak trees, not the overall tree population.Also, the problem doesn't mention any other factors like new trees being planted or any other trees being cut down. It's purely about removing 2 oak trees from the existing 9.I think I've covered all the angles here. The problem is straightforward, and the solution seems solid. I can't think of any other interpretations or complications that might affect the answer. So, I'm pretty confident that the number of oak trees remaining in the park after the workers finish is 7.