💡Okay, so I've got this problem where I need to find all possible values of ( x ) given two equations involving ( x ) and ( y ). The equations are:1. ( x + frac{1}{y} = 10 )2. ( y + frac{1}{x} = frac{5}{12} )Hmm, both ( x ) and ( y ) are nonzero real numbers. I need to find ( x ). Let me think about how to approach this.First, maybe I can express one variable in terms of the other and substitute. Let me try to solve the first equation for ( frac{1}{y} ):From equation 1:( frac{1}{y} = 10 - x )So, ( y = frac{1}{10 - x} ). Okay, that gives me ( y ) in terms of ( x ). Now, let's plug this into equation 2.Substituting ( y = frac{1}{10 - x} ) into equation 2:( frac{1}{10 - x} + frac{1}{x} = frac{5}{12} )Hmm, that looks a bit complicated, but maybe I can combine the fractions. Let's find a common denominator for the left side. The denominators are ( 10 - x ) and ( x ), so the common denominator would be ( x(10 - x) ).So, rewriting the left side:( frac{x}{x(10 - x)} + frac{10 - x}{x(10 - x)} = frac{5}{12} )Combine the numerators:( frac{x + (10 - x)}{x(10 - x)} = frac{5}{12} )Simplify the numerator:( frac{10}{x(10 - x)} = frac{5}{12} )Okay, so now I have:( frac{10}{x(10 - x)} = frac{5}{12} )Let me cross-multiply to solve for ( x ):( 10 times 12 = 5 times x(10 - x) )Calculate both sides:( 120 = 5x(10 - x) )Divide both sides by 5:( 24 = x(10 - x) )Expand the right side:( 24 = 10x - x^2 )Bring all terms to one side to form a quadratic equation:( x^2 - 10x + 24 = 0 )Now, let's solve this quadratic equation. I can try factoring it. Looking for two numbers that multiply to 24 and add up to -10. Hmm, -6 and -4.So, factoring:( (x - 6)(x - 4) = 0 )Therefore, the solutions are:( x = 6 ) or ( x = 4 )Wait, let me check if these solutions make sense in the original equations. Sometimes, when we manipulate equations, we might introduce extraneous solutions.First, check ( x = 6 ):From equation 1: ( 6 + frac{1}{y} = 10 ) implies ( frac{1}{y} = 4 ), so ( y = frac{1}{4} ).Now, plug into equation 2: ( frac{1}{4} + frac{1}{6} = frac{3}{12} + frac{2}{12} = frac{5}{12} ). That works.Next, check ( x = 4 ):From equation 1: ( 4 + frac{1}{y} = 10 ) implies ( frac{1}{y} = 6 ), so ( y = frac{1}{6} ).Now, plug into equation 2: ( frac{1}{6} + frac{1}{4} = frac{2}{12} + frac{3}{12} = frac{5}{12} ). That also works.So both solutions are valid. Therefore, the possible values of ( x ) are 4 and 6.**Final Answer**The possible values of ( x ) are boxed{4} and boxed{6}.

💡Okay, so I have this problem here about a sequence {a_n} where the sum of the first n terms is S_n. They give me that a_1 is 2, and there's a recursive formula: a_{n+1} = 2S_n. I need to find S_7, and there are four options to choose from: 1458, 1460, 2184, and 2186. Hmm, let's see how to approach this.First, let me make sure I understand the problem correctly. The sequence starts with a_1 = 2. Then, each subsequent term a_{n+1} is equal to twice the sum of all the previous terms, which is 2S_n. So, a_2 would be 2 times S_1, which is just a_1, so a_2 = 2*2 = 4. Then, a_3 would be 2 times S_2, which is a_1 + a_2 = 2 + 4 = 6, so a_3 = 2*6 = 12. Hmm, okay, so each term is doubling the sum of all the previous terms.Let me write down the terms step by step to see if I can find a pattern or maybe a formula.Given:- a_1 = 2- a_{n+1} = 2S_nSo, let's compute the first few terms and their sums:1. a_1 = 2 S_1 = a_1 = 22. a_2 = 2S_1 = 2*2 = 4 S_2 = a_1 + a_2 = 2 + 4 = 63. a_3 = 2S_2 = 2*6 = 12 S_3 = S_2 + a_3 = 6 + 12 = 184. a_4 = 2S_3 = 2*18 = 36 S_4 = S_3 + a_4 = 18 + 36 = 545. a_5 = 2S_4 = 2*54 = 108 S_5 = S_4 + a_5 = 54 + 108 = 1626. a_6 = 2S_5 = 2*162 = 324 S_6 = S_5 + a_6 = 162 + 324 = 4867. a_7 = 2S_6 = 2*486 = 972 S_7 = S_6 + a_7 = 486 + 972 = 1458Wait, that gives me S_7 as 1458, which is option A. But hold on, let me check because I might have made a mistake in my calculations.Let me verify each step:- a_1 = 2, S_1 = 2. Correct.- a_2 = 2*S_1 = 4, S_2 = 2 + 4 = 6. Correct.- a_3 = 2*S_2 = 12, S_3 = 6 + 12 = 18. Correct.- a_4 = 2*S_3 = 36, S_4 = 18 + 36 = 54. Correct.- a_5 = 2*S_4 = 108, S_5 = 54 + 108 = 162. Correct.- a_6 = 2*S_5 = 324, S_6 = 162 + 324 = 486. Correct.- a_7 = 2*S_6 = 972, S_7 = 486 + 972 = 1458.Hmm, so according to this, S_7 is 1458, which is option A. But wait, the options also include 2186, which is quite a bit larger. Maybe I'm missing something here.Let me think again. The problem says that a_{n+1} = 2S_n. So, each term is twice the sum of all previous terms. That seems to be a geometric progression because each term is a multiple of the previous one.Looking at the terms I calculated:a_1 = 2a_2 = 4 = 2*2a_3 = 12 = 3*4a_4 = 36 = 3*12a_5 = 108 = 3*36a_6 = 324 = 3*108a_7 = 972 = 3*324Wait a minute, from a_2 onwards, each term is 3 times the previous term. So, starting from a_2, the sequence is geometric with a common ratio of 3. So, a_2 = 4, a_3 = 12, a_4 = 36, etc.So, if I consider the sequence starting from a_2, it's a geometric sequence with first term 4 and ratio 3. Then, the sum S_n can be expressed as a_1 plus the sum of the geometric sequence from a_2 to a_n.Let me formalize this. The sum S_n is:S_n = a_1 + a_2 + a_3 + ... + a_nBut since a_2, a_3, ..., a_n form a geometric sequence with first term 4 and ratio 3, the sum from a_2 to a_n is:Sum = a_2*(r^{n-1} - 1)/(r - 1) = 4*(3^{n-1} - 1)/(3 - 1) = 2*(3^{n-1} - 1)Therefore, S_n = a_1 + 2*(3^{n-1} - 1) = 2 + 2*(3^{n-1} - 1) = 2 + 2*3^{n-1} - 2 = 2*3^{n-1}Wait, that simplifies to S_n = 2*3^{n-1}. Let me test this formula with the values I have:For n=1: S_1 = 2*3^{0} = 2*1 = 2. Correct.For n=2: S_2 = 2*3^{1} = 6. Correct.For n=3: S_3 = 2*3^{2} = 18. Correct.For n=4: S_4 = 2*3^{3} = 54. Correct.For n=5: S_5 = 2*3^{4} = 162. Correct.For n=6: S_6 = 2*3^{5} = 486. Correct.For n=7: S_7 = 2*3^{6} = 2*729 = 1458. Hmm, same result as before.But wait, the options include 2186, which is much larger. Maybe I made a mistake in assuming the sum formula. Let me double-check.Alternatively, perhaps the sequence is not starting the geometric progression at a_2 but earlier. Let me see:Given a_{n+1} = 2S_n, and S_n = S_{n-1} + a_n.So, substituting S_n into the recursive formula:a_{n+1} = 2*(S_{n-1} + a_n) = 2S_{n-1} + 2a_nBut from the recursive formula, a_n = 2S_{n-1} for n >= 2.So, substituting a_n = 2S_{n-1} into the equation:a_{n+1} = 2S_{n-1} + 2*(2S_{n-1}) = 2S_{n-1} + 4S_{n-1} = 6S_{n-1}But S_{n-1} = S_{n-2} + a_{n-1} = S_{n-2} + 2S_{n-2} = 3S_{n-2}Wait, this seems to get complicated. Maybe another approach.Alternatively, let's try to express a_{n+1} in terms of a_n.We have a_{n+1} = 2S_nBut S_n = S_{n-1} + a_nSo, a_{n+1} = 2(S_{n-1} + a_n) = 2S_{n-1} + 2a_nBut from the recursive formula, a_n = 2S_{n-1}So, substituting:a_{n+1} = a_n + 2a_n = 3a_nAh, so a_{n+1} = 3a_n for n >= 1Wait, does that hold?Let me check:From n=1: a_2 = 2S_1 = 2*2 = 4. If a_{n+1}=3a_n, then a_2 should be 3*a_1 = 6, but it's 4. So, that doesn't hold for n=1.Wait, so maybe the relation a_{n+1}=3a_n starts from n=2?Let me check:For n=2: a_3 = 2S_2 = 2*6 = 12. If a_{n+1}=3a_n, then a_3 should be 3*a_2 = 12. That works.For n=3: a_4 = 2S_3 = 2*18 = 36. 3*a_3 = 36. Correct.So, starting from n=2, the sequence follows a_{n+1}=3a_n. So, from a_2 onwards, it's a geometric sequence with ratio 3.Therefore, the sequence is:a_1 = 2a_2 = 4a_3 = 12a_4 = 36a_5 = 108a_6 = 324a_7 = 972So, the sum S_7 is 2 + 4 + 12 + 36 + 108 + 324 + 972.Let me add these up:2 + 4 = 66 + 12 = 1818 + 36 = 5454 + 108 = 162162 + 324 = 486486 + 972 = 1458So, S_7 is 1458, which is option A.But wait, the options also include 2186, which is much larger. Maybe I'm missing something here. Let me think again.Alternatively, perhaps the formula S_n = 2*3^{n-1} is correct, and 3^6 is 729, so 2*729 is 1458. That seems right.But why is 2186 an option? Maybe I made a mistake in the initial assumption.Wait, let me check the recursive formula again. It says a_{n+1} = 2S_n. So, a_2 = 2S_1 = 4, which is correct. Then a_3 = 2S_2 = 12, correct. a_4 = 2S_3 = 36, correct. So, the terms are correct.Therefore, the sum S_7 is indeed 1458, which is option A.Wait, but in the initial calculation, I thought maybe the sum formula was different, but it seems consistent.Alternatively, maybe the problem is asking for something else, but no, it clearly says S_7.Wait, let me check the options again: A:1458, B:1460, C:2184, D:2186.Hmm, 1458 is an option, so maybe that's correct. But I'm confused because 2186 is also an option, which is 2*3^7, but 3^7 is 2187, so 2*2187 is 4374, which is not an option. Wait, no, 3^7 is 2187, so 2*3^6 is 1458, which is S_7.Wait, but if I consider S_n = 2*3^{n-1}, then S_7 = 2*3^6 = 2*729 = 1458. So, that's correct.But why is 2186 an option? Maybe I made a mistake in the formula.Wait, let me think differently. Maybe the sum S_n is not 2*3^{n-1}, but something else.From the recursive formula, a_{n+1} = 2S_n.Also, S_{n} = S_{n-1} + a_n.So, substituting a_n from the recursive formula:a_n = 2S_{n-1}So, S_n = S_{n-1} + 2S_{n-1} = 3S_{n-1}Ah, so S_n = 3S_{n-1}That's a different approach. So, the sum S_n is a geometric sequence with ratio 3.Given S_1 = 2, then S_2 = 3*S_1 = 6, S_3 = 3*S_2 = 18, S_4 = 54, S_5 = 162, S_6 = 486, S_7 = 1458.So, yes, S_7 = 1458, which is option A.Wait, but earlier I thought that the sum S_n = 2*3^{n-1}, which also gives S_7 = 1458.So, both approaches confirm that S_7 is 1458.But then why is 2186 an option? Maybe I'm misunderstanding the problem.Wait, let me read the problem again: "the sum of the first n terms of the sequence {a_n} is S_n, and it satisfies a_1=2 and a_{n+1}=2S_n, then S_7=?"So, the problem is correctly interpreted. Then, S_7 is 1458, which is option A.But in the initial calculation, I thought maybe the sum formula was different, but it seems consistent.Wait, perhaps I made a mistake in the initial step. Let me try to derive the formula again.Given S_n = S_{n-1} + a_nBut a_n = 2S_{n-1} for n >=2So, S_n = S_{n-1} + 2S_{n-1} = 3S_{n-1}Thus, S_n = 3S_{n-1}, which is a geometric sequence with ratio 3.Given S_1 = 2, then S_n = 2*3^{n-1}Therefore, S_7 = 2*3^{6} = 2*729 = 1458.So, yes, 1458 is correct.But then why is 2186 an option? Maybe the problem is asking for something else, but no, it clearly says S_7.Wait, perhaps the problem is in the initial term. Maybe a_1 is considered as the first term, so S_1 = a_1 = 2, S_2 = a_1 + a_2 = 2 + 4 = 6, which is correct.Alternatively, maybe the problem is using a different indexing, but I don't think so.Wait, let me check the options again: A:1458, B:1460, C:2184, D:2186.Hmm, 1458 is an option, so I think that's correct. Maybe 2186 is a distractor, perhaps from a different problem.Alternatively, maybe I made a mistake in the recursive formula.Wait, let me try to compute S_7 again step by step:a_1 = 2, S_1 = 2a_2 = 2*S_1 = 4, S_2 = 2 + 4 = 6a_3 = 2*S_2 = 12, S_3 = 6 + 12 = 18a_4 = 2*S_3 = 36, S_4 = 18 + 36 = 54a_5 = 2*S_4 = 108, S_5 = 54 + 108 = 162a_6 = 2*S_5 = 324, S_6 = 162 + 324 = 486a_7 = 2*S_6 = 972, S_7 = 486 + 972 = 1458Yes, that's consistent. So, S_7 is 1458, which is option A.But wait, in the initial problem statement, the user wrote:"Given that the sum of the first n terms of the sequence {a_n} is S_n, and it satisfies a_1=2 and a_{n+1}=2S_n, then S_7=( )"So, the problem is correctly stated, and the calculations seem correct.Therefore, I think the correct answer is 1458, which is option A.But wait, in the initial problem, the user also wrote:"Please reason step by step, and put your final answer within boxed{}."So, perhaps the assistant made a mistake in the initial thought process, but the correct answer is 1458.Wait, but in the initial assistant's answer, they concluded that S_7 is 2186, which is option D. But according to my calculations, it's 1458.Hmm, maybe the assistant made a mistake. Let me see.In the assistant's thought process, they derived that a_{n+1} = 3a_n, which is correct from n=2 onwards. Then, they considered the sequence as a geometric sequence with a_1=2 and ratio 3, which is incorrect because a_2=4, not 6. So, the sequence is not a pure geometric sequence from the start, but from a_2 onwards, it is.Therefore, the sum S_n is not simply 2*(3^n -1)/(3-1), but rather, it's 2 + 4*(3^{n-1} -1)/(3-1).Wait, let me compute that:Sum from a_2 to a_n is a geometric series with first term 4, ratio 3, and n-1 terms.So, sum = 4*(3^{n-1} -1)/(3-1) = 2*(3^{n-1} -1)Therefore, S_n = a_1 + sum = 2 + 2*(3^{n-1} -1) = 2 + 2*3^{n-1} -2 = 2*3^{n-1}Which is the same as before, so S_n = 2*3^{n-1}Thus, S_7 = 2*3^6 = 2*729 = 1458.So, the assistant's initial thought process was incorrect because they assumed the entire sequence is geometric from a_1, but it's only geometric from a_2 onwards.Therefore, the correct answer is 1458, which is option A.But wait, the assistant concluded 2186, which is 2*3^7, but 3^7 is 2187, so 2*2187 is 4374, which is not an option. Wait, no, 3^7 is 2187, so 2*3^6 is 1458.Wait, perhaps the assistant made a mistake in the formula, thinking that S_n = 2*3^n instead of 2*3^{n-1}.If S_n = 2*3^n, then S_7 = 2*2187 = 4374, which is not an option. But if S_n = 2*3^{n-1}, then S_7 = 1458.Therefore, the correct answer is 1458, which is option A.But in the initial problem, the user wrote:"Given that the sum of the first n terms of the sequence {a_n} is S_n, and it satisfies a_1=2 and a_{n+1}=2S_n, then S_7=( )"So, the problem is correctly stated, and the calculations seem correct.Therefore, I think the correct answer is 1458, which is option A.But wait, in the initial assistant's answer, they concluded 2186, which is option D. So, perhaps the assistant made a mistake.Wait, let me check the assistant's thought process:They wrote:"Given the sequence {a_n} and its sum S_n, we have the initial condition a_1=2 and the recursive formula a_{n+1}=2S_n. Let's find S_7.First, we observe that for n >= 2,a_{n+1}=2S_n and a_n=2S_{n-1}.This leads to the relationship between consecutive terms of the sequence:a_{n+1}-a_n=2S_n - 2S_{n-1}.Given S_n - S_{n-1} = a_n, we simplify the equation to:a_{n+1}-a_n=2a_n.Rearranging, we find the ratio between consecutive terms:a_{n+1}/a_n=3.This indicates that {a_n} is a geometric sequence with the first term a_1=2 and common ratio r=3.The sum of the first n terms of a geometric sequence is given by:S_n = a_1*(1 - r^n)/(1 - r).Substituting a_1=2, r=3, and n=7, we get:S_7 = 2*(1 - 3^7)/(1 - 3) = 2*(1 - 2187)/(-2) = (2*(-2186))/(-2) = 2186.Therefore, the correct answer is D:2186."Wait, but in this thought process, the assistant assumes that the entire sequence {a_n} is geometric from a_1 with ratio 3, which is incorrect because a_2=4, which is 2*2, not 3*2=6. So, the sequence is not geometric from a_1, but from a_2 onwards.Therefore, the assistant's mistake was assuming the entire sequence is geometric, leading to an incorrect sum formula.Thus, the correct answer is 1458, which is option A.

💡Okay, so I have this geometry problem here. It says that lines p and q are parallel. Angle E is 150 degrees, and angle G is 70 degrees. I need to find the measure of angle F. Hmm, let me try to visualize this.First, I remember that when two lines are parallel, the corresponding angles are equal, and the consecutive interior angles are supplementary. That might come in handy here. The Asymptote code is provided, but I'm not super familiar with Asymptote, so I'll try to imagine the diagram based on the description.There are two parallel lines, p and q. Points E, F, and G are mentioned. E is labeled at the top, F at the bottom, and G somewhere else. The angles at E and G are given as 150 degrees and 70 degrees, respectively. I need to find the measure of angle F.Let me think about how these angles relate. Since lines p and q are parallel, any transversal cutting through them will create corresponding angles that are equal. Maybe I can draw a transversal that connects points E and G or something like that.Wait, the Asymptote code shows that there's a segment from E to some point, and another segment from F to G. Maybe E and G are on different lines, and F is where they intersect? I'm not entirely sure, but perhaps drawing some auxiliary lines might help.I remember that sometimes, when dealing with parallel lines and angles, drawing a line parallel to the given lines can help create triangles or other shapes where I can apply angle properties. Maybe I can draw a line from point G that's parallel to lines p and q. Let's call this new line GH, where H is some point on line q.If I draw GH parallel to p and q, then GH is also parallel to both lines. Now, looking at triangle EGH, since GH is parallel to p, the angle at E, which is 150 degrees, and the angle at G should relate somehow. Wait, if GH is parallel to p, then angle EGH should be equal to the angle formed by the transversal at E. But angle E is 150 degrees, so maybe angle EGH is supplementary to that?Hold on, consecutive interior angles are supplementary when two parallel lines are cut by a transversal. So, if angle E is 150 degrees, then angle EGH should be 180 - 150 = 30 degrees. That makes sense because they are on the same side of the transversal and should add up to 180 degrees.Now, looking at triangle EGH, I know two angles: angle E is 150 degrees, and angle EGH is 30 degrees. Wait, no, actually, angle E is at point E, which is on line p, and angle EGH is at point G on line GH. Maybe I need to consider triangle EFG or something else.Wait, maybe I should look at the angles around point G. The angle at G is given as 70 degrees. Since GH is parallel to q, and q is parallel to p, then angle GHF should be equal to angle G, which is 70 degrees. So, angle GHF is 70 degrees.Now, if I look at triangle GHF, I know angle GHF is 70 degrees, and angle EGH is 30 degrees. Wait, no, angle EGH is 30 degrees, but that's at point G. Maybe I need to consider the straight line at point F.Since GH is parallel to q, and F is on q, then the angle at F should be related to angle GHF. If I think about the straight line at F, the angles on a straight line add up to 180 degrees. So, if angle GHF is 70 degrees, then angle F should be 180 - 70 = 110 degrees.Wait, let me make sure I'm not mixing up any points. So, GH is parallel to q, and at point G, the angle is 70 degrees. Since GH is parallel to q, the angle formed at G by the transversal should correspond to angle F. So, angle F is supplementary to angle GHF, which is 70 degrees, making angle F equal to 110 degrees.I think that makes sense. Let me recap:1. Lines p and q are parallel.2. Angle E is 150 degrees, so the consecutive interior angle at G (angle EGH) is 30 degrees.3. Since GH is parallel to q, angle GHF is equal to angle G, which is 70 degrees.4. At point F, the angle is supplementary to angle GHF, so 180 - 70 = 110 degrees.Yeah, that seems right. I don't think I made any mistakes in the reasoning. So, angle F should be 110 degrees.

💡Okay, so I have this problem here about quiz scores in a class. Let me try to understand it step by step. First, the problem says there are n students in the class, and n is greater than 15. The average quiz score for all the students is 10. That means if I add up all the quiz scores of the n students and then divide by n, I get 10. Then, it mentions a group of 15 students who have an average score of 16. So, these 15 students scored higher than the overall average. The question is asking for the mean of the quiz scores of the remaining students in terms of n. That means I need to find the average score for the students who are not in this group of 15, and express it using n.Alright, let me break this down. I know that the average is the total sum divided by the number of items. So, if I can find the total sum of the quiz scores for the entire class and subtract the total sum of the 15 students, I'll get the total sum for the remaining students. Then, I can divide that by the number of remaining students to find their average.Let me write down what I know:1. Total number of students: n2. Average score for all students: 103. Number of students in the group with higher average: 154. Average score for this group: 16First, let me find the total quiz score for all n students. Since the average is 10, the total score is 10 times n, which is 10n.Next, I need the total quiz score for the 15 students who scored higher. Their average is 16, so the total score for them is 16 times 15. Let me calculate that: 16 * 15 = 240. So, these 15 students contributed a total of 240 points.Now, to find the total score for the remaining students, I subtract the total of the 15 students from the overall total. That would be 10n - 240.But how many students are remaining? Since there are n students in total and 15 are in the group, the remaining number of students is n - 15.So, the average score for the remaining students would be the total score of the remaining students divided by the number of remaining students. That would be (10n - 240) divided by (n - 15).Let me write that as a formula:Average score of remaining students = (10n - 240) / (n - 15)Hmm, looking at the answer choices, option A is exactly that: (10n - 240)/(n - 15). So, that seems to be the correct answer.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Total score for all students: 10n. That makes sense because average times number of students is total.2. Total score for the 15 students: 16 * 15 = 240. Yep, that's correct.3. Total score for remaining students: 10n - 240. That seems right because we subtract the higher group's total from the overall total.4. Number of remaining students: n - 15. That's straightforward.5. Average for remaining students: (10n - 240)/(n - 15). Yep, that looks correct.I think I did everything right. So, the answer should be option A.**Final Answer**boxed{A}

💡Okay, so I have this problem: If ( x^2 + y^2 = 4 ), what is the largest possible value of ( |x| + |y| )? Hmm, interesting. I remember that ( x^2 + y^2 = 4 ) is the equation of a circle with radius 2 centered at the origin. So, all the points (x, y) lie on this circle. I need to find the maximum value of ( |x| + |y| ) for points on this circle.First, let me think about what ( |x| + |y| ) represents. It's the sum of the absolute values of x and y. I know that ( |x| + |y| ) is related to the L1 norm, which measures distance in a different way than the usual Euclidean distance (which is what ( x^2 + y^2 ) represents). So, I'm trying to maximize the L1 norm on the circle of radius 2.I remember that for optimization problems like this, especially involving circles and other shapes, using trigonometric substitutions can be helpful. Since ( x^2 + y^2 = 4 ), I can parameterize x and y using sine and cosine. Let me try that.Let me set ( x = 2costheta ) and ( y = 2sintheta ), where ( theta ) is the angle made with the positive x-axis. This substitution should satisfy the equation ( x^2 + y^2 = 4 ) because ( (2costheta)^2 + (2sintheta)^2 = 4(cos^2theta + sin^2theta) = 4(1) = 4 ). Perfect.Now, substituting these into ( |x| + |y| ), I get:[|x| + |y| = |2costheta| + |2sintheta| = 2|costheta| + 2|sintheta| = 2(|costheta| + |sintheta|)]So, I need to maximize ( 2(|costheta| + |sintheta|) ). Since 2 is a constant factor, I can instead focus on maximizing ( |costheta| + |sintheta| ).Let me denote ( f(theta) = |costheta| + |sintheta| ). I need to find the maximum value of ( f(theta) ). Since both ( |costheta| ) and ( |sintheta| ) are non-negative, their sum will also be non-negative.I know that ( costheta ) and ( sintheta ) have maximum values of 1 and minimum values of -1, but since we're taking absolute values, we can consider ( theta ) in the first quadrant where both ( costheta ) and ( sintheta ) are positive. This is because the function ( f(theta) ) is symmetric in all four quadrants. So, without loss of generality, I can assume ( theta ) is in the first quadrant, i.e., ( 0 leq theta leq frac{pi}{2} ).So, in this interval, ( f(theta) = costheta + sintheta ). Now, I need to find the maximum of ( costheta + sintheta ) for ( 0 leq theta leq frac{pi}{2} ).I remember that the maximum of ( acostheta + bsintheta ) is ( sqrt{a^2 + b^2} ). In this case, a = 1 and b = 1, so the maximum should be ( sqrt{1^2 + 1^2} = sqrt{2} ). Therefore, the maximum of ( costheta + sintheta ) is ( sqrt{2} ).Wait, let me verify that. If I take the derivative of ( f(theta) = costheta + sintheta ), the derivative is ( f'(theta) = -sintheta + costheta ). Setting this equal to zero for critical points:[-sintheta + costheta = 0 implies costheta = sintheta implies tantheta = 1 implies theta = frac{pi}{4}]So, at ( theta = frac{pi}{4} ), we have a critical point. Let's evaluate ( f(theta) ) at ( theta = frac{pi}{4} ):[fleft(frac{pi}{4}right) = cosleft(frac{pi}{4}right) + sinleft(frac{pi}{4}right) = frac{sqrt{2}}{2} + frac{sqrt{2}}{2} = sqrt{2}]And at the endpoints:- At ( theta = 0 ): ( f(0) = cos(0) + sin(0) = 1 + 0 = 1 )- At ( theta = frac{pi}{2} ): ( fleft(frac{pi}{2}right) = cosleft(frac{pi}{2}right) + sinleft(frac{pi}{2}right) = 0 + 1 = 1 )So, indeed, the maximum occurs at ( theta = frac{pi}{4} ) and is equal to ( sqrt{2} ).Therefore, going back to the original expression, ( |x| + |y| = 2(|costheta| + |sintheta|) ), and the maximum of ( |costheta| + |sintheta| ) is ( sqrt{2} ). So, the maximum value of ( |x| + |y| ) is ( 2 times sqrt{2} = 2sqrt{2} ).Wait, let me think if there's another way to approach this problem without using trigonometric substitution. Maybe using the Cauchy-Schwarz inequality or some other inequality.I recall that for vectors, the Cauchy-Schwarz inequality states that ( |a cdot b| leq ||a|| cdot ||b|| ). Maybe I can represent ( |x| + |y| ) as a dot product.Let me consider vectors ( (|x|, |y|) ) and ( (1, 1) ). Then, the dot product is ( |x| times 1 + |y| times 1 = |x| + |y| ). The Cauchy-Schwarz inequality tells us that:[|x| + |y| leq sqrt{(|x|^2 + |y|^2)} times sqrt{1^2 + 1^2} = sqrt{x^2 + y^2} times sqrt{2}]Since ( x^2 + y^2 = 4 ), this becomes:[|x| + |y| leq sqrt{4} times sqrt{2} = 2 times sqrt{2} = 2sqrt{2}]So, this confirms the earlier result. The maximum value of ( |x| + |y| ) is ( 2sqrt{2} ).Another approach might be to use the method of Lagrange multipliers, which is a technique from calculus used to find the extrema of a function subject to equality constraints.Let me set up the problem using Lagrange multipliers. We want to maximize ( f(x, y) = |x| + |y| ) subject to the constraint ( g(x, y) = x^2 + y^2 - 4 = 0 ).However, since ( |x| ) and ( |y| ) are not differentiable at x=0 and y=0 respectively, we might need to consider different cases based on the signs of x and y.But since we're looking for the maximum, and the function ( |x| + |y| ) is symmetric in all four quadrants, we can assume without loss of generality that x and y are both non-negative. So, we can consider the first quadrant where x ≥ 0 and y ≥ 0, and then the maximum will be the same in all other quadrants due to symmetry.So, in the first quadrant, the function simplifies to ( f(x, y) = x + y ), and the constraint is ( x^2 + y^2 = 4 ).Now, applying Lagrange multipliers, we set up the equations:[nabla f = lambda nabla g]Calculating the gradients:[nabla f = left( frac{partial f}{partial x}, frac{partial f}{partial y} right) = (1, 1)][nabla g = left( frac{partial g}{partial x}, frac{partial g}{partial y} right) = (2x, 2y)]So, setting ( nabla f = lambda nabla g ):[1 = lambda cdot 2x quad text{and} quad 1 = lambda cdot 2y]From both equations, we have:[2lambda x = 1 quad text{and} quad 2lambda y = 1]Therefore, ( 2lambda x = 2lambda y implies x = y ) (assuming ( lambda neq 0 ), which it can't be because otherwise, the equations would imply 1=0, which is impossible).So, x = y. Substituting back into the constraint ( x^2 + y^2 = 4 ):[x^2 + x^2 = 4 implies 2x^2 = 4 implies x^2 = 2 implies x = sqrt{2}]Since we're in the first quadrant, x is positive, so x = y = ( sqrt{2} ).Therefore, the maximum value of ( x + y ) in the first quadrant is ( sqrt{2} + sqrt{2} = 2sqrt{2} ), which matches our previous results.Just to make sure, let me check another point on the circle. For example, at (2, 0), ( |x| + |y| = 2 + 0 = 2 ). Similarly, at (0, 2), it's also 2. At (1, ( sqrt{3} )), ( |x| + |y| = 1 + sqrt{3} approx 1 + 1.732 = 2.732 ), which is less than ( 2sqrt{2} approx 2.828 ). So, it seems that ( 2sqrt{2} ) is indeed larger.Another point: let's take ( theta = frac{pi}{6} ). Then, x = 2cos(π/6) = 2*(√3/2) = √3 ≈ 1.732, y = 2sin(π/6) = 2*(1/2) = 1. So, ( |x| + |y| = √3 + 1 ≈ 2.732 ), which is still less than ( 2sqrt{2} ).What about ( theta = frac{pi}{3} )? x = 2cos(π/3) = 2*(1/2) = 1, y = 2sin(π/3) = 2*(√3/2) = √3 ≈ 1.732. So, ( |x| + |y| = 1 + √3 ≈ 2.732 ), same as before.So, it seems that as θ approaches π/4, the value of ( |x| + |y| ) increases, reaching the maximum at θ = π/4, and then decreases again as θ moves towards π/2.Therefore, all methods I've tried so far—trigonometric substitution, Cauchy-Schwarz inequality, and Lagrange multipliers—lead me to the conclusion that the maximum value of ( |x| + |y| ) is ( 2sqrt{2} ).Just to recap:1. **Trigonometric Substitution**: Expressed x and y in terms of sine and cosine, transformed the problem into maximizing a function of θ, found the maximum using calculus, and confirmed it with the Cauchy-Schwarz inequality.2. **Cauchy-Schwarz Inequality**: Directly applied the inequality to bound ( |x| + |y| ) and found the maximum value.3. **Lagrange Multipliers**: Set up the optimization problem with constraints, solved the system of equations, and found the critical point where x = y, leading to the maximum value.All these methods consistently give the same result, so I feel confident that ( 2sqrt{2} ) is indeed the correct maximum value.I also thought about specific points on the circle to verify. Points like (2, 0), (0, 2), (1, √3), etc., all gave values less than ( 2sqrt{2} ), which supports the conclusion.In summary, through multiple approaches and checks, I've determined that the largest possible value of ( |x| + |y| ) given ( x^2 + y^2 = 4 ) is ( 2sqrt{2} ).**Final Answer**The largest possible value of ( |x| + |y| ) is boxed{2sqrt{2}}.

💡First, I identify the total number of fruits and the quantities of each known type of fruit.There are 58 fruits in total across five baskets.- One basket contains 18 mangoes.- Another basket contains 12 pawpaws.- The last two baskets contain the same number of kiwis and lemons, respectively. Since there are 9 lemons, there must also be 9 kiwis.Next, I calculate the total number of known fruits:18 mangoes + 12 pawpaws + 9 lemons + 9 kiwis = 48 fruitsTo find the number of pears, I subtract the known fruits from the total:58 total fruits - 48 known fruits = 10 pearsTherefore, there are 10 pears in the basket with pears.