🧠:<think>Alright, so I've got this problem about Metal Gear Solid Snake Eater, which is cool because I remember playing that game a while back. It's about Snake infiltrating a base, which is modeled as a graph with rooms as nodes and corridors as edges. There are two parts to the problem.Starting with part 1: There are 10 rooms, each labeled from 1 to 10. Each room has a security level given by S_i = 2i + 3. I need to find the total security level of the base, which is just the sum of all S_i from i=1 to 10.Okay, so let's break that down. For each room i, the security level is 2i + 3. So for room 1, it's 2*1 + 3 = 5. For room 2, it's 2*2 + 3 = 7, and so on up to room 10. So I can write out each S_i:S1 = 2*1 + 3 = 5S2 = 2*2 + 3 = 7S3 = 2*3 + 3 = 9S4 = 2*4 + 3 = 11S5 = 2*5 + 3 = 13S6 = 2*6 + 3 = 15S7 = 2*7 + 3 = 17S8 = 2*8 + 3 = 19S9 = 2*9 + 3 = 21S10 = 2*10 + 3 = 23So now, to find the total security level, I need to add all these up. Let me list them:5, 7, 9, 11, 13, 15, 17, 19, 21, 23.Hmm, that's an arithmetic sequence where each term increases by 2. The first term is 5, the last term is 23, and there are 10 terms. The formula for the sum of an arithmetic series is (n/2)*(first term + last term). So plugging in the numbers:Sum = (10/2)*(5 + 23) = 5*28 = 140.Wait, let me verify that. Alternatively, I can add them step by step:5 + 7 = 1212 + 9 = 2121 + 11 = 3232 + 13 = 4545 + 15 = 6060 + 17 = 7777 + 19 = 9696 + 21 = 117117 + 23 = 140.Yep, that's 140. So the total security level is 140.Moving on to part 2: Snake starts in room 1 and needs to reach room 10. The corridors (edges) have weights equal to the absolute difference in security levels of the rooms they connect. We need to find the shortest path from room 1 to room 10, minimizing the total weight, which is the sum of the absolute differences along the path.First, I need to figure out the structure of the graph. The problem says it's a connected graph with 10 nodes, but it doesn't specify how the rooms are connected. Hmm, that's a bit of a problem because without knowing the connections, I can't determine the shortest path. Wait, maybe I'm supposed to assume it's a complete graph? Or perhaps it's a linear graph where each room is connected to the next? The problem doesn't specify, so maybe I need to make an assumption here.Looking back at the problem statement: It says the base is structured as a series of interconnected rooms, each with varying levels of security. It's modeled as a graph where each node is a room, and each edge is a corridor. But it doesn't specify the connections. Hmm. Maybe it's a straight line from 1 to 10, each connected in sequence? Or perhaps it's a more complex graph.Wait, in the original Metal Gear Solid: Snake Eater, the base is called the "Shadow Moses Island" and the layout is more like a complex structure with multiple paths. But since the problem doesn't specify, maybe it's just a simple linear graph for the sake of the problem? Or perhaps it's a complete graph where every room is connected to every other room.But if it's a connected graph, it could be any connected graph, so without more information, I might have to assume a specific structure. Alternatively, perhaps the problem is expecting me to realize that the shortest path would be the path that goes through rooms with increasing security levels, minimizing the differences. But without knowing the connections, it's tricky.Wait, maybe the graph is a straight line from 1 to 10, so each room is connected to the next one. So room 1 connected to 2, 2 to 3, ..., 9 to 10. If that's the case, then the path from 1 to 10 is just the straight path, and the total weight would be the sum of the differences between consecutive rooms.Alternatively, if it's a complete graph, then Snake can go directly from 1 to 10, but the weight would be |S10 - S1| = |23 - 5| = 18. But if it's a complete graph, that would be the shortest path. However, if the graph isn't complete, maybe the shortest path is longer.But since the problem doesn't specify the graph structure, I might have to make an assumption. Maybe it's a straight line, so the path is 1-2-3-...-10. Let's go with that for now.So, if it's a straight line, the path is 1 to 2 to 3 to ... to 10. The total weight would be the sum of |S2 - S1| + |S3 - S2| + ... + |S10 - S9|.But wait, since the security levels are increasing (each S_i = 2i + 3, so each subsequent room has a higher security level), the absolute differences are just S_{i+1} - S_i.Calculating each difference:S2 - S1 = 7 - 5 = 2S3 - S2 = 9 - 7 = 2Similarly, each difference is 2. Since there are 9 edges from 1 to 10, the total weight would be 9*2 = 18.But wait, if it's a straight line, the total weight is 18. However, if there are other paths, maybe with fewer edges or smaller differences, the total weight could be less. But without knowing the graph structure, it's hard to say.Alternatively, if the graph is a complete graph, the direct path from 1 to 10 would have a weight of 18, which is the same as the straight line path. So in that case, both paths have the same total weight.But maybe the graph is a tree or something else. Hmm.Wait, another thought: Since the security levels are S_i = 2i + 3, which is an arithmetic progression, the differences between consecutive rooms are constant (2). So if the graph allows moving to any room, the minimal path would be to jump directly from 1 to 10, which has a weight of 18, same as the straight path. But if the graph doesn't allow that, then the straight path is the only way, with total weight 18.But the problem says it's a connected graph, so there must be a path from 1 to 10, but it doesn't specify the edges. So perhaps the minimal possible total weight is 18, achieved by either the direct path or the straight path.Wait, but if the graph allows for other connections, maybe there's a way to have a path with smaller total weight. For example, if there's a connection from 1 to 10, that's 18. If there's a connection from 1 to 5, then 5 to 10, the total weight would be |S5 - S1| + |S10 - S5| = |13 - 5| + |23 - 13| = 8 + 10 = 18. Same total.Alternatively, if there's a connection from 1 to 2, then 2 to 10, the weight would be |7 - 5| + |23 - 7| = 2 + 16 = 18. Still 18.Wait, so regardless of the path, as long as the security levels are in an arithmetic progression, the total weight from 1 to 10 is always 18? Because the differences add up to the same total.Wait, let's test that. Suppose we have a path that goes 1-3-10. The weight would be |S3 - S1| + |S10 - S3| = |9 - 5| + |23 - 9| = 4 + 14 = 18. Same result.Similarly, 1-4-10: |11 - 5| + |23 - 11| = 6 + 12 = 18.So regardless of the path, as long as the security levels are in an arithmetic progression, the total weight from 1 to 10 is always 18. That's interesting.Therefore, the shortest path has a total weight of 18, and any path from 1 to 10 will have the same total weight. So the minimal total weight is 18.But wait, is that always true? Let me think. If the security levels are in an arithmetic progression, then the sum of the differences along any path from 1 to 10 will be equal to S10 - S1, because the differences telescope. For example, in a straight path: (S2 - S1) + (S3 - S2) + ... + (S10 - S9) = S10 - S1.Similarly, in any other path, say 1-3-10: (S3 - S1) + (S10 - S3) = S10 - S1.So regardless of the path, the total weight is always S10 - S1, which is 23 - 5 = 18.Therefore, the minimal total weight is 18, and any path from 1 to 10 will achieve this. So the shortest path can be any path, but the total weight is fixed at 18.But wait, in graph theory, the shortest path is usually the one with the least number of edges or the least total weight. In this case, since the total weight is fixed regardless of the path, any path is equally good in terms of total weight. However, if we consider the number of edges, the direct path from 1 to 10 (if it exists) would be the shortest in terms of the number of edges, but since the total weight is the same, it doesn't matter.But the problem asks for the shortest path to minimize the total security difference encountered, which is the total weight. So since all paths have the same total weight, any path is acceptable. However, the problem might expect us to find a specific path, perhaps the one with the least number of edges, which would be the direct path if it exists.But since the graph is connected, it's possible that there's a direct edge from 1 to 10, making the path just 1-10 with total weight 18. If not, the next best is the straight path 1-2-3-...-10, which also has total weight 18.Therefore, the shortest path is either 1-10 directly or 1-2-3-...-10, both with total weight 18.But since the problem doesn't specify the graph structure, I think the answer is that the total weight is 18, and the path can be any path from 1 to 10, but the minimal total weight is 18.Wait, but the problem says "the corridors between them form a connected graph." It doesn't specify whether it's a tree or has cycles. But regardless, the total weight from 1 to 10 is fixed at 18.So, to sum up:1. Total security level is 140.2. The shortest path has a total weight of 18, and the path can be any path from 1 to 10, but the minimal total weight is 18.But the problem asks to represent the path and compute the total weight. Since the path isn't uniquely determined without knowing the graph structure, but the total weight is fixed, I think the answer is that the total weight is 18, and the path can be any path from 1 to 10.However, perhaps the problem expects us to assume a specific graph structure, like a straight line. If that's the case, the path is 1-2-3-...-10, with total weight 18.Alternatively, if it's a complete graph, the path is 1-10, weight 18.But since the problem doesn't specify, I think the answer is that the total weight is 18, and the path can be any path from 1 to 10.But maybe I should check if the graph is a straight line. Let me think: If it's a straight line, then the path is 1-2-3-...-10, total weight 18. If it's a complete graph, the path is 1-10, total weight 18. If it's some other connected graph, the total weight is still 18.Therefore, regardless of the graph structure, the total weight is 18.So, the answers are:1. Total security level: 140.2. The shortest path has a total weight of 18, and the path can be any path from 1 to 10, but the minimal total weight is 18.But the problem says "determine the shortest path Snake must take to minimize the total security difference encountered. Represent the path and compute the total weight of the shortest path."Since the total weight is fixed, any path is acceptable, but perhaps the problem expects the path with the least number of edges, which would be the direct path if it exists. But without knowing the graph, we can't specify the exact path, only the total weight.Alternatively, maybe the graph is a straight line, so the path is 1-2-3-...-10.But I think the key point is that the total weight is 18, regardless of the path, because of the arithmetic progression of security levels.So, to answer:1. Total security level is 140.2. The shortest path has a total weight of 18. The path can vary depending on the graph structure, but the minimal total weight is 18.But since the problem asks to represent the path, perhaps it's expecting the straight line path, so 1-2-3-...-10, with total weight 18.Alternatively, if the graph allows a direct path, then 1-10 with weight 18.But without knowing the graph, I think the answer is that the total weight is 18, and the path can be any path from 1 to 10.However, in the context of the problem, since it's a connected graph, the minimal path in terms of total weight is 18, achieved by any path from 1 to 10.So, to conclude:1. Total security level: 140.2. The shortest path has a total weight of 18. The path can be any path from room 1 to room 10, but the minimal total weight encountered is 18.But perhaps the problem expects a specific path, like the straight line, so I'll go with that.So, final answers:1. The total security level is 140.2. The shortest path is 1-2-3-4-5-6-7-8-9-10, with a total weight of 18.But wait, the problem says "the corridors between them form a connected graph." It doesn't specify it's a straight line, so maybe it's a different structure. For example, maybe it's a binary tree or something else. But without knowing, I can't specify the exact path.Alternatively, perhaps the graph is a straight line, so the path is 1-2-3-...-10.But to be safe, I think the answer is that the total weight is 18, and the path can be any path from 1 to 10.But the problem asks to "represent the path," so maybe it's expecting a specific sequence. Since the security levels are increasing, perhaps the optimal path is to go through rooms with increasing security levels, minimizing the differences. But since the differences are fixed, any path will have the same total weight.Therefore, the minimal total weight is 18, and the path can be any path from 1 to 10.But to represent a specific path, perhaps the straight line is the most straightforward, so I'll go with that.So, final answers:1. Total security level: 140.2. The shortest path is 1-2-3-4-5-6-7-8-9-10, with a total weight of 18.</think>

🧠:<think>Alright, so I've got these two problems to solve, both related to cryptography and algorithmic complexity. Let me try to break them down one by one.Starting with the first problem about cryptographic hash functions. The question is about calculating the probability of a collision after a certain number of hash operations. The hash function H outputs a 256-bit hash, and we need to find the probability when n = 2^64 operations are done. The collision resistance is supposed to be at least 2^128, so I need to see if this probability meets that requirement.I remember that the birthday problem is related to collision probabilities. The formula for the probability of a collision after n hashes is approximately n^2 / (2 * 2^m), where m is the number of bits in the hash. Here, m is 256, so the denominator becomes 2^(257). Plugging in n = 2^64, the numerator becomes (2^64)^2 = 2^128. So the probability is 2^128 / 2^257, which simplifies to 1 / 2^129. That's a very small probability, much less than 1/(2^128), which is the collision resistance target. So it seems like the probability is acceptable, but I need to make sure I'm not missing anything here.Wait, actually, the collision resistance is the difficulty of finding a collision, which is about 2^(m/2). For a 256-bit hash, that's 2^128. So if n is 2^64, the expected number of collisions is roughly n^2 / 2^256. Plugging in n=2^64, that's (2^128) / (2^256) = 1 / 2^128. So the probability is about 1/2^128, which is exactly the collision resistance. Hmm, so maybe the initial calculation was a bit off because I included an extra factor of 2 in the denominator. Let me double-check.The exact formula for the probability of at least one collision is approximately 1 - e^(-n^2 / (2 * 2^256)). For large n, this can be approximated as n^2 / (2 * 2^256). So with n=2^64, it's (2^128) / (2 * 2^256) = 1 / (2^129). So the probability is 1/(2^129), which is half of the collision resistance target. That still meets the requirement since it's less than 1/(2^128). So I think that's correct.Moving on to the second problem about elliptic curve cryptography and Pollard's Rho algorithm. The question is about the time complexity of solving the ECDLP using Pollard's Rho. The curve is defined over a finite field F_p with p ≈ 2^256. The point Q is kP, and we need to find k.I recall that Pollard's Rho algorithm has a time complexity of O(√n), where n is the order of the group. In ECC, the order is roughly p, so the complexity should be O(√p). Since p is about 2^256, √p is 2^128. So the time complexity is O(2^128). But wait, in practice, Pollard's Rho is probabilistic and has some constants involved, but asymptotically, it's dominated by the square root term.So in terms of p, the complexity is O(√p). But since p is 2^256, √p is 2^128, so the time complexity is O(2^128) operations. That makes sense because ECC's security is based on the difficulty of ECDLP, which is why it's considered secure for large p.I should also consider if there are any optimizations or better algorithms, but as far as I know, Pollard's Rho is still the best known algorithm for ECDLP, and its complexity is indeed O(√p). So I think that's the answer.Wait, but sometimes people talk about the complexity in terms of bit operations or something else. But since the question asks in terms of p, I think O(√p) is the right way to express it. So yeah, that should be the answer.Final Answer1. The probability of a collision is boxed{dfrac{1}{2^{129}}}.2. The time complexity of solving the ECDLP using Pollard's Rho is boxed{O(2^{128})}.</think>

🧠:<think>Okay, so I have this problem about a roller derby athlete who's creating a new training regimen. She's using a circular track with a radius of 10 meters. She skates along an inward spiral path described by the polar equation ( r(theta) = 10 - 0.5theta ). There are two parts to the problem: first, finding the angular displacement ( theta ) when she reaches the center, and second, calculating the total distance she skates along this spiral.Starting with the first part: finding the angular displacement when she reaches the center. Hmm, so the center of the track would be where the radial distance ( r ) is zero. That makes sense because the center is at ( r = 0 ). So, I need to solve for ( theta ) when ( r(theta) = 0 ).Given the equation ( r(theta) = 10 - 0.5theta ), I can set that equal to zero and solve for ( theta ):( 0 = 10 - 0.5theta )Let me rearrange this equation to solve for ( theta ). Subtract 10 from both sides:( -10 = -0.5theta )Then, divide both sides by -0.5:( theta = (-10)/(-0.5) )Calculating that, ( (-10)/(-0.5) = 20 ). So, ( theta = 20 ) radians. That seems straightforward. So, the angular displacement when she reaches the center is 20 radians.Wait, let me just verify that. If ( theta = 20 ), then plugging back into the equation:( r(20) = 10 - 0.5*20 = 10 - 10 = 0 ). Yep, that checks out. So, part one is done, ( theta = 20 ) radians.Moving on to the second part: finding the total distance she skated along the spiral until she reaches the center. Hmm, so this is the length of the spiral from the starting point to the center.I remember that the formula for the length of a polar curve ( r(theta) ) from ( theta = a ) to ( theta = b ) is given by the integral:( L = int_{a}^{b} sqrt{ left( frac{dr}{dtheta} right)^2 + r^2 } dtheta )So, in this case, ( r(theta) = 10 - 0.5theta ), and we need to find the length from ( theta = 0 ) to ( theta = 20 ) radians.First, let's compute ( frac{dr}{dtheta} ). The derivative of ( r ) with respect to ( theta ) is:( frac{dr}{dtheta} = frac{d}{dtheta} (10 - 0.5theta) = -0.5 )So, that's a constant, which simplifies things.Now, plug ( r(theta) ) and ( frac{dr}{dtheta} ) into the arc length formula:( L = int_{0}^{20} sqrt{ (-0.5)^2 + (10 - 0.5theta)^2 } dtheta )Simplify the expression under the square root:( (-0.5)^2 = 0.25 )So,( L = int_{0}^{20} sqrt{ 0.25 + (10 - 0.5theta)^2 } dtheta )Hmm, this integral looks a bit complicated, but maybe we can simplify it.Let me denote ( u = 10 - 0.5theta ). Then, ( du/dtheta = -0.5 ), so ( dtheta = -2 du ). Let's see how the limits change. When ( theta = 0 ), ( u = 10 - 0 = 10 ). When ( theta = 20 ), ( u = 10 - 0.5*20 = 10 - 10 = 0 ). So, the integral becomes:( L = int_{u=10}^{u=0} sqrt{0.25 + u^2} (-2 du) )The negative sign flips the limits:( L = 2 int_{0}^{10} sqrt{0.25 + u^2} du )So, now we have:( L = 2 int_{0}^{10} sqrt{u^2 + 0.25} du )This integral is a standard form. The integral of ( sqrt{u^2 + a^2} du ) is:( frac{u}{2} sqrt{u^2 + a^2} + frac{a^2}{2} lnleft( u + sqrt{u^2 + a^2} right) + C )In our case, ( a = 0.5 ), since ( a^2 = 0.25 ). So, applying this formula:( int sqrt{u^2 + 0.25} du = frac{u}{2} sqrt{u^2 + 0.25} + frac{0.25}{2} lnleft( u + sqrt{u^2 + 0.25} right) + C )Simplify:( = frac{u}{2} sqrt{u^2 + 0.25} + frac{0.125}{1} lnleft( u + sqrt{u^2 + 0.25} right) + C )So, evaluating from 0 to 10:First, plug in ( u = 10 ):( frac{10}{2} sqrt{10^2 + 0.25} + 0.125 lnleft( 10 + sqrt{10^2 + 0.25} right) )Simplify:( 5 sqrt{100 + 0.25} + 0.125 lnleft( 10 + sqrt{100.25} right) )Calculate ( sqrt{100.25} ). Hmm, 100.25 is 100 + 0.25, so sqrt(100.25) is sqrt(100 + 0.25). Since 10^2 = 100, and 10.0125^2 ≈ 100.25. Wait, actually, 10.0125^2 = (10 + 0.0125)^2 = 100 + 2*10*0.0125 + 0.0125^2 = 100 + 0.25 + 0.00015625 ≈ 100.25015625. So, sqrt(100.25) ≈ 10.0125.But maybe I can write it as ( sqrt{100.25} = sqrt{401/4} = sqrt{401}/2 ). Because 100.25 is 401/4. So, sqrt(401)/2 ≈ 20.02499/2 ≈ 10.0125. So, that's consistent.So, back to the expression:( 5 * (sqrt(401)/2) + 0.125 * ln(10 + sqrt(401)/2) )Simplify:( (5/2) sqrt(401) + 0.125 ln(10 + sqrt(401)/2) )Similarly, plug in ( u = 0 ):( frac{0}{2} sqrt{0 + 0.25} + 0.125 ln(0 + sqrt(0 + 0.25)) )Simplify:( 0 + 0.125 ln(0 + 0.5) )Which is:( 0.125 ln(0.5) )So, putting it all together, the integral from 0 to 10 is:( [ (5/2) sqrt(401) + 0.125 ln(10 + sqrt(401)/2) ] - [ 0.125 ln(0.5) ] )Simplify the logarithmic terms:( 0.125 [ ln(10 + sqrt(401)/2) - ln(0.5) ] )Using logarithm properties, ( ln(a) - ln(b) = ln(a/b) ):( 0.125 lnleft( frac{10 + sqrt(401)/2}{0.5} right) )Simplify the fraction inside the log:( frac{10 + sqrt(401)/2}{0.5} = 2*(10 + sqrt(401)/2) = 20 + sqrt(401) )So, the logarithmic term becomes:( 0.125 ln(20 + sqrt(401)) )Therefore, the integral from 0 to 10 is:( (5/2) sqrt(401) + 0.125 ln(20 + sqrt(401)) )But remember, the total length ( L ) is twice this integral:( L = 2 * [ (5/2) sqrt(401) + 0.125 ln(20 + sqrt(401)) ] )Simplify:( L = 5 sqrt(401) + 0.25 ln(20 + sqrt(401)) )So, that's the exact expression for the total distance. Now, to get a numerical value, I can compute this.First, compute ( sqrt(401) ). Let's see, 20^2 = 400, so sqrt(401) ≈ 20.02499.So, 5 * 20.02499 ≈ 100.12495.Next, compute ( ln(20 + sqrt(401)) ). Since sqrt(401) ≈ 20.02499, so 20 + 20.02499 ≈ 40.02499.So, ln(40.02499). Let me compute that. I know that ln(40) is approximately 3.688879454, and ln(40.02499) is slightly larger. Let me compute it more accurately.Using a calculator approximation:ln(40.02499) ≈ ln(40) + (0.02499)/40 ≈ 3.688879454 + 0.00062475 ≈ 3.689504204.So, approximately 3.6895.Then, 0.25 * 3.6895 ≈ 0.922375.So, adding the two parts together:100.12495 + 0.922375 ≈ 101.047325 meters.So, approximately 101.05 meters.Wait, let me double-check my calculations, especially the integral part.Wait, when I did the substitution, I had ( u = 10 - 0.5theta ), so ( du = -0.5 dtheta ), hence ( dtheta = -2 du ). Then, the limits went from 10 to 0, which became 0 to 10 after flipping, and the integral became 2 times the integral from 0 to 10 of sqrt(u^2 + 0.25) du.Yes, that seems correct.Then, the integral of sqrt(u^2 + a^2) du is (u/2)sqrt(u^2 + a^2) + (a^2 / 2) ln(u + sqrt(u^2 + a^2)).So, with a = 0.5, that's correct.Plugging in u=10: 5*sqrt(100.25) + 0.125*ln(10 + sqrt(100.25)).Similarly, u=0: 0 + 0.125*ln(0.5).So, subtracting, we get 5*sqrt(100.25) + 0.125*(ln(10 + sqrt(100.25)) - ln(0.5)).Which simplifies to 5*sqrt(100.25) + 0.125*ln((10 + sqrt(100.25))/0.5).Which is 5*sqrt(100.25) + 0.125*ln(20 + 2*sqrt(100.25)).Wait, hold on, earlier I thought sqrt(100.25) is sqrt(401)/2, which is correct because 100.25 = 401/4, so sqrt(401)/2.So, 2*sqrt(100.25) = sqrt(401). So, 10 + sqrt(100.25) = 10 + sqrt(401)/2.Then, (10 + sqrt(401)/2)/0.5 = 20 + sqrt(401). So, that part is correct.So, the logarithmic term is ln(20 + sqrt(401)), which is approximately ln(40.02499) ≈ 3.6895.So, 0.125 * 3.6895 ≈ 0.4611875, but wait, no, wait:Wait, the expression was:After substitution, the integral was 2 times [ (5/2) sqrt(401) + 0.125 ln(20 + sqrt(401)) ]Wait, no, let's retrace:Wait, the integral from 0 to 10 was:(5/2) sqrt(401) + 0.125 ln(20 + sqrt(401)).Then, L = 2*(that integral), so:2*(5/2 sqrt(401)) = 5 sqrt(401)2*(0.125 ln(...)) = 0.25 ln(...)So, yes, 5 sqrt(401) + 0.25 ln(20 + sqrt(401)).So, 5*sqrt(401) ≈ 5*20.02499 ≈ 100.124950.25*ln(20 + sqrt(401)) ≈ 0.25*ln(40.02499) ≈ 0.25*3.6895 ≈ 0.922375Adding together: 100.12495 + 0.922375 ≈ 101.047325 meters.So, approximately 101.05 meters.Wait, but let me check if I can get a more precise value for sqrt(401). Let's compute sqrt(401):We know that 20^2 = 400, so sqrt(401) is 20 + (1)/(2*20) approximately, using the binomial approximation.So, sqrt(400 + 1) ≈ 20 + 1/(40) = 20.025. So, that's consistent with our earlier approximation.Similarly, ln(40.02499). Let's compute it more accurately.We know that ln(40) ≈ 3.688879454.Compute ln(40.02499):Using the Taylor series expansion around 40:ln(40 + x) ≈ ln(40) + x/40 - x^2/(2*40^2) + ...Here, x = 0.02499.So,ln(40.02499) ≈ ln(40) + 0.02499/40 - (0.02499)^2/(2*1600)Compute each term:First term: ln(40) ≈ 3.688879454Second term: 0.02499 / 40 ≈ 0.00062475Third term: (0.02499)^2 / (2*1600) ≈ (0.0006245) / 3200 ≈ 0.000000195So, subtracting the third term:≈ 3.688879454 + 0.00062475 - 0.000000195 ≈ 3.689504009So, ln(40.02499) ≈ 3.689504Therefore, 0.25 * ln(40.02499) ≈ 0.25 * 3.689504 ≈ 0.922376So, adding to 5*sqrt(401):5*20.02499 ≈ 100.12495Total L ≈ 100.12495 + 0.922376 ≈ 101.047326 meters.So, approximately 101.05 meters.Is that the total distance? Hmm, let me think if I did everything correctly.Wait, another way to approach this is to parametrize the spiral and compute the arc length.Given that ( r(theta) = 10 - 0.5theta ), and ( theta ) goes from 0 to 20 radians.The formula for arc length in polar coordinates is indeed ( L = int_{a}^{b} sqrt{ (dr/dtheta)^2 + r^2 } dtheta ).We computed ( dr/dtheta = -0.5 ), so ( (dr/dtheta)^2 = 0.25 ).Thus, the integrand becomes ( sqrt{0.25 + (10 - 0.5theta)^2 } ).Which is what we had earlier.So, substitution was correct, and the integral was evaluated properly.Therefore, the total distance is approximately 101.05 meters.But let me see if I can express this in terms of exact expressions.We had:( L = 5 sqrt{401} + frac{1}{4} ln(20 + sqrt{401}) )So, that's an exact expression. If I want to write it in terms of radicals and logarithms, that's as simplified as it gets.Alternatively, if I compute it numerically, it's approximately 101.05 meters.Wait, just to make sure, let me compute the integral numerically using another method, maybe using a calculator or approximate integration.But since I don't have a calculator here, perhaps I can use another substitution or see if the integral can be expressed differently.Alternatively, perhaps using hyperbolic functions? Wait, not sure.Alternatively, maybe I can approximate the integral numerically.Wait, the integral is ( int_{0}^{10} sqrt{u^2 + 0.25} du ). Let me approximate this integral numerically.We can use the trapezoidal rule or Simpson's rule.But since it's a smooth function, Simpson's rule might be more accurate.Let me divide the interval [0,10] into, say, 4 subintervals, each of width 2.5.Compute the function at 0, 2.5, 5, 7.5, 10.Compute ( f(u) = sqrt{u^2 + 0.25} )At u=0: sqrt(0 + 0.25) = 0.5At u=2.5: sqrt(6.25 + 0.25) = sqrt(6.5) ≈ 2.5495At u=5: sqrt(25 + 0.25) = sqrt(25.25) ≈ 5.0249At u=7.5: sqrt(56.25 + 0.25) = sqrt(56.5) ≈ 7.5166At u=10: sqrt(100 + 0.25) = sqrt(100.25) ≈ 10.0125Now, applying Simpson's rule:Integral ≈ (h/3) [ f(0) + 4f(2.5) + 2f(5) + 4f(7.5) + f(10) ]Where h = 2.5So,≈ (2.5/3) [ 0.5 + 4*2.5495 + 2*5.0249 + 4*7.5166 + 10.0125 ]Compute each term:0.54*2.5495 = 10.1982*5.0249 = 10.04984*7.5166 = 30.066410.0125Add them up:0.5 + 10.198 = 10.69810.698 + 10.0498 = 20.747820.7478 + 30.0664 = 50.814250.8142 + 10.0125 = 60.8267Multiply by (2.5)/3:≈ (2.5 / 3) * 60.8267 ≈ (0.833333) * 60.8267 ≈ 50.6889So, the integral from 0 to10 is approximately 50.6889.But earlier, our exact expression gave:(5/2) sqrt(401) + 0.125 ln(20 + sqrt(401)) ≈ 50.0624 + 0.125*3.6895 ≈ 50.0624 + 0.4612 ≈ 50.5236Wait, but Simpson's rule gave us 50.6889, which is a bit higher.Hmm, so there's a discrepancy here. Wait, but actually, our exact expression was:Integral from 0 to10: (5/2) sqrt(401) + 0.125 ln(20 + sqrt(401)) ≈ 50.0624 + 0.4612 ≈ 50.5236But Simpson's rule gave us 50.6889.Wait, so which one is correct?Wait, perhaps I made a mistake in the exact expression.Wait, let me recast the integral:Wait, when I did the substitution, I had:( L = 2 int_{0}^{10} sqrt{u^2 + 0.25} du )But in the exact calculation, I had:After substitution, the integral became 2 times [ (5/2) sqrt(401) + 0.125 ln(20 + sqrt(401)) ]Wait, no, actually, let me retrace.Wait, when I did the substitution, I had:( L = 2 int_{0}^{10} sqrt{u^2 + 0.25} du )Then, the integral of sqrt(u^2 + a^2) du is:( frac{u}{2} sqrt{u^2 + a^2} + frac{a^2}{2} ln(u + sqrt{u^2 + a^2}) )So, with a = 0.5, the integral from 0 to10 is:[ (10/2)*sqrt(100 + 0.25) + (0.25)/2 * ln(10 + sqrt(100 + 0.25)) ] - [ 0 + (0.25)/2 * ln(0 + sqrt(0 + 0.25)) ]Simplify:[5*sqrt(100.25) + 0.125*ln(10 + sqrt(100.25))] - [0 + 0.125*ln(0.5)]Which is:5*sqrt(100.25) + 0.125*ln(10 + sqrt(100.25)) - 0.125*ln(0.5)Which is:5*sqrt(100.25) + 0.125*ln( (10 + sqrt(100.25))/0.5 )Which is:5*sqrt(100.25) + 0.125*ln(20 + 2*sqrt(100.25))Wait, 2*sqrt(100.25) is sqrt(401), because sqrt(100.25) = sqrt(401)/2.So, 2*sqrt(100.25) = sqrt(401).Therefore, the expression becomes:5*sqrt(100.25) + 0.125*ln(20 + sqrt(401))But sqrt(100.25) is sqrt(401)/2, so 5*sqrt(100.25) = (5/2)*sqrt(401)So, the integral from 0 to10 is:(5/2)*sqrt(401) + 0.125*ln(20 + sqrt(401))Thus, L = 2*(that integral) = 5*sqrt(401) + 0.25*ln(20 + sqrt(401))So, that's correct.Now, computing this numerically:sqrt(401) ≈ 20.024995*sqrt(401) ≈ 5*20.02499 ≈ 100.12495ln(20 + sqrt(401)) ≈ ln(20 + 20.02499) ≈ ln(40.02499) ≈ 3.68950.25*3.6895 ≈ 0.922375Adding together: 100.12495 + 0.922375 ≈ 101.047325 meters.But when I did Simpson's rule with 4 intervals, I got 50.6889 for the integral from 0 to10, which when multiplied by 2 gives 101.3778 meters.Hmm, so there's a discrepancy between the exact expression and Simpson's rule.Wait, perhaps my Simpson's rule approximation is too crude with only 4 intervals.Let me try with more intervals for better accuracy.Alternatively, maybe I can use a calculator to compute the integral numerically.But since I don't have a calculator, perhaps I can accept that the exact expression is approximately 101.05 meters, and the Simpson's rule with 4 intervals gave 101.3778, which is a bit higher, but considering it's an approximation with only 4 intervals, it's not too far off.Alternatively, perhaps I can use a better approximation method.Alternatively, maybe I can compute the integral numerically using another technique.Alternatively, perhaps I can use the fact that the integral of sqrt(u^2 + a^2) du is (u/2)sqrt(u^2 + a^2) + (a^2 / 2) ln(u + sqrt(u^2 + a^2)).So, using that exact formula, we have the integral from 0 to10 as:(5/2)*sqrt(401) + 0.125*ln(20 + sqrt(401)) ≈ 50.0624 + 0.4612 ≈ 50.5236Then, L = 2*50.5236 ≈ 101.0472 meters.So, that's consistent with our earlier calculation.Therefore, I think the exact value is approximately 101.05 meters.Therefore, the total distance skated by the athlete is approximately 101.05 meters.So, summarizing:1. The angular displacement when she reaches the center is 20 radians.2. The total distance skated is approximately 101.05 meters.I think that's the answer.Final Answer1. The angular displacement is boxed{20} radians.2. The total distance skated is approximately boxed{101.05} meters.</think>

🧠:<think>Alright, so Mr. Thompson is having some trouble with his new smart thermostat, and he needs help figuring out two things: the maximum and minimum temperatures his thermostat will set during the day, and the times when his energy consumption is at its minimum. Let me try to break this down step by step.First, let's tackle the temperature function. The function given is:[T(t) = 10 + 5sinleft(frac{pi}{12}t - frac{pi}{4}right)]I remember that the sine function oscillates between -1 and 1. So, if I can figure out the amplitude and the vertical shift, I can find the maximum and minimum temperatures.Looking at the function, the amplitude is 5 because it's multiplied by the sine function. The vertical shift is 10, which is added outside the sine function. So, the maximum temperature should be the vertical shift plus the amplitude, and the minimum should be the vertical shift minus the amplitude.Calculating that, the maximum temperature would be 10 + 5 = 15 degrees Celsius, and the minimum would be 10 - 5 = 5 degrees Celsius. That seems straightforward.But wait, let me double-check. The sine function does indeed have a range of [-1, 1], so multiplying by 5 gives a range of [-5, 5], and adding 10 shifts it up to [5, 15]. Yep, that looks right. So, the thermostat will vary the temperature between 5°C and 15°C over the 24-hour period.Now, moving on to the second part about energy consumption. The energy-saving mode uses a quadratic function:[E(t) = aT(t)^2 + bT(t) + c]where ( a = -0.01 ), ( b = 0.3 ), and ( c = 2 ). We need to find the time(s) ( t ) within 24 hours when ( E(t) ) is minimized.Hmm, quadratic functions have either a minimum or maximum depending on the coefficient ( a ). Since ( a = -0.01 ) is negative, this parabola opens downward, meaning it has a maximum point, not a minimum. Wait, that seems contradictory because we're asked to find the minimum energy consumption. Maybe I need to think differently.Wait, perhaps I'm misunderstanding. The function ( E(t) ) is quadratic in terms of ( T(t) ), which itself is a sinusoidal function of time. So, ( E(t) ) is a composition of functions, not a simple quadratic in ( t ). That complicates things a bit.So, to find the minimum of ( E(t) ), I might need to take the derivative of ( E(t) ) with respect to ( t ) and set it equal to zero. Let's try that.First, let's write out ( E(t) ):[E(t) = -0.01T(t)^2 + 0.3T(t) + 2]Since ( T(t) ) is given, I can substitute it in:[E(t) = -0.01left(10 + 5sinleft(frac{pi}{12}t - frac{pi}{4}right)right)^2 + 0.3left(10 + 5sinleft(frac{pi}{12}t - frac{pi}{4}right)right) + 2]That looks a bit messy, but maybe I can simplify it before taking the derivative.Let me denote ( theta = frac{pi}{12}t - frac{pi}{4} ) to make the expression cleaner.So, ( T(t) = 10 + 5sintheta ), and ( E(t) = -0.01(10 + 5sintheta)^2 + 0.3(10 + 5sintheta) + 2 ).Expanding ( (10 + 5sintheta)^2 ):[(10 + 5sintheta)^2 = 100 + 100sintheta + 25sin^2theta]So, substituting back into ( E(t) ):[E(t) = -0.01(100 + 100sintheta + 25sin^2theta) + 0.3(10 + 5sintheta) + 2]Let's compute each term:First term: ( -0.01 times 100 = -1 )Second term: ( -0.01 times 100sintheta = -1sintheta )Third term: ( -0.01 times 25sin^2theta = -0.25sin^2theta )Fourth term: ( 0.3 times 10 = 3 )Fifth term: ( 0.3 times 5sintheta = 1.5sintheta )Sixth term: ( +2 )Now, combining all these:[E(t) = (-1 - 1sintheta - 0.25sin^2theta) + (3 + 1.5sintheta) + 2]Simplify term by term:Constants: -1 + 3 + 2 = 4Sin terms: -1sintheta + 1.5sintheta = 0.5sinthetaSin squared term: -0.25sin^2thetaSo, overall:[E(t) = 4 + 0.5sintheta - 0.25sin^2theta]But ( theta = frac{pi}{12}t - frac{pi}{4} ), so substituting back:[E(t) = 4 + 0.5sinleft(frac{pi}{12}t - frac{pi}{4}right) - 0.25sin^2left(frac{pi}{12}t - frac{pi}{4}right)]Hmm, this is still a bit complicated, but maybe we can express it in terms of a single trigonometric function. Alternatively, perhaps taking the derivative is manageable.Let me denote ( phi = frac{pi}{12}t - frac{pi}{4} ), so ( dphi/dt = pi/12 ).Then, ( E(t) = 4 + 0.5sinphi - 0.25sin^2phi )To find the minimum, take derivative of E with respect to t:( dE/dt = dE/dphi times dphi/dt )Compute ( dE/dphi ):( dE/dphi = 0.5cosphi - 0.5 times 2sinphicosphi times (-0.25) )Wait, hold on. Let me compute it step by step.First, derivative of 4 is 0.Derivative of ( 0.5sinphi ) with respect to ( phi ) is ( 0.5cosphi ).Derivative of ( -0.25sin^2phi ) with respect to ( phi ) is ( -0.25 times 2sinphicosphi = -0.5sinphicosphi ).So, overall:( dE/dphi = 0.5cosphi - 0.5sinphicosphi )Factor out 0.5cosphi:( dE/dphi = 0.5cosphi (1 - sinphi) )Therefore, the derivative with respect to t is:( dE/dt = 0.5cosphi (1 - sinphi) times pi/12 )To find critical points, set ( dE/dt = 0 ):So,( 0.5cosphi (1 - sinphi) times pi/12 = 0 )Since ( 0.5pi/12 ) is a non-zero constant, we can ignore it for setting the equation to zero. So,( cosphi (1 - sinphi) = 0 )This gives two possibilities:1. ( cosphi = 0 )2. ( 1 - sinphi = 0 ) => ( sinphi = 1 )Let's solve each case.Case 1: ( cosphi = 0 )Solutions are ( phi = pi/2 + kpi ), where k is integer.Case 2: ( sinphi = 1 )Solutions are ( phi = pi/2 + 2kpi )Wait, but ( phi = pi/2 + 2kpi ) is actually a subset of the solutions for Case 1. So, the critical points occur at ( phi = pi/2 + kpi ).But let's write out the solutions for ( phi ):( phi = pi/2 + kpi )But ( phi = frac{pi}{12}t - frac{pi}{4} ), so:( frac{pi}{12}t - frac{pi}{4} = pi/2 + kpi )Solving for t:Add ( pi/4 ) to both sides:( frac{pi}{12}t = pi/2 + pi/4 + kpi = (3pi/4) + kpi )Multiply both sides by 12/π:( t = 12/π times (3π/4 + kπ) = 12/π times π(3/4 + k) = 12(3/4 + k) = 9 + 12k )But since t is within 0 to 24 hours, let's find k such that t is in this interval.For k=0: t=9For k=1: t=21For k=2: t=33, which is beyond 24.For k=-1: t= -3, which is negative.So, critical points at t=9 and t=21.Now, we need to determine whether these points are minima or maxima.To do that, we can use the second derivative test or analyze the behavior around these points.Alternatively, since we have only two critical points, we can evaluate E(t) at these points and also check the endpoints t=0 and t=24 to see which gives the minimum.But before that, let me think if there are any other critical points. Wait, our derivative set to zero gave us only t=9 and t=21. So, these are the only critical points.But let me verify by plugging in t=9 and t=21 into the original E(t) function.First, let's compute T(t) at t=9 and t=21.At t=9:( T(9) = 10 + 5sinleft(frac{pi}{12} times 9 - frac{pi}{4}right) )Compute the argument:( frac{pi}{12} times 9 = frac{3pi}{4} )So,( frac{3pi}{4} - frac{pi}{4} = frac{2pi}{4} = frac{pi}{2} )Thus,( T(9) = 10 + 5sin(pi/2) = 10 + 5(1) = 15 )°CSimilarly, at t=21:( T(21) = 10 + 5sinleft(frac{pi}{12} times 21 - frac{pi}{4}right) )Compute the argument:( frac{pi}{12} times 21 = frac{7pi}{4} )So,( frac{7pi}{4} - frac{pi}{4} = frac{6pi}{4} = frac{3pi}{2} )Thus,( T(21) = 10 + 5sin(3pi/2) = 10 + 5(-1) = 5 )°CSo, at t=9, T=15°C, and at t=21, T=5°C.Now, let's compute E(t) at these points.First, E(t) is given by:[E(t) = -0.01T(t)^2 + 0.3T(t) + 2]At t=9, T=15:[E(9) = -0.01(15)^2 + 0.3(15) + 2 = -0.01(225) + 4.5 + 2 = -2.25 + 4.5 + 2 = 4.25]At t=21, T=5:[E(21) = -0.01(5)^2 + 0.3(5) + 2 = -0.01(25) + 1.5 + 2 = -0.25 + 1.5 + 2 = 3.25]So, E(9)=4.25 and E(21)=3.25.Now, let's check the endpoints t=0 and t=24.At t=0:( T(0) = 10 + 5sinleft(0 - frac{pi}{4}right) = 10 + 5sin(-pi/4) = 10 - 5frac{sqrt{2}}{2} approx 10 - 3.5355 = 6.4645 )°CE(0):[E(0) = -0.01(6.4645)^2 + 0.3(6.4645) + 2 approx -0.01(41.81) + 1.93935 + 2 approx -0.4181 + 1.93935 + 2 approx 3.52125]At t=24:( T(24) = 10 + 5sinleft(frac{pi}{12} times 24 - frac{pi}{4}right) = 10 + 5sin(2pi - pi/4) = 10 + 5sin(7pi/4) = 10 + 5(-sqrt{2}/2) approx 10 - 3.5355 = 6.4645 )°CE(24):Same as E(0), since T(24)=T(0):E(24) ≈ 3.52125So, summarizing:E(0) ≈ 3.52E(9) = 4.25E(21) = 3.25E(24) ≈ 3.52So, the minimum E(t) occurs at t=21 with E=3.25, and the maximum at t=9 with E=4.25.But wait, hold on. The question asks for the time(s) when E(t) is at its minimum. So, t=21 is the time when E(t) is minimized.But let me think again. The function E(t) is quadratic in T(t), and since a is negative, the parabola opens downward, meaning that E(t) will have a maximum at the vertex of the parabola. However, since T(t) is a sinusoidal function, the composition might have multiple maxima and minima.But in our case, we found that the critical points occur at t=9 and t=21, where T(t) is at its maximum and minimum, respectively. Then, evaluating E(t) at these points, we found that E(t) is higher at the maximum temperature and lower at the minimum temperature.So, it seems that E(t) is minimized when T(t) is minimized, which occurs at t=21. Therefore, the energy consumption is at its minimum at t=21 hours, which is 9 PM.But just to be thorough, let me check another point. For example, let's pick t=3.At t=3:( T(3) = 10 + 5sinleft(frac{pi}{12} times 3 - frac{pi}{4}right) = 10 + 5sinleft(frac{pi}{4} - frac{pi}{4}right) = 10 + 5sin(0) = 10 )°CE(3):[E(3) = -0.01(10)^2 + 0.3(10) + 2 = -1 + 3 + 2 = 4]So, E(3)=4, which is higher than E(21)=3.25.Similarly, let's check t=15.At t=15:( T(15) = 10 + 5sinleft(frac{pi}{12} times 15 - frac{pi}{4}right) = 10 + 5sinleft(frac{5pi}{4} - frac{pi}{4}right) = 10 + 5sinleft(piright) = 10 + 0 = 10 )°CE(15):Same as E(3), since T=10:E(15)=4So, E(t) is 4 at t=3 and t=15, which is higher than at t=21.Therefore, it seems that t=21 is indeed the time when E(t) is minimized.But just to make sure, let's think about the behavior of E(t). Since E(t) is a quadratic function of T(t), and since the quadratic has a maximum (because a is negative), the minimum of E(t) would occur at the extremes of T(t). Since T(t) varies between 5 and 15, and E(t) is a downward opening parabola in terms of T(t), the minimum E(t) would occur at the endpoints of T(t)'s range, i.e., at T=5 and T=15.But when we computed E(t) at T=15, E=4.25, and at T=5, E=3.25. So, the minimum E(t) is at T=5, which occurs at t=21.Therefore, the energy consumption is minimized at t=21.So, summarizing:1. The maximum temperature is 15°C, and the minimum is 5°C.2. The energy consumption is minimized at t=21 hours.I think that's it. I don't see any mistakes in my reasoning, but let me just recap:- For the temperature function, the amplitude is 5, vertical shift is 10, so max 15, min 5.- For energy consumption, E(t) is quadratic in T(t), which is sinusoidal. Took derivative, found critical points at t=9 and t=21. Evaluated E(t) at these points and endpoints, found that E(t) is minimized at t=21.Yes, that seems solid.Final Answer1. The maximum temperature is boxed{15} degrees Celsius and the minimum temperature is boxed{5} degrees Celsius.2. The energy consumption is minimized at time boxed{21} hours.</think>