🧠:<think>Okay, so I have this problem about a geologist studying the formation of a cuesta, which is a landform with a gently dipping rock layer and a steep escarpment. The problem has two parts. Let me try to tackle them one by one.Starting with part 1: The erosion rate of the escarpment is proportional to the angle of inclination, θ(t), at time t. The differential equation given is dθ/dt = -k sin(θ(t)), where k is a positive constant. I need to solve this differential equation to find θ(t) in terms of θ₀, k, and t.Hmm, okay. So this is a first-order ordinary differential equation. It looks like a separable equation because I can write it as dθ/dt = -k sinθ, which suggests that I can separate the variables θ and t.Let me write it down:dθ/dt = -k sinθSo, to separate variables, I can divide both sides by sinθ and multiply both sides by dt:dθ / sinθ = -k dtThat gives me:∫ (1/sinθ) dθ = ∫ -k dtNow, integrating both sides. The left side integral is ∫ cscθ dθ, which I remember is ln |tan(θ/2)| + C. The right side is straightforward: ∫ -k dt = -k t + C.So putting it together:ln |tan(θ/2)| = -k t + CNow, I need to solve for θ(t). Let me exponentiate both sides to get rid of the natural log:|tan(θ/2)| = e^{-k t + C} = e^C e^{-k t}Since e^C is just another constant, let's call it A. So:tan(θ/2) = A e^{-k t}Now, applying the initial condition. At t = 0, θ = θ₀. So let's plug that in:tan(θ₀ / 2) = A e^{0} = ATherefore, A = tan(θ₀ / 2). So substituting back:tan(θ/2) = tan(θ₀ / 2) e^{-k t}Now, to solve for θ(t), I can take the arctangent of both sides:θ/2 = arctan(tan(θ₀ / 2) e^{-k t})So,θ(t) = 2 arctan(tan(θ₀ / 2) e^{-k t})Hmm, that seems a bit complicated, but I think that's the solution. Let me check if this makes sense. When t = 0, θ(0) = 2 arctan(tan(θ₀ / 2)) = θ₀, which is correct. As t increases, the argument of arctan decreases exponentially, so θ(t) should decrease towards zero, which also makes sense because as the escarpment erodes, the angle becomes less steep.Alright, so that should be the solution for part 1.Moving on to part 2: The height h(t) of the cuesta's escarpment decreases over time due to erosion and is modeled by the exponential decay function h(t) = h₀ e^{-λ t}, where h₀ is the initial height and λ is a positive constant. I need to determine the time t_c at which the height h(t) is reduced to half of its initial value h₀, and express t_c in terms of λ.Okay, so I need to find t_c such that h(t_c) = h₀ / 2.Given h(t) = h₀ e^{-λ t}, so setting h(t_c) = h₀ / 2:h₀ / 2 = h₀ e^{-λ t_c}Divide both sides by h₀:1/2 = e^{-λ t_c}Take the natural logarithm of both sides:ln(1/2) = -λ t_cSimplify ln(1/2) is equal to -ln 2, so:- ln 2 = -λ t_cMultiply both sides by -1:ln 2 = λ t_cTherefore, t_c = (ln 2) / λSo that's the time when the height is reduced to half its initial value.Let me just recap: For part 1, I solved the differential equation by separating variables, integrated both sides, applied the initial condition, and expressed θ(t) in terms of arctangent. For part 2, it was a straightforward exponential decay problem where I set h(t_c) = h₀ / 2 and solved for t_c using logarithms.I think that covers both parts. I don't see any mistakes in my reasoning, so I'm confident with these solutions.Final Answer1. (boxed{theta(t) = 2 arctanleft( tanleft( frac{theta_0}{2} right) e^{-kt} right)})2. (boxed{t_c = frac{ln 2}{lambda}})</think>

🧠:<think>Okay, so I have this problem where a political analyst is looking at the effectiveness of military operations and media impact. There are two sub-problems here, each with their own formulas. Let me try to tackle them one by one.Starting with Sub-problem 1: The probability of success ( P(S) ) is given by the function:[ P(S) = frac{T cdot I^2 cdot e^{S}}{a + b cdot T cdot I cdot e^{S}} ]The constants are ( a = 1000 ) and ( b = 0.5 ). The variables are ( T = 1500 ) troops, ( I = 0.8 ), and ( S = 0.5 ). I need to plug these values into the formula.First, let me write down all the known values:- ( T = 1500 )- ( I = 0.8 )- ( S = 0.5 )- ( a = 1000 )- ( b = 0.5 )So, I need to compute the numerator and the denominator separately.Calculating the Numerator:The numerator is ( T cdot I^2 cdot e^{S} ).Let me compute each part step by step.1. ( I^2 = 0.8^2 = 0.64 )2. ( e^{S} = e^{0.5} ). I remember that ( e^{0.5} ) is approximately 1.6487. Let me double-check that. Yes, ( e^{0.5} ) is about 1.64872.3. Now, multiply them all together: ( 1500 times 0.64 times 1.64872 )Let me compute ( 1500 times 0.64 ) first. 1500 * 0.64 is 960. Then, 960 * 1.64872. Let me calculate that.960 * 1.64872:First, 960 * 1 = 960960 * 0.6 = 576960 * 0.04 = 38.4960 * 0.008 = 7.68960 * 0.00072 = approximately 0.6912Adding them all together:960 + 576 = 15361536 + 38.4 = 1574.41574.4 + 7.68 = 1582.081582.08 + 0.6912 ≈ 1582.7712So, the numerator is approximately 1582.7712.Calculating the Denominator:The denominator is ( a + b cdot T cdot I cdot e^{S} ).Again, let's break it down.1. ( b cdot T cdot I cdot e^{S} ) - ( b = 0.5 ) - ( T = 1500 ) - ( I = 0.8 ) - ( e^{S} = 1.64872 ) So, compute step by step:First, multiply ( 0.5 times 1500 = 750 ).Then, 750 * 0.8 = 600.Next, 600 * 1.64872 ≈ 600 * 1.64872.Calculating that:600 * 1 = 600600 * 0.6 = 360600 * 0.04 = 24600 * 0.008 = 4.8600 * 0.00072 = 0.432Adding them up:600 + 360 = 960960 + 24 = 984984 + 4.8 = 988.8988.8 + 0.432 ≈ 989.232So, ( b cdot T cdot I cdot e^{S} ≈ 989.232 )Now, add ( a = 1000 ) to this:Denominator = 1000 + 989.232 = 1989.232Putting it all together:Now, ( P(S) = frac{1582.7712}{1989.232} )Let me compute this division.First, approximate:1582.7712 / 1989.232 ≈ ?Let me see, 1989.232 goes into 1582.7712 less than once. So, it's approximately 0.795.Wait, let me compute it more accurately.Let me use a calculator approach:Divide numerator and denominator by 1000 to simplify:1.5827712 / 1.989232 ≈ ?Compute 1.5827712 ÷ 1.989232.Let me see:1.989232 * 0.795 ≈ 1.989232 * 0.8 = 1.5913856But 1.5913856 is slightly more than 1.5827712, so maybe 0.795 is a bit high.Let me try 0.795 * 1.989232:0.7 * 1.989232 = 1.39246240.09 * 1.989232 = 0.179030880.005 * 1.989232 = 0.00994616Adding them up: 1.3924624 + 0.17903088 = 1.57149328 + 0.00994616 ≈ 1.58143944That's very close to 1.5827712.So, 0.795 gives approximately 1.58143944, which is just a bit less than 1.5827712.The difference is 1.5827712 - 1.58143944 ≈ 0.00133176.So, to find how much more we need beyond 0.795:Let me compute 0.00133176 / 1.989232 ≈ 0.00067.So, total is approximately 0.795 + 0.00067 ≈ 0.79567.Therefore, ( P(S) ≈ 0.7957 ) or approximately 79.57%.Wait, let me verify with another method.Alternatively, compute 1582.7712 ÷ 1989.232.Let me write it as:1989.232 ) 1582.7712Since 1989.232 is larger than 1582.7712, the result is less than 1.Compute 1582.7712 / 1989.232:Multiply numerator and denominator by 100000 to eliminate decimals:15827712 / 198923200Simplify:Divide numerator and denominator by 16:15827712 ÷ 16 = 989232198923200 ÷ 16 = 12432700So, 989232 / 12432700 ≈ ?Compute 989232 ÷ 12432700:Approximate 989232 / 12432700 ≈ 0.07957Wait, that can't be. Wait, 989232 / 12432700 is approximately 0.07957, but that would mean 0.07957, which is about 7.957%.But that contradicts the earlier result. Hmm, something's wrong here.Wait, no, I think I messed up the decimal places when multiplying by 100000.Wait, original numbers:Numerator: 1582.7712Denominator: 1989.232Multiply numerator and denominator by 100000:Numerator becomes 158277120Denominator becomes 198923200So, 158277120 / 198923200Simplify:Divide numerator and denominator by 16:158277120 ÷ 16 = 9892320198923200 ÷ 16 = 12432700So, 9892320 / 12432700 ≈ ?Compute 9892320 ÷ 12432700.Let me compute 9892320 ÷ 12432700.Divide numerator and denominator by 100: 98923.2 / 124327Compute 98923.2 ÷ 124327 ≈ ?Well, 124327 * 0.8 = 99461.6Which is slightly higher than 98923.2.So, 0.8 gives 99461.6, which is 98923.2 + 538.4.So, the difference is 538.4.So, 538.4 / 124327 ≈ 0.00433.So, subtract that from 0.8: 0.8 - 0.00433 ≈ 0.79567.So, approximately 0.79567, which is about 0.7957 or 79.57%.So, that matches my initial calculation.Therefore, ( P(S) ≈ 0.7957 ) or 79.57%.So, approximately 79.57% probability of success.Wait, but let me check with a calculator for more precision.Alternatively, use logarithms or exponentials.But since I don't have a calculator, I think 79.57% is a reasonable approximation.So, moving on to Sub-problem 2.Sub-problem 2: The public perception ( P(P) ) is modeled by:[ P(P) = frac{M cdot G^3}{c + d cdot M cdot G} ]Given constants ( c = 200 ) and ( d = 0.3 ). The variables are ( M = 0.6 ) and ( G = 0.7 ).So, plug in these values.Calculating the Numerator:Numerator is ( M cdot G^3 ).Compute ( G^3 = 0.7^3 = 0.343 ).Then, multiply by ( M = 0.6 ):0.6 * 0.343 = 0.2058.So, numerator is 0.2058.Calculating the Denominator:Denominator is ( c + d cdot M cdot G ).Compute ( d cdot M cdot G = 0.3 * 0.6 * 0.7 ).First, 0.3 * 0.6 = 0.18.Then, 0.18 * 0.7 = 0.126.So, denominator is ( c + 0.126 = 200 + 0.126 = 200.126 ).Putting it all together:( P(P) = frac{0.2058}{200.126} )Compute this division.0.2058 ÷ 200.126 ≈ ?Well, 200.126 goes into 0.2058 approximately 0.001028 times.Because 200.126 * 0.001 = 0.200126Subtract that from 0.2058: 0.2058 - 0.200126 = 0.005674Now, 200.126 * 0.000028 ≈ 0.0056035So, total is approximately 0.001 + 0.000028 ≈ 0.001028.Therefore, ( P(P) ≈ 0.001028 ) or about 0.1028%.Wait, that seems really low. Let me check my calculations again.Wait, numerator is 0.2058, denominator is 200.126.So, 0.2058 / 200.126.Let me compute 0.2058 ÷ 200.126.Convert to fractions:0.2058 / 200.126 ≈ (2058 / 10000) / (200126 / 1000) = (2058 / 10000) * (1000 / 200126) = (2058 * 1000) / (10000 * 200126) = 2058000 / 2001260000 ≈ 0.001028.Yes, so approximately 0.001028, which is 0.1028%.That seems extremely low, but given the formula, it's correct because the denominator is much larger than the numerator.Alternatively, maybe I made a mistake in interpreting the formula.Wait, let me check the formula again:[ P(P) = frac{M cdot G^3}{c + d cdot M cdot G} ]Yes, that's correct. So, with M=0.6, G=0.7, c=200, d=0.3.So, numerator is 0.6*(0.7)^3 = 0.6*0.343=0.2058.Denominator is 200 + 0.3*0.6*0.7=200 + 0.126=200.126.So, 0.2058 / 200.126 ≈ 0.001028.So, yes, that's correct.Therefore, public perception is approximately 0.1028%, which is very low.But considering the formula, it's possible because the denominator is much larger.Alternatively, maybe the formula is intended to have higher values, but with the given constants, it results in a low value.So, I think that's the answer.Final AnswerSub-problem 1: boxed{0.796}Sub-problem 2: boxed{0.00103}</think>

🧠:<think>Alright, so I've got this problem about a diplomat analyzing their country's strategic positioning. It's split into two parts, both involving vectors. Let me try to unpack each part step by step.Starting with part 1: We have two vectors, v and w, representing economic influence and military presence, respectively. Their sum is u, which needs to be orthogonal to the vector k = (1, -1). Hmm, orthogonality in vectors means their dot product is zero. So, I need to find the relationship between a, b, c, and d such that u is orthogonal to k.First, let's write down what u is. Since u = v + w, that means u = (a + c, b + d). Now, the vector k is (1, -1). For u and k to be orthogonal, their dot product must be zero. The dot product of u and k is (a + c)*1 + (b + d)*(-1) = (a + c) - (b + d). So, setting that equal to zero:(a + c) - (b + d) = 0Simplifying that, we get:a + c = b + dSo, the relationship between the components is that the sum of a and c equals the sum of b and d. That seems straightforward. So, part 1 is done.Moving on to part 2: The diplomat wants to maximize the magnitude of u, which is the overall influence. The constraints are that the magnitude of v is less than or equal to 5, and the magnitude of w is less than or equal to 3. So, we need to set up an optimization problem to find the maximum possible ||u||, given ||v|| ≤ 5 and ||w|| ≤ 3.Let me recall that the magnitude of a vector u = (x, y) is sqrt(x² + y²). So, ||u|| = sqrt((a + c)² + (b + d)²). We need to maximize this.But since u = v + w, we can also think of this in terms of vector addition. The maximum magnitude of u would be when v and w are in the same direction, right? Because when vectors are aligned, their magnitudes add up. So, the maximum ||u|| would be ||v|| + ||w||, which is 5 + 3 = 8. But wait, is that always the case?Hold on, there's a catch here. The vectors v and w might not necessarily be in the same direction. Moreover, in part 1, we had a condition that u must be orthogonal to k = (1, -1). So, u has to lie along the line that's perpendicular to k, which is the line y = x, since k is (1, -1). So, the direction of u is fixed along y = x.Therefore, to maximize ||u||, we need to maximize the projection of v + w along the direction of y = x. But since u is fixed to be orthogonal to k, we can't just align v and w in any direction; they have to be such that their sum is along y = x.Wait, maybe I should think about this using the concept of vector addition with constraints.Given that u must be orthogonal to k, which is (1, -1), so u must lie along the direction (1, 1) because that's the orthogonal direction. So, u is a scalar multiple of (1, 1). Therefore, u can be written as t*(1, 1), where t is a scalar.So, u = (t, t). Then, ||u|| = sqrt(t² + t²) = sqrt(2t²) = |t|*sqrt(2). To maximize ||u||, we need to maximize |t|.But u = v + w, so v + w = (t, t). Therefore, v = (a, b) and w = (c, d) must satisfy a + c = t and b + d = t.Also, we have the constraints ||v|| ≤ 5 and ||w|| ≤ 3. So, sqrt(a² + b²) ≤ 5 and sqrt(c² + d²) ≤ 3.So, the problem reduces to finding the maximum t such that there exist vectors v and w with ||v|| ≤ 5, ||w|| ≤ 3, and v + w = (t, t).Alternatively, we can think of this as finding the maximum t where (t, t) is in the Minkowski sum of the two disks of radii 5 and 3. The Minkowski sum would be a disk of radius 8, but constrained along the line y = x.Wait, but the Minkowski sum of two disks is another disk whose radius is the sum of the radii, so 5 + 3 = 8. But since u must lie along (1,1), the maximum t would be 8 / sqrt(2), because the vector (t, t) has magnitude t*sqrt(2). Wait, no, that might not be correct.Wait, let me clarify. If we have two vectors v and w, each constrained within their own disks, the maximum ||v + w|| is the sum of their maximum magnitudes, which is 5 + 3 = 8. However, in our case, u must lie along (1,1). So, the maximum t is such that (t, t) is within the Minkowski sum of the two disks.But the Minkowski sum of two disks is another disk with radius 8, centered at the origin. So, the vector (t, t) must lie within this disk. The maximum t occurs when (t, t) is in the direction of (1,1), which is the same direction as the vector we're considering. So, the maximum t would be when (t, t) is on the boundary of the Minkowski sum disk.The magnitude of (t, t) is t*sqrt(2), and the maximum magnitude of the Minkowski sum is 8. Therefore, t*sqrt(2) = 8, so t = 8 / sqrt(2) = 4*sqrt(2). Therefore, the maximum ||u|| is 8, but wait, no. Wait, ||u|| is t*sqrt(2), which is 8, so t = 4*sqrt(2). So, the maximum ||u|| is 8.Wait, that seems conflicting. Let me think again.If u = v + w, and ||v|| ≤ 5, ||w|| ≤ 3, then the maximum ||u|| is indeed 5 + 3 = 8, achieved when v and w are in the same direction. However, in our case, u must be orthogonal to k, which constrains u to lie along (1,1). So, the maximum ||u|| is the maximum projection of the Minkowski sum disk onto the (1,1) direction.The projection of the Minkowski sum disk (radius 8) onto the (1,1) direction is 8*sqrt(2)/2 = 4*sqrt(2). Wait, no, the projection of a disk of radius R onto a line is 2R. Wait, no, the maximum projection is R*sqrt(2) for a disk in 2D. Wait, I'm getting confused.Let me recall that the maximum value of the dot product of u with a unit vector in the (1,1) direction is equal to the magnitude of u. Since u must lie along (1,1), the maximum ||u|| is the maximum possible when v and w are aligned in that direction.But since u = v + w, and both v and w can be in any direction, but their sum must be along (1,1). So, to maximize ||u||, we need to align both v and w along (1,1) as much as possible.So, let me parameterize v and w along (1,1). Let’s say v = s*(1,1) and w = r*(1,1), where s and r are scalars. Then, u = (s + r)*(1,1). The magnitudes are ||v|| = |s|*sqrt(2) ≤ 5, so |s| ≤ 5 / sqrt(2). Similarly, ||w|| = |r|*sqrt(2) ≤ 3, so |r| ≤ 3 / sqrt(2).Then, ||u|| = |s + r|*sqrt(2). To maximize this, we set s and r to their maximum values in the positive direction. So, s = 5 / sqrt(2), r = 3 / sqrt(2). Then, s + r = 8 / sqrt(2), so ||u|| = (8 / sqrt(2)) * sqrt(2) = 8.Wait, that makes sense. So, the maximum ||u|| is 8, achieved when both v and w are aligned along (1,1) to their maximum allowed magnitudes.But wait, is that always possible? Because in part 1, we had the condition that u must be orthogonal to k, which is (1, -1). So, u must lie along (1,1), which is exactly what we're doing here. So, yes, this is consistent.Therefore, the maximum ||u|| is 8, achieved when v and w are both in the direction of (1,1), with ||v|| = 5 and ||w|| = 3.But let me double-check. Suppose v is (5/sqrt(2), 5/sqrt(2)) and w is (3/sqrt(2), 3/sqrt(2)). Then, u = v + w = (8/sqrt(2), 8/sqrt(2)) = 4*sqrt(2)*(1,1). The magnitude of u is sqrt( (8/sqrt(2))² + (8/sqrt(2))² ) = sqrt(64/2 + 64/2) = sqrt(64) = 8. Perfect, that checks out.So, putting it all together, the maximum ||u|| is 8, achieved when both v and w are aligned along the direction (1,1) to their maximum allowed magnitudes.But wait, is there a possibility that by not aligning them perfectly, we could get a larger ||u||? For example, if v and w are not exactly aligned, but their sum still lies along (1,1). Hmm, but since we're trying to maximize the projection along (1,1), aligning them as much as possible would give the maximum sum. Any deviation from that alignment would reduce the total projection.Therefore, the maximum is indeed achieved when both v and w are aligned along (1,1).So, to summarize part 2, the optimization problem is to maximize ||u|| = sqrt( (a + c)^2 + (b + d)^2 ) subject to sqrt(a² + b²) ≤ 5 and sqrt(c² + d²) ≤ 3, and the condition from part 1 that a + c = b + d. The maximum ||u|| is 8, achieved when v and w are both in the direction of (1,1) with their maximum magnitudes.Wait, but in part 1, the condition was a + c = b + d, which is equivalent to u being orthogonal to k. So, in part 2, we have to consider that u must satisfy that condition, which is already baked into the problem. So, when we set u = (t, t), that satisfies the orthogonality condition.Therefore, the optimization is constrained to u being along (1,1), and we found that the maximum ||u|| is 8.I think that's it. So, the relation from part 1 is a + c = b + d, and the maximum ||u|| is 8, achieved when v and w are aligned along (1,1) to their maximum magnitudes.</think>

🧠:<think>Alright, so I have this problem where a travel blogger wants to plan a circular trip around the Maldives, visiting 7 major atolls. The goal is twofold: first, calculate the total distance traveled given their coordinates, and second, determine the optimal path that minimizes the travel cost, where the cost is proportional to the square of the Euclidean distance.Starting with the first part, calculating the total distance. I know that the Euclidean distance between two points ((x_i, y_i)) and ((x_{i+1}, y_{i+1})) is given by the formula:[text{Distance} = sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2}]Since the trip is circular, the traveler starts and ends at the first atoll, so I need to compute the distance from the 7th atoll back to the first one as well. That means I'll have 7 segments in total: from 1 to 2, 2 to 3, ..., 7 to 1.To find the total distance, I need to sum up each of these individual distances. Let me denote the coordinates as ((x_1, y_1), (x_2, y_2), ..., (x_7, y_7)). So, the total distance (D) would be:[D = sum_{i=1}^{7} sqrt{(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2}]Where (x_8 = x_1) and (y_8 = y_1) to complete the loop.For the second part, determining the optimal path to minimize the travel cost. The cost is proportional to the square of the Euclidean distance, so the cost between two atolls is:[text{Cost} = k times [(x_{i+1} - x_i)^2 + (y_{i+1} - y_i)^2]]Where (k) is the proportionality constant. Since we're looking to minimize the total cost, we need to find the permutation of the atolls (visiting each exactly once and returning to the start) that results in the smallest total cost.This sounds exactly like the Traveling Salesman Problem (TSP). The TSP is a well-known problem in combinatorial optimization. The challenge is to find the shortest possible route that visits each city (or in this case, atoll) exactly once and returns to the origin city.Given that there are 7 atolls, the number of possible permutations is ((7-1)! = 720). While this is manageable with brute force for small numbers, 720 is still a significant number, but with modern computing power, it might be feasible.However, for larger numbers, we would need more advanced algorithms like dynamic programming, branch and bound, or heuristic methods such as the nearest neighbor, genetic algorithms, or simulated annealing. But since 7 is manageable, maybe a brute force approach is acceptable here.But wait, the problem mentions "advanced optimization techniques or algorithms," so perhaps it's expecting a more sophisticated method than brute force.Let me recall that the TSP can be formulated as an integer linear programming problem. The formulation involves decision variables (x_{ij}) which are 1 if the path goes from city (i) to city (j), and 0 otherwise. The objective is to minimize the total cost, which in this case is the sum over all edges of (x_{ij}) multiplied by the squared distance between (i) and (j).The constraints would include:1. Each city must be entered exactly once.2. Each city must be exited exactly once.3. The subtour elimination constraints to prevent forming smaller cycles within the tour.But setting up and solving an ILP for TSP can be complex, especially for someone who might not have experience with optimization software.Alternatively, another approach is to use the Held-Karp algorithm, which is a dynamic programming approach for solving the TSP. It has a time complexity of (O(n^2 2^n)), which for (n=7) would be (7^2 times 2^7 = 49 times 128 = 6272) operations. That's manageable even manually, though it's a bit tedious.But perhaps the problem expects recognizing that it's a TSP and knowing that it can be solved with such methods, rather than actually computing it step by step.Wait, but the problem says "formulate and solve" the optimization problem. So maybe I need to set up the mathematical model.Let me try to formalize it.Let me denote:- (n = 7) atolls.- Let (c_{ij} = k times [(x_j - x_i)^2 + (y_j - y_i)^2]) be the cost of traveling from atoll (i) to atoll (j).- Decision variable (x_{ij}) is 1 if the path goes from (i) to (j), else 0.The objective function is:[text{Minimize} sum_{i=1}^{7} sum_{j=1}^{7} c_{ij} x_{ij}]Subject to:1. Each atoll is entered exactly once:[sum_{i=1}^{7} x_{ij} = 1 quad text{for all } j = 1, 2, ..., 7]2. Each atoll is exited exactly once:[sum_{j=1}^{7} x_{ij} = 1 quad text{for all } i = 1, 2, ..., 7]3. Subtour elimination constraints:To prevent the solution from forming smaller cycles, we can use the Miller-Tucker-Zemlin (MTZ) constraints. These introduce a variable (u_i) for each city (i), representing the order in which the city is visited.The MTZ constraints are:[u_i - u_j + n x_{ij} leq n - 1 quad text{for all } i neq j]And (u_1 = 0) since we start at atoll 1.This ensures that if we go from (i) to (j), then (u_j geq u_i + 1), preventing subtours.So, putting it all together, the ILP formulation is:Minimize (sum_{i=1}^{7} sum_{j=1}^{7} c_{ij} x_{ij})Subject to:1. (sum_{i=1}^{7} x_{ij} = 1) for all (j)2. (sum_{j=1}^{7} x_{ij} = 1) for all (i)3. (u_i - u_j + 7 x_{ij} leq 6) for all (i neq j)4. (u_1 = 0)5. (x_{ij} in {0,1}) for all (i, j)6. (u_i in mathbb{Z}) for all (i)This is a standard TSP formulation with the MTZ constraints. Solving this ILP would give the optimal path.However, solving this manually would be quite time-consuming. In practice, one would use optimization software like CPLEX, Gurobi, or even open-source solvers like SCIP or GLPK to input this model and obtain the solution.Alternatively, if exact methods are too time-consuming or not feasible, heuristic methods could be employed. For instance, the nearest neighbor heuristic starts at a point and repeatedly visits the nearest unvisited point until all are visited, then returns to the start. However, this doesn't guarantee the optimal solution, but it's quick.Another heuristic is the 2-opt algorithm, which iteratively reverses segments of the route to reduce the total cost. This can often find a near-optimal solution, especially after multiple iterations.Given that the problem mentions "advanced optimization techniques," it's likely expecting the formulation as a TSP and perhaps the mention of using an exact method like the Held-Karp algorithm or an ILP solver, rather than a heuristic.But in summary, for the first part, it's straightforward: compute each segment's distance and sum them up. For the second part, it's recognizing it as a TSP and setting up the appropriate optimization model, possibly using an ILP with MTZ constraints or another exact method to solve it.I think I've covered both parts. The first is computational, just applying the distance formula. The second is more about modeling and knowing the appropriate optimization techniques to apply.Final Answer1. The total distance traveled is (boxed{D}), where (D) is the sum of the Euclidean distances between consecutive atolls, including the return to the starting point.2. The optimal path can be determined by solving the Traveling Salesman Problem using advanced optimization techniques such as integer linear programming with Miller-Tucker-Zemlin constraints or the Held-Karp algorithm. The exact sequence of atolls for the minimal cost path is (boxed{text{Optimal Path}}).</think>