🧠:<think>Alright, so I'm trying to help Alex explore Fibonacci polynomials and the golden ratio. Let me start by understanding the first part about Fibonacci polynomials.Problem 1(a): Derive the explicit formula for ( F_n(x) ) in terms of ( x ).Okay, so the Fibonacci polynomials are defined by the recurrence relation:[F_n(x) = xF_{n-1}(x) + F_{n-2}(x)]with initial conditions ( F_0(x) = 0 ) and ( F_1(x) = 1 ).Hmm, this looks similar to the standard Fibonacci sequence but with a variable ( x ) instead of a constant. I remember that for linear recursions, we can use characteristic equations to find explicit formulas.Let me set up the characteristic equation for this recurrence. If I assume a solution of the form ( r^n ), plugging into the recurrence gives:[r^n = x r^{n-1} + r^{n-2}]Dividing both sides by ( r^{n-2} ) (assuming ( r neq 0 )):[r^2 = x r + 1]So the characteristic equation is:[r^2 - x r - 1 = 0]Solving this quadratic equation, the roots are:[r = frac{x pm sqrt{x^2 + 4}}{2}]Let me denote these roots as ( alpha = frac{x + sqrt{x^2 + 4}}{2} ) and ( beta = frac{x - sqrt{x^2 + 4}}{2} ).Since the recurrence is linear and homogeneous, the general solution is:[F_n(x) = A alpha^n + B beta^n]where ( A ) and ( B ) are constants determined by the initial conditions.Now, applying the initial conditions:For ( n = 0 ):[F_0(x) = 0 = A alpha^0 + B beta^0 = A + B]So, ( A + B = 0 ) which implies ( B = -A ).For ( n = 1 ):[F_1(x) = 1 = A alpha + B beta]But since ( B = -A ), substitute:[1 = A alpha - A beta = A (alpha - beta)]Compute ( alpha - beta ):[alpha - beta = frac{x + sqrt{x^2 + 4}}{2} - frac{x - sqrt{x^2 + 4}}{2} = frac{2 sqrt{x^2 + 4}}{2} = sqrt{x^2 + 4}]So,[1 = A sqrt{x^2 + 4} implies A = frac{1}{sqrt{x^2 + 4}}]Therefore, ( B = -frac{1}{sqrt{x^2 + 4}} ).Putting it all together, the explicit formula is:[F_n(x) = frac{alpha^n - beta^n}{sqrt{x^2 + 4}}]Substituting ( alpha ) and ( beta ):[F_n(x) = frac{1}{sqrt{x^2 + 4}} left( left( frac{x + sqrt{x^2 + 4}}{2} right)^n - left( frac{x - sqrt{x^2 + 4}}{2} right)^n right)]That should be the explicit formula for ( F_n(x) ).Problem 1(b): Evaluate ( F_5(x) ) and interpret its significance.Alright, let's compute ( F_5(x) ) using the recurrence relation.Given:- ( F_0(x) = 0 )- ( F_1(x) = 1 )Compute step by step:- ( F_2(x) = xF_1(x) + F_0(x) = x*1 + 0 = x )- ( F_3(x) = xF_2(x) + F_1(x) = x*x + 1 = x^2 + 1 )- ( F_4(x) = xF_3(x) + F_2(x) = x*(x^2 + 1) + x = x^3 + x + x = x^3 + 2x )- ( F_5(x) = xF_4(x) + F_3(x) = x*(x^3 + 2x) + (x^2 + 1) = x^4 + 2x^2 + x^2 + 1 = x^4 + 3x^2 + 1 )So, ( F_5(x) = x^4 + 3x^2 + 1 ).Interpretation: Fibonacci polynomials, like the standard Fibonacci sequence, follow a recursive pattern. Each term is built from the previous two, multiplied by ( x ) and added. The result here, ( x^4 + 3x^2 + 1 ), shows how the polynomials can grow in complexity as ( n ) increases. This might relate to how natural patterns, like the arrangement of leaves, can have increasing complexity yet follow a consistent underlying rule, reflecting the interconnectedness and hidden structures in nature that Alex is exploring.Problem 2(a): Prove that the ratio ( frac{F_{n+1}}{F_n} ) approaches ( phi ) as ( n ) tends to infinity.Okay, so the standard Fibonacci sequence is defined by ( F_n = F_{n-1} + F_{n-2} ) with ( F_0 = 0 ), ( F_1 = 1 ). The golden ratio ( phi = frac{1 + sqrt{5}}{2} ) is known to be the limit of the ratio of consecutive Fibonacci numbers.Let me denote ( r_n = frac{F_{n+1}}{F_n} ). We can write the recurrence relation in terms of ( r_n ).From the Fibonacci recurrence:[F_{n+1} = F_n + F_{n-1}]Divide both sides by ( F_n ):[r_n = 1 + frac{F_{n-1}}{F_n} = 1 + frac{1}{r_{n-1}}]So, the ratio satisfies the recurrence:[r_n = 1 + frac{1}{r_{n-1}}]Assuming the limit ( L = lim_{n to infty} r_n ) exists, then taking limits on both sides:[L = 1 + frac{1}{L}]Multiply both sides by ( L ):[L^2 = L + 1]Which leads to:[L^2 - L - 1 = 0]Solving this quadratic equation:[L = frac{1 pm sqrt{5}}{2}]Since the ratio ( r_n ) is positive, we take the positive root:[L = frac{1 + sqrt{5}}{2} = phi]Therefore, the ratio ( frac{F_{n+1}}{F_n} ) approaches ( phi ) as ( n ) tends to infinity.Problem 2(b): Approximate ( F_{20} ) using the result from part (a) and compare it to the actual value.First, let's recall that as ( n ) becomes large, ( F_n ) can be approximated using Binet's formula:[F_n approx frac{phi^n}{sqrt{5}}]But since we're using the ratio ( frac{F_{n+1}}{F_n} approx phi ), we can also express ( F_{n} approx frac{F_{n+1}}{phi} ). However, for approximation, using Binet's formula is more straightforward.Compute ( phi^{20} ) divided by ( sqrt{5} ).First, calculate ( phi ):[phi = frac{1 + sqrt{5}}{2} approx frac{1 + 2.23607}{2} approx 1.61803]Compute ( phi^{20} ):Let me compute step by step:- ( phi^1 approx 1.61803 )- ( phi^2 approx 2.61803 )- ( phi^3 approx 4.23607 )- ( phi^4 approx 6.85401 )- ( phi^5 approx 11.09017 )- ( phi^6 approx 17.94425 )- ( phi^7 approx 29.03442 )- ( phi^8 approx 46.97867 )- ( phi^9 approx 76.01309 )- ( phi^{10} approx 122.99176 )- ( phi^{11} approx 199.00485 )- ( phi^{12} approx 321.99661 )- ( phi^{13} approx 521.00146 )- ( phi^{14} approx 842.99807 )- ( phi^{15} approx 1364.00000 )- ( phi^{16} approx 2206.99807 )- ( phi^{17} approx 3571.00000 )- ( phi^{18} approx 5777.99807 )- ( phi^{19} approx 9349.00000 )- ( phi^{20} approx 15126.99807 )So, ( phi^{20} approx 15127 ).Now, compute ( frac{phi^{20}}{sqrt{5}} ):[sqrt{5} approx 2.23607]So,[frac{15127}{2.23607} approx 6765.0]Wait, actually, let me compute that division more accurately.Compute ( 15127 / 2.23607 ):- 2.23607 * 6765 ≈ 2.23607 * 6000 = 13416.42; 2.23607 * 765 ≈ 1700. So total ≈ 13416.42 + 1700 ≈ 15116.42, which is close to 15127.So, ( frac{phi^{20}}{sqrt{5}} approx 6765 ).But let's check the actual value of ( F_{20} ). The Fibonacci sequence goes:( F_0 = 0 )( F_1 = 1 )( F_2 = 1 )( F_3 = 2 )( F_4 = 3 )( F_5 = 5 )( F_6 = 8 )( F_7 = 13 )( F_8 = 21 )( F_9 = 34 )( F_{10} = 55 )( F_{11} = 89 )( F_{12} = 144 )( F_{13} = 233 )( F_{14} = 377 )( F_{15} = 610 )( F_{16} = 987 )( F_{17} = 1597 )( F_{18} = 2584 )( F_{19} = 4181 )( F_{20} = 6765 )So, the actual ( F_{20} ) is 6765, which matches our approximation. Therefore, the approximation using the golden ratio is accurate.Summary of Thoughts:For the Fibonacci polynomials, I used the characteristic equation method to derive the explicit formula, which is similar to Binet's formula but generalized for polynomials. Evaluating ( F_5(x) ) gave a quartic polynomial, showing how complexity increases with ( n ), mirroring natural patterns.For the golden ratio, I proved the limit of the ratio of consecutive Fibonacci numbers by setting up the recurrence relation for the ratio and solving the resulting quadratic equation. Using this, I approximated ( F_{20} ) and found it matches the actual value, demonstrating the deep connection between the Fibonacci sequence and the golden ratio.Alex's exploration shows how mathematical patterns, like Fibonacci polynomials and the golden ratio, are not just abstract concepts but are deeply embedded in nature, providing a sense of order and interconnectedness that can give meaning and purpose in understanding the world.Final Answer1. (a) The explicit formula for ( F_n(x) ) is ( boxed{F_n(x) = frac{1}{sqrt{x^2 + 4}} left( left( frac{x + sqrt{x^2 + 4}}{2} right)^n - left( frac{x - sqrt{x^2 + 4}}{2} right)^n right)} ). (b) ( F_5(x) = boxed{x^4 + 3x^2 + 1} ).2. (a) The ratio ( frac{F_{n+1}}{F_n} ) approaches ( phi ) as ( n ) tends to infinity, proven by solving the characteristic equation of the recurrence relation. (b) The approximation for ( F_{20} ) is ( boxed{6765} ), which matches the actual value.</think>

🧠:<think>Okay, so I have this problem about optimizing the defense of a critical base. It's divided into two sub-problems. Let me try to tackle them one by one.Starting with Sub-problem 1: I need to find the minimum spanning tree (MST) of a given graph. The graph has vertices V and edges E with specific weights. The edges and their weights are given as:(1,2) with weight 4,(1,3) with weight 7,(2,3) with weight 1,(2,4) with weight 3,(3,4) with weight 2,(3,5) with weight 8,(4,5) with weight 6.I remember that an MST is a subset of the edges that connects all the vertices together, without any cycles, and with the minimum possible total edge weight. There are a couple of algorithms to find the MST, like Kruskal's and Prim's. Since the graph isn't too big, maybe Kruskal's algorithm would be straightforward here.Kruskal's algorithm works by sorting all the edges from the lowest weight to the highest and then adding them one by one, making sure that adding the edge doesn't form a cycle. If it doesn't form a cycle, we include it in the MST.Let me list all the edges in order of increasing weight:1. (2,3) with weight 12. (3,4) with weight 23. (2,4) with weight 34. (4,5) with weight 65. (1,2) with weight 46. (1,3) with weight 77. (3,5) with weight 8Wait, hold on, after (2,4) with weight 3, the next edge is (4,5) with weight 6, but (1,2) is 4 which is less than 6. So actually, the correct order should be:1. (2,3) - 12. (3,4) - 23. (2,4) - 34. (1,2) - 45. (4,5) - 66. (1,3) - 77. (3,5) - 8Yes, that makes more sense. So let's go step by step.Start with the smallest edge: (2,3) with weight 1. Add this to the MST. Now, vertices 2 and 3 are connected.Next edge: (3,4) with weight 2. Adding this connects vertex 4 to the existing tree (since 3 is already connected). So now, vertices 2,3,4 are connected.Next edge: (2,4) with weight 3. But wait, vertices 2 and 4 are already connected through 2-3-4. So adding this edge would form a cycle. So we skip this edge.Next edge: (1,2) with weight 4. Adding this connects vertex 1 to the tree. Now, vertices 1,2,3,4 are connected.Next edge: (4,5) with weight 6. Adding this connects vertex 5 to the tree. Now, all vertices 1,2,3,4,5 are connected.Wait, but let me check if all vertices are connected. So far, we have edges (2,3), (3,4), (1,2), (4,5). That connects all vertices. So we don't need to add any more edges.Wait, hold on, let me verify. Let's see:Edges in MST:(2,3) connects 2 and 3.(3,4) connects 3 and 4.(1,2) connects 1 and 2.(4,5) connects 4 and 5.So yes, all vertices 1,2,3,4,5 are connected. So the MST is formed with these four edges.Now, let's calculate the total weight. Sum of the weights:1 (from 2,3) + 2 (from 3,4) + 4 (from 1,2) + 6 (from 4,5) = 1 + 2 + 4 + 6 = 13.Wait, is that correct? Let me double-check.Alternatively, maybe I missed an edge. Let me recount:Edges added:1. (2,3) - 12. (3,4) - 23. (1,2) - 44. (4,5) - 6Total: 1 + 2 + 4 + 6 = 13.But wait, the graph has 5 vertices, so the MST should have 4 edges, which we have. So yes, the total weight is 13.But let me think again. Is there a possibility of a different MST with a lower total weight? Because sometimes, especially with multiple edges of the same weight, there can be different MSTs. But in this case, the edges are all unique weights, so the MST should be unique.Wait, let me try another approach. Maybe using Prim's algorithm to confirm.Prim's algorithm starts with an arbitrary vertex and adds the smallest edge that connects the tree to a new vertex.Let's start with vertex 1.From vertex 1, the edges are (1,2) with weight 4 and (1,3) with weight 7. The smallest is (1,2) with weight 4. So add edge (1,2). Now, vertices 1 and 2 are in the tree.From vertices 1 and 2, look for the smallest edge connecting to a new vertex. From 1: (1,3) -7. From 2: (2,3)-1, (2,4)-3. The smallest is (2,3)-1. Add this edge. Now, vertex 3 is added.From vertices 1,2,3: look for edges connecting to new vertices. From 1: (1,3)-7 (already considered). From 2: (2,4)-3. From 3: (3,4)-2, (3,5)-8. The smallest is (3,4)-2. Add this edge. Now, vertex 4 is added.From vertices 1,2,3,4: look for edges connecting to new vertices. From 4: (4,5)-6. From 3: (3,5)-8. The smallest is (4,5)-6. Add this edge. Now, vertex 5 is added.So the edges added are (1,2)-4, (2,3)-1, (3,4)-2, (4,5)-6. Total weight: 4 + 1 + 2 + 6 = 13. Same as before.So that confirms the total weight is indeed 13.Alright, so Sub-problem 1 is solved. The total weight of the MST is 13.Moving on to Sub-problem 2: Using the MST from Sub-problem 1, I need to determine the minimum number of drones needed to cover all vertices. Each drone can monitor all adjacent vertices connected by an edge in the MST.So, essentially, this is a vertex cover problem on the MST. A vertex cover is a set of vertices such that every edge in the graph is incident to at least one vertex in the set. But in this case, it's slightly different because each drone can monitor all adjacent vertices, meaning that if a drone is placed at a vertex, it covers that vertex and all its neighbors.Wait, actually, the problem says: "each drone can monitor all adjacent vertices connected by an edge in the MST." So, does that mean that placing a drone at a vertex covers that vertex and all its neighbors? Or does it mean that the drone can monitor the edges connected to it, thereby covering the vertices connected by those edges?Wait, the wording is: "monitor all adjacent vertices connected by an edge in the MST." So, if a drone is placed at vertex u, it can monitor all vertices adjacent to u in the MST. So, it covers u and all its neighbors.Therefore, the problem reduces to finding the minimum number of vertices such that every vertex is either in the set or adjacent to a vertex in the set. This is known as a dominating set problem.The dominating set problem is NP-hard, but since the graph is small, maybe we can find it manually.First, let's draw the MST to visualize it better.From the edges in the MST:(1,2), (2,3), (3,4), (4,5).So, the MST is a straight line: 1-2-3-4-5.So, it's a path graph with vertices 1,2,3,4,5 connected in a straight line.Now, we need to place drones such that every vertex is either a drone or adjacent to a drone.What's the minimum number of drones needed?In a path graph, the minimum dominating set can be found by selecting every other vertex.For example, in a path of 5 vertices: 1-2-3-4-5.If we place drones at vertices 2 and 4:- Drone at 2 covers vertices 1,2,3.- Drone at 4 covers vertices 3,4,5.But wait, vertex 3 is covered by both, but all vertices are covered.Alternatively, placing drones at 1 and 3 and 5:- Drone at 1 covers 1 and 2.- Drone at 3 covers 2,3,4.- Drone at 5 covers 4 and 5.But that uses 3 drones, which is more than the previous option.Alternatively, placing drones at 2 and 4 as above covers all with 2 drones.Wait, let me check:- Drone at 2: covers 1,2,3.- Drone at 4: covers 3,4,5.So, yes, all vertices are covered. So, 2 drones suffice.Is it possible to do it with just 1 drone? Let's see.If we place a drone at vertex 3:- Covers 2,3,4.But vertices 1 and 5 are not covered. So, we need at least one more drone.Alternatively, placing a drone at 2:- Covers 1,2,3.But vertices 4 and 5 are not covered.Similarly, placing a drone at 4:- Covers 3,4,5.But vertices 1 and 2 are not covered.So, 1 drone is insufficient.Therefore, the minimum number of drones needed is 2.Now, identifying the vertices where the drones should be placed.From the above, placing drones at vertices 2 and 4 covers all vertices.Alternatively, another option could be placing drones at vertices 1 and 3 and 5, but that's 3 drones, which is more.Alternatively, placing drones at 3 and 5:- Drone at 3 covers 2,3,4.- Drone at 5 covers 4,5.But vertex 1 is not covered. So, we need to add a drone at 1 as well, making it 3 drones.Similarly, placing drones at 1 and 4:- Drone at 1 covers 1,2.- Drone at 4 covers 3,4,5.But vertex 2 is covered by 1, vertex 3 is covered by 4, vertex 4 is covered, vertex 5 is covered. So, yes, this also covers all vertices with 2 drones.So, another possible placement is at vertices 1 and 4.Wait, let me verify:- Drone at 1: covers 1,2.- Drone at 4: covers 3,4,5.So, vertex 3 is covered by 4, vertex 2 is covered by 1, vertex 5 is covered by 4. So, yes, all vertices are covered.So, there are multiple solutions: placing drones at 2 and 4, or at 1 and 4, or at 2 and 5.Wait, placing at 2 and 5:- Drone at 2: covers 1,2,3.- Drone at 5: covers 4,5.But vertex 4 is not covered by 5? Wait, in the MST, vertex 4 is connected to 3 and 5. So, if a drone is at 5, it covers 4 and 5.Wait, no: the drone at 5 covers 4 and 5. So, yes, vertex 4 is covered.So, placing drones at 2 and 5 also covers all vertices.Similarly, placing drones at 1 and 3:- Drone at 1: covers 1,2.- Drone at 3: covers 2,3,4.But vertex 5 is not covered. So, we need another drone at 5, making it 3.So, the minimal number is 2, and the possible placements are:- 2 and 4,- 1 and 4,- 2 and 5.But wait, let's check if placing at 1 and 5 covers all:- Drone at 1: covers 1,2.- Drone at 5: covers 4,5.But vertex 3 is not covered. So, we need a drone at 3 as well, making it 3.So, that's not minimal.Similarly, placing at 3 and 5:- Drone at 3: covers 2,3,4.- Drone at 5: covers 4,5.But vertex 1 is not covered. So, need a drone at 1, making it 3.So, the minimal number is indeed 2, and the possible placements are:- 2 and 4,- 1 and 4,- 2 and 5.Wait, but let me think again. Is there a way to cover all with just 2 drones without overlapping too much?Alternatively, placing at 3 and 4:- Drone at 3: covers 2,3,4.- Drone at 4: covers 3,4,5.But vertex 1 is not covered. So, need a drone at 1, making it 3.So, no.Alternatively, placing at 2 and 3:- Drone at 2: covers 1,2,3.- Drone at 3: covers 2,3,4.But vertex 5 is not covered. So, need a drone at 5, making it 3.So, no.Therefore, the minimal number is 2, and the possible placements are:- 2 and 4,- 1 and 4,- 2 and 5.But wait, in the MST, the structure is 1-2-3-4-5. So, the distance between 1 and 4 is 3 edges, but in terms of coverage, placing drones at 2 and 4 or 1 and 4 or 2 and 5.But perhaps the most symmetric way is placing at 2 and 4, as they are central points.Alternatively, placing at 1 and 4 might leave a gap between 2 and 3, but no, because drone at 1 covers 2, and drone at 4 covers 3.Similarly, placing at 2 and 5: drone at 2 covers 1,2,3; drone at 5 covers 4,5.So, all vertices are covered.Therefore, the minimal number is 2, and possible placements are vertices 2 and 4, or 1 and 4, or 2 and 5.But the question says "identify the vertices where the drones should be placed." It doesn't specify if there's a unique solution or multiple. So, perhaps all possible minimal solutions should be mentioned.But maybe the answer expects the minimal number and one possible placement.Alternatively, perhaps the optimal placement is at 2 and 4, as they are central.But let me think again. In a path graph, the minimum dominating set can be found by selecting every other vertex starting from the second one.So, for 1-2-3-4-5, selecting 2 and 4 would cover all.Alternatively, selecting 1 and 3 and 5, but that's more.So, yes, 2 and 4 is a minimal dominating set.Alternatively, another minimal dominating set is 1 and 4.Wait, let me check:- Drone at 1: covers 1,2.- Drone at 4: covers 3,4,5.So, vertex 3 is covered by 4, vertex 2 is covered by 1, vertex 5 is covered by 4. So, yes, all covered.Similarly, drone at 2 and 5:- Drone at 2: covers 1,2,3.- Drone at 5: covers 4,5.So, vertex 4 is covered by 5, vertex 3 is covered by 2, vertex 1 is covered by 2, vertex 5 is covered by 5. So, all covered.So, there are three possible minimal dominating sets: {2,4}, {1,4}, {2,5}.But perhaps the answer expects the minimal number and one possible placement.Alternatively, maybe the problem expects the minimal number and the specific vertices, but since there are multiple solutions, perhaps any one is acceptable.But let me see if there's a standard way to choose. In some cases, the dominating set might prefer central nodes to minimize the number, but in this case, all options are equally minimal.So, perhaps the answer is 2 drones, placed at vertices 2 and 4.Alternatively, the problem might accept any of the possible pairs.But to be thorough, I should mention that the minimal number is 2, and possible placements include {2,4}, {1,4}, or {2,5}.But since the question says "identify the vertices where the drones should be placed," it might expect a specific answer, perhaps the one with the smallest vertex numbers or something. But I think any correct pair is acceptable.Alternatively, perhaps the answer is 2 drones, placed at vertices 2 and 4.I think that's a reasonable answer.So, to recap:Sub-problem 1: The total weight of the MST is 13.Sub-problem 2: The minimum number of drones needed is 2, and they can be placed at vertices 2 and 4 (or other equivalent pairs).But let me just confirm once more.In the MST, the structure is a straight line: 1-2-3-4-5.To cover all vertices with the fewest drones, each drone can cover itself and its neighbors.If we place a drone at 2, it covers 1,2,3.If we place a drone at 4, it covers 3,4,5.So, together, they cover all vertices.Alternatively, placing at 1 and 4:- Drone at 1 covers 1,2.- Drone at 4 covers 3,4,5.So, vertex 3 is covered by 4, vertex 2 is covered by 1, vertex 5 is covered by 4.Yes, all covered.Similarly, placing at 2 and 5:- Drone at 2 covers 1,2,3.- Drone at 5 covers 4,5.So, vertex 4 is covered by 5, vertex 3 is covered by 2, vertex 1 is covered by 2, vertex 5 is covered by 5.Yes, all covered.So, all three options are valid.But perhaps the answer expects the pair {2,4} as it's the most central.Alternatively, the answer might accept any of the three.But since the problem says "identify the vertices," perhaps any correct pair is acceptable.But to be safe, I'll go with {2,4} as the answer.So, final answers:Sub-problem 1: Total weight of MST is 13.Sub-problem 2: Minimum number of drones is 2, placed at vertices 2 and 4.</think>

🧠:<think>Alright, so I have this problem about a Democrat in Wyoming trying to analyze voting patterns and demographics, specifically looking at how Evangelical Christians influence voting behavior. There are two sub-problems here, and I need to tackle them step by step.Starting with Sub-problem 1. The population is divided into Democrats and Republicans, with proportions ( P_D ) and ( P_R ), respectively, and they add up to 1. So, ( P_D + P_R = 1 ). They want to examine the impact of Evangelical Christians on voting behavior. Let me parse the variables given: ( E ) is the proportion of Evangelical Christians among Republicans, and ( V_E ) is the proportion of Evangelical Christians who vote Republican. Recent surveys show that 30% of Wyoming's population identifies as Evangelical Christians. So, the total proportion of Evangelicals in the state is 0.3.Wait, hold on. Is ( E ) the proportion of Evangelicals among Republicans or the proportion of Republicans who are Evangelicals? The problem says, "Let ( E ) represent the proportion of Evangelical Christians among Republicans." So, that would mean ( E ) is the proportion of Republicans who are Evangelical. So, if ( P_R ) is the proportion of Republicans in the population, then the proportion of the entire population that is both Republican and Evangelical would be ( P_R times E ).Similarly, ( V_E ) is the proportion of Evangelical Christians who vote Republican. So, if 30% of the population are Evangelicals, then the number of Evangelicals voting Republican is ( 0.3 times V_E ).But the question is to derive an expression for the proportion of Evangelical Christians who vote Republican in terms of ( P_R ) and ( V_E ). Wait, that seems a bit confusing. Let me re-read the problem."Derive an expression for the proportion of Evangelical Christians who vote Republican in terms of ( P_R ) and ( V_E )."Hmm. So, ( V_E ) is already defined as the proportion of Evangelical Christians who vote Republican. So, is the question asking for an expression for ( V_E ) in terms of ( P_R ) and something else?Wait, maybe I misread. Let me check again."Let ( E ) represent the proportion of Evangelical Christians among Republicans, and let ( V_E ) represent the proportion of Evangelical Christians who vote for Republican candidates. If recent surveys show that 30% of Wyoming's population identifies as Evangelical Christians, derive an expression for the proportion of Evangelical Christians who vote Republican in terms of ( P_R ) and ( V_E )."Wait, that seems redundant because ( V_E ) is already the proportion of Evangelical Christians who vote Republican. So, perhaps I'm misunderstanding the problem.Alternatively, maybe the problem is asking for the proportion of the total population that are Evangelical Christians voting Republican, expressed in terms of ( P_R ) and ( V_E ). Let me think.Total population: 100%. 30% are Evangelicals. So, total number of Evangelicals is 0.3. The proportion of Evangelicals who vote Republican is ( V_E ). So, the number of Evangelicals voting Republican is ( 0.3 times V_E ).But how does that relate to ( P_R )?Wait, ( P_R ) is the proportion of Republicans in the population. So, the number of Republicans is ( P_R times N ), where ( N ) is the total population. The number of Republicans who are Evangelicals is ( E times P_R times N ).But the number of Evangelicals voting Republican is ( V_E times 0.3 times N ). So, equating these two expressions because both represent the number of Evangelical Republicans:( E times P_R times N = V_E times 0.3 times N )We can cancel ( N ) from both sides:( E times P_R = V_E times 0.3 )So, solving for ( V_E ):( V_E = frac{E times P_R}{0.3} )But wait, the problem says to express the proportion of Evangelical Christians who vote Republican in terms of ( P_R ) and ( V_E ). But ( V_E ) is already that proportion. So, perhaps the question is to express ( V_E ) in terms of ( E ) and ( P_R ), but given that 30% are Evangelicals.Wait, maybe I need to think differently. The proportion of Evangelical Christians who vote Republican is ( V_E ), but we can also express this proportion in terms of the overlap between Evangelicals and Republicans.So, the number of Evangelical Republicans is ( E times P_R times N ), as above. The total number of Evangelicals is 0.3N. Therefore, the proportion of Evangelicals who are Republican is ( frac{E times P_R}{0.3} ). So, that would be ( V_E ).Thus, ( V_E = frac{E P_R}{0.3} ). So, that's the expression.But the problem says to derive an expression for the proportion of Evangelical Christians who vote Republican in terms of ( P_R ) and ( V_E ). Wait, but ( V_E ) is that proportion. So, maybe I'm overcomplicating.Alternatively, perhaps the problem is asking for the proportion of the population that are Evangelical Republicans, which would be ( E times P_R ). But that's not in terms of ( V_E ).Wait, maybe the question is to express ( V_E ) in terms of ( E ) and ( P_R ). Since we have ( V_E = frac{E P_R}{0.3} ), as above.But the problem says "derive an expression for the proportion of Evangelical Christians who vote Republican in terms of ( P_R ) and ( V_E )." Hmm, that seems circular because ( V_E ) is the proportion we're trying to express.Wait, perhaps the problem is misworded. Maybe it's supposed to express ( E ) in terms of ( P_R ) and ( V_E ). Let me check.No, the problem says: "derive an expression for the proportion of Evangelical Christians who vote Republican in terms of ( P_R ) and ( V_E )." But ( V_E ) is already that proportion. So, maybe it's a trick question where the expression is just ( V_E ).Alternatively, perhaps the problem is asking for the proportion of the total population that are Evangelical Republicans, which would be ( E times P_R ). But that's not in terms of ( V_E ).Wait, let me think again. If 30% are Evangelicals, and ( V_E ) is the proportion of them who vote Republican, then the number of Evangelical Republicans is ( 0.3 times V_E ). On the other hand, the number of Evangelical Republicans is also ( E times P_R ). Therefore, ( E times P_R = 0.3 times V_E ). So, solving for ( V_E ), we get ( V_E = frac{E P_R}{0.3} ). So, the proportion of Evangelical Christians who vote Republican is ( V_E = frac{E P_R}{0.3} ).But the question is to express the proportion of Evangelical Christians who vote Republican, which is ( V_E ), in terms of ( P_R ) and ( V_E ). Wait, that doesn't make sense because ( V_E ) is already the proportion. So, perhaps the question is to express ( V_E ) in terms of ( E ) and ( P_R ), which we have done: ( V_E = frac{E P_R}{0.3} ).But the problem says "in terms of ( P_R ) and ( V_E )", which is confusing because ( V_E ) is the variable we're solving for. Maybe it's a typo, and they meant to express ( E ) in terms of ( P_R ) and ( V_E ). In that case, from ( E P_R = 0.3 V_E ), we get ( E = frac{0.3 V_E}{P_R} ).But the problem specifically says to express the proportion of Evangelical Christians who vote Republican, which is ( V_E ), in terms of ( P_R ) and ( V_E ). That seems redundant. Maybe the question is to express ( V_E ) in terms of ( E ) and ( P_R ), which we have as ( V_E = frac{E P_R}{0.3} ).So, perhaps the answer is ( V_E = frac{E P_R}{0.3} ).Moving on to Sub-problem 2. The Democrat wants to understand the correlation between religious affiliation and party support. The probability ( P(D|E) ) of being a Democrat given that a person is Evangelical is proportional to the ratio of Democrats to Evangelicals, with a proportionality constant ( k ). Given the total population ( N ), and the number of Evangelical Christians who are Democrats is ( N_E ), derive the expression for ( k ) in terms of ( P_D ), ( P_R ), ( E ), ( V_E ), and ( N ). Additionally, determine how changes in ( V_E ) affect ( P(D|E) ).First, let's parse the given information. ( P(D|E) ) is the probability that someone is a Democrat given they are Evangelical. It's proportional to the ratio of Democrats to Evangelicals, with proportionality constant ( k ). So, mathematically, this can be written as:( P(D|E) = k times frac{P_D}{E} )Wait, but ( E ) is the proportion of Evangelicals among Republicans, not the proportion of Evangelicals in the entire population. Wait, no, earlier in Sub-problem 1, we were told that 30% of the population are Evangelicals, so the proportion of Evangelicals in the entire population is 0.3, not ( E ). ( E ) is the proportion of Republicans who are Evangelical.So, the total proportion of Evangelicals is 0.3, which is given. So, the ratio of Democrats to Evangelicals would be ( frac{P_D}{0.3} ), since ( P_D ) is the proportion of Democrats in the population, and 0.3 is the proportion of Evangelicals.But the problem says ( P(D|E) ) is proportional to the ratio of Democrats to Evangelicals. So, ( P(D|E) = k times frac{P_D}{0.3} ).But we also know that ( P(D|E) ) is equal to the number of Evangelical Democrats divided by the total number of Evangelicals. So, ( P(D|E) = frac{N_E}{0.3 N} ), since ( N_E ) is the number of Evangelical Democrats, and the total number of Evangelicals is ( 0.3 N ).Therefore, we have:( frac{N_E}{0.3 N} = k times frac{P_D}{0.3} )Simplifying, multiply both sides by 0.3:( frac{N_E}{N} = k P_D )So, ( k = frac{N_E}{N P_D} )But ( N_E ) is the number of Evangelical Democrats, which can also be expressed as ( P(D|E) times 0.3 N ). Wait, but we already have ( P(D|E) = frac{N_E}{0.3 N} ), so ( N_E = P(D|E) times 0.3 N ).But substituting back into the expression for ( k ):( k = frac{P(D|E) times 0.3 N}{N P_D} = frac{0.3 P(D|E)}{P_D} )But from earlier, ( P(D|E) = k times frac{P_D}{0.3} ), so substituting:( k = frac{0.3 times (k times frac{P_D}{0.3})}{P_D} )Simplifying:( k = frac{0.3 times k times frac{P_D}{0.3}}{P_D} = frac{k P_D}{P_D} = k )Which is a tautology, so that approach might not be helpful.Alternatively, let's go back to the initial equation:( P(D|E) = k times frac{P_D}{0.3} )But we also have ( P(D|E) = frac{N_E}{0.3 N} ). So, equate the two:( frac{N_E}{0.3 N} = k times frac{P_D}{0.3} )Multiply both sides by 0.3:( frac{N_E}{N} = k P_D )Therefore, ( k = frac{N_E}{N P_D} )But ( N_E ) is the number of Evangelical Democrats, which can be expressed as ( P(D|E) times 0.3 N ). Wait, but that's circular because ( P(D|E) ) is defined in terms of ( N_E ).Alternatively, perhaps we can express ( N_E ) in terms of other variables. From Sub-problem 1, we know that the number of Evangelical Republicans is ( E P_R N ), and the total number of Evangelicals is 0.3 N. Therefore, the number of Evangelical Democrats is ( 0.3 N - E P_R N ).So, ( N_E = 0.3 N - E P_R N = N (0.3 - E P_R) )Substituting into the expression for ( k ):( k = frac{N (0.3 - E P_R)}{N P_D} = frac{0.3 - E P_R}{P_D} )But from Sub-problem 1, we have ( V_E = frac{E P_R}{0.3} ), so ( E P_R = 0.3 V_E ). Substituting into the expression for ( k ):( k = frac{0.3 - 0.3 V_E}{P_D} = frac{0.3 (1 - V_E)}{P_D} )So, ( k = frac{0.3 (1 - V_E)}{P_D} )Now, the second part of Sub-problem 2 asks how changes in ( V_E ) affect ( P(D|E) ). From the earlier equation:( P(D|E) = k times frac{P_D}{0.3} )But we have ( k = frac{0.3 (1 - V_E)}{P_D} ), so substituting:( P(D|E) = frac{0.3 (1 - V_E)}{P_D} times frac{P_D}{0.3} = 1 - V_E )So, ( P(D|E) = 1 - V_E ). Therefore, as ( V_E ) increases, ( P(D|E) ) decreases, and vice versa. So, they are inversely related.To summarize:Sub-problem 1: The proportion of Evangelical Christians who vote Republican is ( V_E = frac{E P_R}{0.3} ).Sub-problem 2: The proportionality constant ( k ) is ( k = frac{0.3 (1 - V_E)}{P_D} ), and ( P(D|E) = 1 - V_E ), meaning that as ( V_E ) increases, ( P(D|E) ) decreases.But wait, let me double-check the Sub-problem 2 derivation. We had ( N_E = 0.3 N - E P_R N ), which is correct because total Evangelicals are 0.3N, and Evangelical Republicans are ( E P_R N ). So, Evangelical Democrats are the remainder.Then, ( k = frac{N_E}{N P_D} = frac{0.3 - E P_R}{P_D} ). Since ( E P_R = 0.3 V_E ), substituting gives ( k = frac{0.3 - 0.3 V_E}{P_D} = frac{0.3 (1 - V_E)}{P_D} ). That seems correct.Then, ( P(D|E) = k times frac{P_D}{0.3} = frac{0.3 (1 - V_E)}{P_D} times frac{P_D}{0.3} = 1 - V_E ). So, yes, that's correct.Therefore, the final expressions are:Sub-problem 1: ( V_E = frac{E P_R}{0.3} )Sub-problem 2: ( k = frac{0.3 (1 - V_E)}{P_D} ), and ( P(D|E) = 1 - V_E ), so as ( V_E ) increases, ( P(D|E) ) decreases.I think that's the solution.</think>

🧠:<think>Alright, so I have this problem about a digital artist who's working on an algorithm to adjust the brightness of pixels in a digital art piece. The problem has two parts, and I need to tackle them one by one. Let me start with the first part.Problem 1: Calculating Adjusted Intensity A(1,1)Okay, the formula given is:[ A(i, j) = frac{1}{9} sum_{k=-1}^{1} sum_{l=-1}^{1} I(i+k, j+l) cdot W(k, l) ]And the weight matrix W is:[W = begin{bmatrix}0.5 & 1 & 0.5 1 & 2 & 1 0.5 & 1 & 0.5end{bmatrix}]The initial intensity grid I is:[I = begin{bmatrix}100 & 150 & 200 150 & 250 & 150 200 & 150 & 100end{bmatrix}]I need to calculate A(1,1). Hmm, so in the grid, positions are (i,j), with i and j starting at 1? Or are they starting at 0? Wait, in the matrix, the first row is 100, 150, 200, so probably (1,1) is the center pixel, which is 250. Let me confirm.Yes, in the grid, (1,1) would be the center of the 3x3 grid, so that's 250. So, to compute A(1,1), I need to consider all the surrounding pixels, apply the weights, sum them up, and then divide by 9.Let me write out the grid with indices for clarity.Assuming the grid is 3x3, with positions (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3). So, the center is (2,2). Wait, hold on, maybe I misread.Wait, the grid is given as:Row 1: 100, 150, 200Row 2: 150, 250, 150Row 3: 200, 150, 100So, if it's a 3x3 grid, the positions are (1,1)=100, (1,2)=150, (1,3)=200, (2,1)=150, (2,2)=250, (2,3)=150, (3,1)=200, (3,2)=150, (3,3)=100.So, the center is (2,2)=250. But the question is asking for A(1,1). Wait, (1,1) is the top-left corner, which is 100. So, do I need to compute the adjusted intensity for the top-left pixel?But wait, if I apply the formula, for (i,j)=(1,1), then k and l go from -1 to 1. So, the neighboring pixels would be (1-1,1-1)=(0,0), which is outside the grid. Similarly, (1-1,1)=(0,1), which is also outside. So, how do we handle the edges?Hmm, the problem statement doesn't specify, but in image processing, when you're at the edge, you usually either ignore the out-of-bound pixels, or mirror them, or replicate the edge. But since the weight matrix is 3x3, and the grid is 3x3, maybe the question assumes that (1,1) is the center? Wait, no, because in a 3x3 grid, (1,1) is the top-left.Alternatively, maybe the grid is 1-based, so (1,1) is the center? Wait, no, in a 3x3 grid, (2,2) is the center if it's 1-based. Hmm, this is confusing.Wait, let me check the formula again. It says:[ A(i, j) = frac{1}{9} sum_{k=-1}^{1} sum_{l=-1}^{1} I(i+k, j+l) cdot W(k, l) ]So, for (i,j)=(1,1), k and l go from -1 to 1, so we need to access I(0,0), I(0,1), I(0,2), I(1,0), I(1,1), I(1,2), I(2,0), I(2,1), I(2,2). But in the given grid, the indices go from 1 to 3 in both directions. So, I(0,0) is out of bounds. Similarly, I(0,1), I(0,2), I(1,0), I(2,0) are all out of bounds.So, how do we handle that? Maybe we just ignore the out-of-bound pixels? Or perhaps the grid is considered to be surrounded by zeros? Or maybe the artist is using a different indexing where (1,1) is the center.Wait, perhaps the grid is 3x3, and (1,1) is the center. So, in that case, the grid is:(1,1)=250, (1,2)=150, (1,3)=100(2,1)=150, (2,2)=250, (2,3)=150(3,1)=200, (3,2)=150, (3,3)=100Wait, no, that doesn't make sense because the first row is 100,150,200. Hmm, maybe the grid is given as rows, so row 1 is top row, row 2 is middle, row 3 is bottom.So, (1,1)=100, (1,2)=150, (1,3)=200(2,1)=150, (2,2)=250, (2,3)=150(3,1)=200, (3,2)=150, (3,3)=100So, (1,1) is top-left, (2,2) is center.So, if we compute A(1,1), we need to look at the 3x3 neighborhood around (1,1). But (1,1) is at the corner, so the neighborhood would include pixels outside the grid.So, how do we handle that? The problem statement doesn't specify, but perhaps we can assume that the grid is extended by mirroring or replication. Alternatively, maybe the grid is large enough that (1,1) is not at the edge, but in this case, it's a 3x3 grid, so (1,1) is definitely at the edge.Wait, maybe the grid is 5x5, but the problem says 3x3 grid. Hmm.Alternatively, perhaps the artist is using a different approach, like only considering the existing pixels, but that would complicate the calculation.Wait, maybe the question is assuming that (1,1) is the center, so the grid is 3x3, and (1,1) is the center. But in the given grid, (1,1)=100, which is the top-left. So, perhaps the grid is 3x3, but the indices are 0-based? So, (0,0)=100, (0,1)=150, etc. Then, (1,1) would be the center, which is 250.Wait, but the problem says "a 3x3 grid with the initial intensities as follows", and the grid is given as 3 rows, each with 3 numbers. So, probably, (1,1) is the top-left, (1,2) is top-middle, etc.But then, computing A(1,1) would require accessing pixels outside the grid. So, perhaps the artist is using a different approach, like padding the grid with zeros or replicating the edge.Wait, the problem doesn't specify, so maybe I should assume that the grid is large enough that (1,1) is not at the edge, but in this case, it's a 3x3 grid, so (1,1) is at the edge. Hmm.Alternatively, perhaps the grid is considered to be toroidal, meaning that the edges wrap around. But that seems complicated.Wait, maybe the question is just expecting me to compute the sum using the given 3x3 grid, even if some pixels are out of bounds, but treating them as zero? Or perhaps the question is assuming that (1,1) is the center, so the grid is 5x5, but the given grid is 3x3. Hmm, this is confusing.Wait, perhaps the question is miswritten, and (1,1) is the center. Because otherwise, computing A(1,1) would require out-of-bound pixels, which are not provided.Alternatively, maybe the grid is 5x5, but only the central 3x3 is given, and the rest are zero. But that's an assumption.Wait, maybe I should proceed by assuming that the grid is 3x3, and (1,1) is the center. So, the grid is:(1,1)=250, (1,2)=150, (1,3)=100(2,1)=150, (2,2)=250, (2,3)=150(3,1)=200, (3,2)=150, (3,3)=100Wait, no, that's not matching the given grid. The given grid is:Row 1: 100, 150, 200Row 2: 150, 250, 150Row 3: 200, 150, 100So, if it's 1-based, (1,1)=100, (1,2)=150, (1,3)=200(2,1)=150, (2,2)=250, (2,3)=150(3,1)=200, (3,2)=150, (3,3)=100So, (1,1) is top-left, (2,2) is center.So, to compute A(1,1), we need to consider the 3x3 neighborhood around (1,1). But (1,1) is at the corner, so the neighborhood would include (0,0), (0,1), (0,2), (1,0), (1,1), (1,2), (2,0), (2,1), (2,2). But in the given grid, (0,0) is out of bounds, as well as (0,1), (0,2), (1,0), (2,0). So, how do we handle that?Since the problem doesn't specify, maybe we can assume that the grid is extended by mirroring or replication. Alternatively, perhaps the grid is considered to be surrounded by zeros.Wait, in image processing, when dealing with edges, sometimes they use zero-padding, which means that out-of-bound pixels are considered as zero. Alternatively, they might replicate the edge pixel values.But since the problem doesn't specify, maybe I should assume that the grid is large enough that (1,1) is not at the edge, but in this case, it's a 3x3 grid, so (1,1) is at the edge. Hmm.Alternatively, perhaps the question is expecting me to compute the sum using only the available pixels, but that would complicate the calculation.Wait, maybe the question is miswritten, and (1,1) is the center, so the grid is 5x5, but the given grid is 3x3. Hmm, this is confusing.Alternatively, perhaps the grid is considered to be toroidal, meaning that the edges wrap around. So, (0,0) would be (3,3), (0,1)=(3,2), etc. But that seems complicated.Wait, maybe the question is just expecting me to compute the sum using the given 3x3 grid, even if some pixels are out of bounds, but treating them as zero? Or perhaps the question is assuming that (1,1) is the center, so the grid is 5x5, but the given grid is 3x3. Hmm, this is confusing.Wait, perhaps I should proceed by assuming that the grid is 3x3, and (1,1) is the center. So, the grid is:(1,1)=250, (1,2)=150, (1,3)=100(2,1)=150, (2,2)=250, (2,3)=150(3,1)=200, (3,2)=150, (3,3)=100But in the given grid, (1,1)=100, which is the top-left. So, this is conflicting.Wait, maybe the grid is 0-based. So, (0,0)=100, (0,1)=150, (0,2)=200(1,0)=150, (1,1)=250, (1,2)=150(2,0)=200, (2,1)=150, (2,2)=100In that case, (1,1) is the center, which is 250. So, to compute A(1,1), we need to consider the 3x3 neighborhood around (1,1), which would be:(0,0)=100, (0,1)=150, (0,2)=200(1,0)=150, (1,1)=250, (1,2)=150(2,0)=200, (2,1)=150, (2,2)=100So, that's the entire grid. So, in this case, A(1,1) would be the sum of all these multiplied by the weights, divided by 9.Wait, but the weight matrix is:[W = begin{bmatrix}0.5 & 1 & 0.5 1 & 2 & 1 0.5 & 1 & 0.5end{bmatrix}]So, the weight for (0,0) is 0.5, (0,1)=1, (0,2)=0.5(1,0)=1, (1,1)=2, (1,2)=1(2,0)=0.5, (2,1)=1, (2,2)=0.5So, let's compute the sum:Sum = (100 * 0.5) + (150 * 1) + (200 * 0.5) +(150 * 1) + (250 * 2) + (150 * 1) +(200 * 0.5) + (150 * 1) + (100 * 0.5)Let me compute each term:First row:100 * 0.5 = 50150 * 1 = 150200 * 0.5 = 100Sum of first row: 50 + 150 + 100 = 300Second row:150 * 1 = 150250 * 2 = 500150 * 1 = 150Sum of second row: 150 + 500 + 150 = 800Third row:200 * 0.5 = 100150 * 1 = 150100 * 0.5 = 50Sum of third row: 100 + 150 + 50 = 300Total sum: 300 + 800 + 300 = 1400Then, A(1,1) = 1400 / 9 ≈ 155.555...But wait, if (1,1) is the center, then the calculation is as above. But if (1,1) is the top-left, then we have to consider that some pixels are out of bounds. Since the problem didn't specify, but given that the grid is 3x3, and (1,1) is the center in a 3x3 grid, I think that's the intended approach.So, A(1,1) = 1400 / 9 ≈ 155.555...But let me double-check the calculation:First row:100 * 0.5 = 50150 * 1 = 150200 * 0.5 = 100Total: 50 + 150 + 100 = 300Second row:150 * 1 = 150250 * 2 = 500150 * 1 = 150Total: 150 + 500 + 150 = 800Third row:200 * 0.5 = 100150 * 1 = 150100 * 0.5 = 50Total: 100 + 150 + 50 = 300Sum: 300 + 800 + 300 = 1400Divide by 9: 1400 / 9 ≈ 155.555...So, approximately 155.56.But let me check if I have the weights correct. The weight matrix is:Row 1: 0.5, 1, 0.5Row 2: 1, 2, 1Row 3: 0.5, 1, 0.5Yes, that's correct.So, the adjusted intensity A(1,1) is approximately 155.56.But since the problem might expect an exact value, 1400/9 is 155 and 5/9, which is approximately 155.555...So, I can write it as 155.56 or as a fraction.But let me see if I made a mistake in the calculation.Wait, 100 * 0.5 is 50, 150 *1 is 150, 200*0.5 is 100. That's 300.150*1=150, 250*2=500, 150*1=150. That's 800.200*0.5=100, 150*1=150, 100*0.5=50. That's 300.Total: 300 + 800 + 300 = 1400.Yes, that's correct.So, 1400 / 9 = 155.555...So, I think that's the answer.Problem 2: Determining the Maximum Allowable Constant Factor CThe artist wants to process a 1920x1080 image in under 2 seconds on a device that can do 10 million operations per second. Each pixel adjustment involves 18 arithmetic operations.We need to formulate an inequality to find the maximum allowable constant factor C that can be used to speed up the processing time, and find the minimum value of C that satisfies the constraint.Hmm, okay. So, first, let's understand what's being asked.The artist wants to process the image in under 2 seconds. The device can perform 10 million operations per second. Each pixel requires 18 operations. The image has 1920x1080 pixels.So, total operations needed: 1920 * 1080 * 18.But the artist wants to speed up the processing time by a factor of C. So, the total operations would be divided by C.Wait, but the question is about the maximum allowable constant factor C that can be used to speed up the processing time. So, higher C means more speedup, but we need to ensure that the total time is under 2 seconds.Wait, let me think.Total operations without any speedup: 1920 * 1080 * 18.If we can speed up by a factor of C, then the effective operations become (1920 * 1080 * 18) / C.The device can perform 10 million operations per second, so the time taken would be:Time = (1920 * 1080 * 18) / (C * 10^7)We need this time to be less than 2 seconds.So, the inequality is:(1920 * 1080 * 18) / (C * 10^7) < 2We need to solve for C.So, let's write that:(1920 * 1080 * 18) / (C * 10^7) < 2Multiply both sides by C * 10^7:1920 * 1080 * 18 < 2 * C * 10^7Then, divide both sides by 2 * 10^7:(1920 * 1080 * 18) / (2 * 10^7) < CSo, C > (1920 * 1080 * 18) / (2 * 10^7)Compute the left-hand side:First, compute 1920 * 1080:1920 * 1080 = Let's compute 1920 * 1000 = 1,920,0001920 * 80 = 153,600So, total is 1,920,000 + 153,600 = 2,073,600Then, multiply by 18:2,073,600 * 18Compute 2,073,600 * 10 = 20,736,0002,073,600 * 8 = 16,588,800Total: 20,736,000 + 16,588,800 = 37,324,800Then, divide by (2 * 10^7):37,324,800 / 20,000,000 = 1.86624So, C > 1.86624Therefore, the minimum value of C that satisfies the constraint is just above 1.86624. Since C must be a constant factor, we can round it up to the next whole number if necessary, but the question doesn't specify. So, the minimum value is approximately 1.86624.But let me double-check the calculations.Compute 1920 * 1080:1920 * 1000 = 1,920,0001920 * 80 = 153,600Total: 1,920,000 + 153,600 = 2,073,600Yes.2,073,600 * 18:Let me compute 2,073,600 * 10 = 20,736,0002,073,600 * 8 = 16,588,800Total: 20,736,000 + 16,588,800 = 37,324,800Yes.37,324,800 / 20,000,000 = 1.86624Yes.So, C must be greater than approximately 1.86624. Therefore, the minimum value of C is just over 1.86624. If we need an exact value, we can write it as 1.86624, but since it's a constant factor, it's likely acceptable to present it as a decimal or a fraction.Alternatively, we can express it as a fraction:37,324,800 / 20,000,000 = 373248 / 200000 = 373248 ÷ 200000Divide numerator and denominator by 16:373248 ÷16=23328, 200000 ÷16=12500So, 23328 / 12500 = 1.86624Yes, so it's 1.86624.Therefore, the minimum value of C is approximately 1.86624.But let me think again: the question says "formulate an inequality to determine the maximum allowable constant factor C that can be used to speed up the processing time, and find the minimum value of C that satisfies this constraint."Wait, so the maximum allowable C is the smallest C that makes the time less than 2 seconds. So, the minimum C is 1.86624, meaning that C must be at least 1.86624 to meet the time constraint.So, the minimum value of C is approximately 1.86624.But let me check if I interpreted the speedup correctly. If C is the speedup factor, then time becomes original_time / C. So, original_time is (1920*1080*18)/10^7 seconds.Compute original_time:1920*1080=2,073,6002,073,600*18=37,324,80037,324,800 / 10^7 = 3.73248 seconds.So, original time is 3.73248 seconds.To make it under 2 seconds, we need to speed it up by a factor of C, so:3.73248 / C < 2Therefore, C > 3.73248 / 2 = 1.86624Yes, that's correct.So, the minimum value of C is 1.86624.Therefore, the artist needs a speedup factor of at least approximately 1.86624 to process the image in under 2 seconds.Final Answer1. The adjusted intensity ( A(1, 1) ) is boxed{dfrac{1400}{9}}.2. The minimum value of ( C ) is boxed{1.86624}.</think>