🧠:<think>Okay, so I have this problem where an FBI agent is trying to decrypt some messages. The encryption function is given as ( E(x) = (ax + b) mod p ), and they've intercepted two messages. I need to find the constants ( a ) and ( b ) using the given values. Let me write down what I know:- ( x_1 = 5 ), ( y_1 = 17 )- ( x_2 = 10 ), ( y_2 = 29 )- ( p = 31 )So, the encryption function is linear, right? It's a modular linear function. That means if I can set up two equations based on the given messages, I can solve for ( a ) and ( b ).Let me write the equations:1. ( a times 5 + b equiv 17 mod 31 )2. ( a times 10 + b equiv 29 mod 31 )Hmm, so I have two equations with two unknowns. Maybe I can subtract the first equation from the second to eliminate ( b ). Let's try that.Subtracting equation 1 from equation 2:( (10a + b) - (5a + b) equiv 29 - 17 mod 31 )Simplify:( 5a equiv 12 mod 31 )Okay, so now I have ( 5a equiv 12 mod 31 ). I need to solve for ( a ). To do that, I need the modular inverse of 5 modulo 31. The inverse of 5 is a number ( c ) such that ( 5c equiv 1 mod 31 ).Let me find ( c ). I can use the extended Euclidean algorithm for this.Compute GCD of 5 and 31:31 divided by 5 is 6 with a remainder of 1.So, 31 = 5*6 + 1Then, 5 = 1*5 + 0So, GCD is 1, which means the inverse exists.Now, working backwards:1 = 31 - 5*6So, 1 ≡ -5*6 mod 31Which means -6 is the inverse of 5 mod 31. But we can write -6 as 25 mod 31 because 31 - 6 = 25.Therefore, the inverse of 5 mod 31 is 25.So, multiplying both sides of ( 5a equiv 12 mod 31 ) by 25:( a equiv 12 * 25 mod 31 )Calculate 12*25:12*25 = 300Now, 300 divided by 31: 31*9=279, 300-279=21. So, 300 ≡ 21 mod 31.Thus, ( a ≡ 21 mod 31 ). So, ( a = 21 ).Now, plug ( a = 21 ) back into one of the original equations to find ( b ). Let's use the first equation:( 21*5 + b ≡ 17 mod 31 )Calculate 21*5: 105105 mod 31: 31*3=93, 105-93=12. So, 105 ≡ 12 mod 31.So, 12 + b ≡ 17 mod 31Subtract 12 from both sides:b ≡ 17 - 12 mod 31b ≡ 5 mod 31So, ( b = 5 ).Wait, let me double-check with the second equation to make sure I didn't make a mistake.Second equation: ( 21*10 + 5 mod 31 )21*10=210210 mod 31: 31*6=186, 210-186=24. So, 24 + 5=29 mod 31.Which matches ( y_2 = 29 ). Perfect, that checks out.So, the constants are ( a = 21 ) and ( b = 5 ).Moving on to the second part. The agent models the criminal network as a directed graph ( G ) with ( n ) nodes and ( m ) edges. The adjacency matrix ( A ) is known. They define matrix ( M = (I - A)^{-1} ). I need to explain how ( M ) can be used to identify influential individuals, specifically focusing on centrality.Hmm, okay. So, ( M = (I - A)^{-1} ). I remember that in graph theory, the matrix ( (I - A)^{-1} ) is related to the concept of the influence matrix or something like that. It's used in the context of identifying the influence of each node in a network.Wait, let me think. If ( A ) is the adjacency matrix, then ( I - A ) is the identity matrix minus the adjacency matrix. Inverting that gives ( M ). I think this is related to the concept of the Moore-Penrose pseudoinverse or something else?Alternatively, in the context of linear algebra, ( (I - A)^{-1} ) can be expressed as a Neumann series if the spectral radius of ( A ) is less than 1. So, ( M = I + A + A^2 + A^3 + dots ). This is similar to the idea of considering all possible paths in the graph.So, each entry ( M_{ij} ) in the matrix ( M ) represents the total number of walks from node ( i ) to node ( j ), considering all possible lengths. Therefore, if we sum the entries in each row of ( M ), it gives a measure of how influential each node is in terms of reaching other nodes through various paths.Wait, but in the context of centrality, there are different types: degree centrality, betweenness, closeness, eigenvector, etc. Here, since ( M ) is related to the number of walks, it might be similar to eigenvector centrality or something else.Alternatively, if we consider the matrix ( M ), the entries ( M_{ij} ) can represent the influence of node ( i ) on node ( j ). So, if we sum over all ( j ), we get the total influence of node ( i ). The node with the highest total influence would be the most influential.Alternatively, maybe it's about the number of times a node is involved in all possible walks, which would relate to its centrality.Wait, another thought: in the context of social networks, the matrix ( (I - A)^{-1} ) is used in the calculation of the Katz centrality. Katz centrality measures the influence of a node in a network by considering the number of walks starting from that node, with longer walks being damped by a factor. If ( alpha ) is the damping factor, then the Katz centrality is given by ( (I - alpha A)^{-1} mathbf{1} ), where ( mathbf{1} ) is a vector of ones.In our case, the matrix ( M = (I - A)^{-1} ) is similar, but without the damping factor. So, it might be that the entries of ( M ) represent the total number of walks from each node, and thus, the sum of each row would give a measure of centrality.Alternatively, if we take the vector ( mathbf{c} = M mathbf{1} ), where ( mathbf{1} ) is a vector of ones, then each entry ( c_i ) would be the sum of all walks starting from node ( i ), which would indicate how influential node ( i ) is.Therefore, the individual with the highest centrality would correspond to the node with the highest value in this vector ( mathbf{c} ). So, by computing ( M ) and then multiplying by a vector of ones, we can identify the most influential individual in the network.Alternatively, if we consider the diagonal entries of ( M ), they might represent the number of walks that start and end at the same node, which could relate to the node's influence on itself, but I think the more relevant measure is the total influence on all other nodes.So, in summary, the matrix ( M = (I - A)^{-1} ) allows us to compute the total influence or number of walks from each node to all others. By summing the entries in each row, we can determine the centrality of each node, and the node with the highest sum is the most influential.Wait, but I should make sure that ( I - A ) is invertible. Since ( A ) is the adjacency matrix of a directed graph, it's not necessarily invertible. However, in this case, since the agent is using ( M = (I - A)^{-1} ), it must be that ( I - A ) is invertible. This usually requires that the spectral radius of ( A ) is less than 1, which might be the case if the graph doesn't have cycles with too much influence or something like that.But maybe in the context of the problem, we can assume that ( I - A ) is invertible, perhaps because the graph is a DAG or something. Anyway, moving on.So, to identify influential individuals, we compute ( M ), then for each node, sum its row in ( M ), and the node with the highest sum is the most central or influential.Alternatively, if we consider the columns, it might represent the influence on each node, but I think rows are more about the influence coming out from each node.Wait, actually, in matrix multiplication terms, ( M ) is such that ( M = I + A + A^2 + A^3 + dots ). So, each entry ( M_{ij} ) is the number of walks from ( i ) to ( j ). Therefore, if we sum over all ( j ), we get the total number of walks starting from ( i ), which is a measure of how influential ( i ) is.So, the sum of each row in ( M ) gives the total influence of each node, and the node with the highest sum is the most influential.Therefore, the matrix ( M ) can be used to identify influential individuals by calculating the row sums, which represent the total number of walks originating from each node, and the highest row sum corresponds to the individual with the highest centrality.I think that makes sense. So, in conclusion, once ( a ) and ( b ) are found, the matrix ( M ) is computed, and by analyzing its row sums, the agent can identify the most influential individuals in the criminal network.Final Answer1. The constants are ( a = boxed{21} ) and ( b = boxed{5} ).2. The matrix ( M ) can be used to identify influential individuals by calculating the row sums, which represent the total number of walks originating from each node. The individual with the highest row sum has the highest centrality.boxed{21} and boxed{5}</think>

🧠:<think>Okay, so I have this problem about a pharmacologist working on a new drug. The first part is about analyzing the stability of equilibrium points in a system of differential equations. Let me try to break this down step by step.First, the system is given by two differential equations:1. dD/dt = -k1 D(t) P(t) + k2 (D0 - D(t))2. dP/dt = -k1 D(t) P(t) + k3 (P0 - P(t))Here, D(t) is the drug concentration, P(t) is the protein concentration, and k1, k2, k3 are rate constants. D0 and P0 are the initial concentrations.I need to analyze the stability of the equilibrium points. So, first, I should find the equilibrium points by setting dD/dt and dP/dt to zero.Let me write that down:At equilibrium:- k1 D P + k2 (D0 - D) = 0- k1 D P + k3 (P0 - P) = 0Wait, actually, the equations are:dD/dt = -k1 D P + k2 (D0 - D) = 0dP/dt = -k1 D P + k3 (P0 - P) = 0So, setting them equal to zero:1. -k1 D P + k2 (D0 - D) = 02. -k1 D P + k3 (P0 - P) = 0Let me denote equation 1 as Eq1 and equation 2 as Eq2.From Eq1: -k1 D P + k2 D0 - k2 D = 0Similarly, from Eq2: -k1 D P + k3 P0 - k3 P = 0Let me rearrange both equations:From Eq1: k2 D0 = k1 D P + k2 DFrom Eq2: k3 P0 = k1 D P + k3 PSo, both right-hand sides have k1 D P, so I can set them equal:k2 D0 - k2 D = k3 P0 - k3 PWait, let me think again.Wait, from Eq1: k2 D0 = k1 D P + k2 D => k1 D P = k2 D0 - k2 DFrom Eq2: k3 P0 = k1 D P + k3 P => k1 D P = k3 P0 - k3 PTherefore, since both equal to k1 D P, we can set them equal:k2 D0 - k2 D = k3 P0 - k3 PSo, k2 D0 - k2 D = k3 P0 - k3 PLet me rearrange terms:k2 D0 - k3 P0 = k2 D - k3 PHmm, this seems a bit tricky. Maybe I can express P in terms of D or vice versa.Alternatively, perhaps I can subtract the two equations.Wait, let's see:From Eq1: k1 D P = k2 (D0 - D)From Eq2: k1 D P = k3 (P0 - P)Therefore, k2 (D0 - D) = k3 (P0 - P)So, k2 D0 - k2 D = k3 P0 - k3 PWhich can be rewritten as:k2 D0 - k3 P0 = k2 D - k3 PHmm, so this is a linear equation relating D and P.Let me denote this as Eq3: k2 D0 - k3 P0 = k2 D - k3 PSo, perhaps I can express P in terms of D or D in terms of P.Let me solve for P:From Eq3: k2 D0 - k3 P0 = k2 D - k3 PBring terms involving P to one side:k3 P = k2 D - (k2 D0 - k3 P0)So,k3 P = k2 D - k2 D0 + k3 P0Divide both sides by k3:P = (k2 / k3) D - (k2 / k3) D0 + P0So, P = (k2 / k3) (D - D0) + P0Alternatively, P = (k2 / k3) D + (P0 - (k2 / k3) D0)Let me denote this as Eq4: P = (k2 / k3) D + C, where C = P0 - (k2 / k3) D0Now, plug this expression for P into one of the original equilibrium equations, say Eq1.From Eq1: k1 D P = k2 (D0 - D)Substitute P from Eq4:k1 D [ (k2 / k3) D + C ] = k2 (D0 - D)Let me compute this:k1 D [ (k2 / k3) D + (P0 - (k2 / k3) D0) ] = k2 D0 - k2 DLet me expand the left-hand side:k1 (k2 / k3) D^2 + k1 (P0 - (k2 / k3) D0) D = k2 D0 - k2 DBring all terms to one side:k1 (k2 / k3) D^2 + k1 (P0 - (k2 / k3) D0) D - k2 D0 + k2 D = 0This is a quadratic equation in D:A D^2 + B D + C = 0Where:A = k1 k2 / k3B = k1 (P0 - (k2 / k3) D0) + k2C = -k2 D0Wait, let me compute each coefficient:A = k1 * (k2 / k3) = (k1 k2) / k3B = k1 (P0 - (k2 / k3) D0) + k2C = -k2 D0So, the quadratic equation is:[(k1 k2)/k3] D^2 + [k1 (P0 - (k2 / k3) D0) + k2] D - k2 D0 = 0This seems a bit complicated. Maybe I can factor it or find a way to simplify.Alternatively, perhaps I can consider specific cases or look for possible solutions.Wait, another approach: let's consider that at equilibrium, the rates of change are zero, so the terms involving binding must balance the terms involving production or degradation.Alternatively, perhaps I can consider that the system might have multiple equilibrium points, but we are interested in their stability.But before that, perhaps I can find the equilibrium points by solving the quadratic equation.So, let's denote:A = (k1 k2)/k3B = k1 (P0 - (k2 / k3) D0) + k2C = -k2 D0So, quadratic equation: A D^2 + B D + C = 0The solutions are:D = [-B ± sqrt(B^2 - 4AC)] / (2A)But this might get messy. Maybe I can assume that there's a unique equilibrium point, but perhaps not.Alternatively, maybe I can consider that if the system has only one equilibrium point, then that's the only solution.Wait, perhaps I can think about the case where D = D0 and P = P0. Let me check if that's an equilibrium.If D = D0 and P = P0, then:From Eq1: -k1 D0 P0 + k2 (D0 - D0) = -k1 D0 P0 + 0 = -k1 D0 P0 ≠ 0 unless k1=0 or D0=0 or P0=0, which is not the case.So, D = D0 and P = P0 is not an equilibrium unless trivially.Alternatively, perhaps the equilibrium is when D and P are such that the binding term balances the other terms.Alternatively, perhaps I can consider the case where the binding term is zero, but that would require D=0 or P=0, which might not be the case.Alternatively, perhaps I can think about the system in terms of conservation laws or steady-state approximations.Wait, perhaps I can consider that in the absence of the drug, the protein would be at P0, and in the absence of the protein, the drug would be at D0. But with both present, they interact.Alternatively, perhaps I can consider that the system might have two equilibrium points: one where the drug and protein are zero, but that's trivial, and another where they are at some non-zero concentrations.Wait, but in the equations, the terms k2 (D0 - D) and k3 (P0 - P) suggest that D tends to D0 and P tends to P0 in the absence of the other, but when they interact, their concentrations are reduced due to binding.Wait, perhaps the equilibrium points are when the binding rate equals the production/degradation rates.Alternatively, perhaps I can think about the system as a competition between the binding term and the production/degradation terms.But maybe I should proceed with solving the quadratic equation.So, let's write the quadratic equation again:[(k1 k2)/k3] D^2 + [k1 (P0 - (k2 / k3) D0) + k2] D - k2 D0 = 0Let me denote this as:A D^2 + B D + C = 0Where:A = (k1 k2)/k3B = k1 (P0 - (k2 / k3) D0) + k2C = -k2 D0Now, the discriminant is D = B^2 - 4ACLet me compute D:D = [k1 (P0 - (k2 / k3) D0) + k2]^2 - 4 * (k1 k2 / k3) * (-k2 D0)Simplify:D = [k1 (P0 - (k2 / k3) D0) + k2]^2 + 4 (k1 k2 / k3)(k2 D0)This is positive, so there are two real roots.Therefore, there are two possible equilibrium points for D, and correspondingly for P.But which one is stable?To determine stability, I need to analyze the Jacobian matrix of the system at the equilibrium points and find the eigenvalues.The Jacobian matrix J is given by:J = [ d(dD/dt)/dD d(dD/dt)/dP ] [ d(dP/dt)/dD d(dP/dt)/dP ]Compute the partial derivatives:From dD/dt = -k1 D P + k2 (D0 - D)So,d(dD/dt)/dD = -k1 P - k2d(dD/dt)/dP = -k1 DSimilarly, from dP/dt = -k1 D P + k3 (P0 - P)So,d(dP/dt)/dD = -k1 Pd(dP/dt)/dP = -k1 D - k3Therefore, the Jacobian matrix is:[ -k1 P - k2 -k1 D ][ -k1 P -k1 D - k3 ]At the equilibrium points, we can evaluate this matrix.The eigenvalues of J will determine the stability. If both eigenvalues have negative real parts, the equilibrium is stable (asymptotically stable). If any eigenvalue has a positive real part, it's unstable.So, let's denote the Jacobian at equilibrium as:J = [ a b ] [ c d ]Where:a = -k1 P - k2b = -k1 Dc = -k1 Pd = -k1 D - k3The trace Tr = a + d = (-k1 P - k2) + (-k1 D - k3) = -k1 (P + D) - (k2 + k3)The determinant Det = a d - b c = [(-k1 P - k2)(-k1 D - k3)] - [(-k1 D)(-k1 P)]Let me compute Det:First term: (-k1 P - k2)(-k1 D - k3) = (k1 P + k2)(k1 D + k3) = k1^2 P D + k1 P k3 + k1 D k2 + k2 k3Second term: (-k1 D)(-k1 P) = k1^2 D PSo, Det = [k1^2 P D + k1 P k3 + k1 D k2 + k2 k3] - [k1^2 D P] = k1 P k3 + k1 D k2 + k2 k3Therefore, Det = k1 k3 P + k1 k2 D + k2 k3Now, for stability, we need both eigenvalues to have negative real parts. For a 2x2 system, this requires:1. Tr < 02. Det > 0So, let's check these conditions.First, Tr = -k1 (P + D) - (k2 + k3) < 0Since k1, k2, k3 are positive rate constants, and P and D are concentrations, which are positive, so Tr is negative.Second, Det = k1 k3 P + k1 k2 D + k2 k3 > 0All terms are positive, so Det is positive.Therefore, both conditions are satisfied, meaning that any equilibrium point is a stable node.Wait, but earlier we found that there are two equilibrium points. So, does that mean both are stable? Or is one stable and the other unstable?Wait, but in a two-dimensional system, if both eigenvalues are negative, the equilibrium is a stable node. If one eigenvalue is positive and the other negative, it's a saddle point. If both are positive, it's an unstable node.But in our case, for both equilibrium points, the Jacobian has Tr < 0 and Det > 0, so both eigenvalues have negative real parts, meaning both equilibrium points are stable nodes.But that seems counterintuitive because typically, such systems have one stable and one unstable equilibrium.Wait, perhaps I made a mistake in the determinant calculation.Let me recompute the determinant:Det = a d - b ca = -k1 P - k2d = -k1 D - k3b = -k1 Dc = -k1 PSo,a d = (-k1 P - k2)(-k1 D - k3) = (k1 P + k2)(k1 D + k3) = k1^2 P D + k1 P k3 + k1 D k2 + k2 k3b c = (-k1 D)(-k1 P) = k1^2 D PTherefore, Det = a d - b c = (k1^2 P D + k1 P k3 + k1 D k2 + k2 k3) - k1^2 P D = k1 P k3 + k1 D k2 + k2 k3Yes, that's correct.So, Det is positive, and Tr is negative, so both eigenvalues have negative real parts, meaning both equilibrium points are stable.But that seems odd because in many systems, you have one stable and one unstable equilibrium.Wait, perhaps I need to consider the nature of the system. Maybe the system can have multiple stable equilibria, but in reality, it's more likely that only one is stable.Alternatively, perhaps I made a mistake in assuming that both equilibrium points are valid.Wait, let's think about the quadratic equation for D. We have two solutions, but perhaps only one of them is positive, as concentrations can't be negative.So, let's check the solutions:D = [-B ± sqrt(B^2 - 4AC)] / (2A)Given that A = (k1 k2)/k3 > 0C = -k2 D0 < 0So, the quadratic equation is A D^2 + B D + C = 0, with A > 0, C < 0Therefore, the product of the roots is C/A = (-k2 D0) / (k1 k2 / k3) = (-k2 D0 k3) / (k1 k2) = (-D0 k3)/k1 < 0So, one root is positive, and the other is negative.Since concentration can't be negative, only the positive root is valid.Therefore, there is only one valid equilibrium point.Wait, that makes more sense. So, even though the quadratic equation gives two roots, only the positive one is physically meaningful.Therefore, the system has only one equilibrium point, which is stable because Tr < 0 and Det > 0.Wait, but earlier I thought there were two equilibrium points, but now I see that only one is valid because the other would give negative concentrations.Therefore, the system has a unique equilibrium point which is stable.So, the conclusion is that the system reaches a stable equilibrium under the given conditions, and the equilibrium point is unique.Now, moving to the second part of the problem.The pharmacologist uses the Hill equation to model the binding affinity:θ(D) = D^n / (K_d^n + D^n)Where θ(D) is the fraction bound, K_d is the dissociation constant, and n is the Hill coefficient.We need to find the EC50, which is the concentration where 50% of the protein is bound, i.e., θ(D) = 0.5.So, set θ(D) = 0.5:0.5 = D^n / (K_d^n + D^n)Multiply both sides by denominator:0.5 (K_d^n + D^n) = D^nMultiply out:0.5 K_d^n + 0.5 D^n = D^nSubtract 0.5 D^n from both sides:0.5 K_d^n = 0.5 D^nMultiply both sides by 2:K_d^n = D^nTake nth root:K_d = DTherefore, EC50 = K_dWait, that seems too straightforward. Let me double-check.Given θ(D) = D^n / (K_d^n + D^n) = 0.5So,D^n / (K_d^n + D^n) = 1/2Cross-multiplying:2 D^n = K_d^n + D^nSubtract D^n:D^n = K_d^nTherefore, D = K_dSo, EC50 = K_dHmm, that's correct. So, the EC50 is equal to the dissociation constant K_d.Now, the next part is to determine the sensitivity of EC50 with respect to changes in n, i.e., compute d(EC50)/dn.But wait, from above, EC50 = K_d, which is independent of n. Therefore, d(EC50)/dn = 0.But that seems odd because in reality, the Hill coefficient affects the steepness of the curve, but EC50 is the concentration at 50% binding, which in this case is always K_d, regardless of n.Wait, let me think again. The Hill equation is θ(D) = D^n / (K_d^n + D^n). So, when D = K_d, θ(D) = (K_d^n)/(K_d^n + K_d^n) = 1/2, regardless of n. So, indeed, EC50 is always K_d, so it doesn't depend on n. Therefore, the derivative d(EC50)/dn is zero.But perhaps I'm missing something. Maybe the question is considering a different definition of EC50, but according to the Hill equation as given, EC50 is indeed K_d, and it's independent of n.Alternatively, perhaps the question is considering a different form of the Hill equation, but as given, it's θ(D) = D^n / (K_d^n + D^n), so EC50 is K_d.Therefore, the sensitivity is zero.So, summarizing:1. The system has a unique stable equilibrium point.2. EC50 = K_d, and d(EC50)/dn = 0.But wait, let me think again about the first part. Earlier, I concluded that there's only one valid equilibrium point because the other root is negative, but let me confirm that.Given the quadratic equation for D:A D^2 + B D + C = 0With A = (k1 k2)/k3 > 0C = -k2 D0 < 0So, the product of the roots is C/A = (-k2 D0) / (k1 k2 / k3) = (-D0 k3)/k1 < 0Therefore, one root is positive, the other is negative. Since D represents concentration, only the positive root is valid. Therefore, there's only one equilibrium point, which is stable because the Jacobian has Tr < 0 and Det > 0.Therefore, the system reaches a stable equilibrium under the given conditions.So, the conditions on k1, k2, k3 are that they are positive rate constants, which they are by definition, so the system will always reach a stable equilibrium.Therefore, the answer to part 1 is that the system has a unique stable equilibrium point, and it is guaranteed to reach it given positive rate constants.For part 2, EC50 = K_d, and its derivative with respect to n is zero.So, putting it all together.</think>

🧠:<think>Alright, so I have this problem where I need to analyze the efficiency of different job roles in a project team. The team consists of developers, testers, and project managers. The project managers help by reducing the time needed for developers and testers by 20% and 30% respectively. First, let me try to understand what the problem is asking. There are two sub-problems. Sub-problem 1: Calculate the total number of tasks completed by developers and testers over the first 5 weeks, considering the time reduction by project managers.Sub-problem 2: Determine the minimum number of weeks required to reach a goal of 1000 tasks, again considering the time reductions.Let me tackle Sub-problem 1 first.So, for developers, their task completion rate is given by D(t) = 5t² - 3t + 2 tasks per week. Similarly, testers have a rate of T(t) = 4t³ - 6t² + 3t tasks per week. But wait, the project managers are facilitating by reducing the time needed. So, does that mean that the effective rate is increased? Because if the time needed is reduced, they can complete tasks faster. So, for developers, their effective rate would be increased by 20%, and for testers by 30%.Hmm, let me think. If the time is reduced by 20%, that means they can do 1 / (1 - 0.2) = 1.25 times the original rate. Similarly, for testers, it's 1 / (1 - 0.3) = approximately 1.4286 times the original rate.Yes, that makes sense. Because if you take 20% less time, you can do 25% more tasks in the same period. So, the effective rate is multiplied by 1.25 for developers and 1.4286 for testers.So, the effective developer rate becomes D_eff(t) = D(t) * 1.25, and the effective tester rate is T_eff(t) = T(t) * (10/7) since 1/0.7 ≈ 1.4286.Wait, 1/0.7 is approximately 1.42857, which is 10/7. So, yes, that's correct.So, for each week t, the number of tasks completed by developers is D(t) * 1.25, and by testers is T(t) * (10/7).But wait, is the time reduction applied per week or overall? The problem says "facilitate the completion of tasks by reducing the time needed by 20% for developers and 30% for testers." So, I think it's a constant reduction, so each week, their effective rate is increased by those factors.Therefore, for each week from t=1 to t=5, I need to compute D(t) * 1.25 and T(t) * (10/7), and then sum them up over the 5 weeks.Wait, but actually, the functions D(t) and T(t) are given as tasks per week. So, if the time is reduced, does that mean that the rate is increased? For example, if a developer takes 20% less time per task, they can complete 25% more tasks per week. So, yes, the rate is multiplied by 1.25.Similarly, testers can complete 1 / 0.7 ≈ 1.4286 times their original rate.So, effectively, the total tasks completed by developers over 5 weeks would be the sum from t=1 to t=5 of D(t) * 1.25, and similarly for testers, sum from t=1 to t=5 of T(t) * (10/7).Then, the total tasks would be the sum of both.Alternatively, maybe the functions D(t) and T(t) are cumulative? Wait, no, the problem says "tasks per week", so D(t) is the rate at week t. So, to get the total tasks, we need to integrate over time, but since it's discrete weeks, we can sum the rates.Wait, but in the problem statement, it's not specified whether t is continuous or discrete. Hmm. It says "t is the number of weeks since the project started." So, t is an integer, 1, 2, 3, etc. So, it's discrete. Therefore, for each week t, the number of tasks completed is D(t) * 1.25 for developers and T(t) * (10/7) for testers.Therefore, to find the total over 5 weeks, we need to compute the sum from t=1 to t=5 of D(t) * 1.25 and the sum from t=1 to t=5 of T(t) * (10/7), then add them together.So, let's compute that.First, let's compute D(t) for t=1 to 5:D(t) = 5t² - 3t + 2t=1: 5(1) - 3(1) + 2 = 5 - 3 + 2 = 4t=2: 5(4) - 3(2) + 2 = 20 - 6 + 2 = 16t=3: 5(9) - 3(3) + 2 = 45 - 9 + 2 = 38t=4: 5(16) - 3(4) + 2 = 80 - 12 + 2 = 70t=5: 5(25) - 3(5) + 2 = 125 - 15 + 2 = 112Now, multiply each by 1.25:t=1: 4 * 1.25 = 5t=2: 16 * 1.25 = 20t=3: 38 * 1.25 = 47.5t=4: 70 * 1.25 = 87.5t=5: 112 * 1.25 = 140Now, sum these up:5 + 20 = 2525 + 47.5 = 72.572.5 + 87.5 = 160160 + 140 = 300So, total tasks by developers: 300Now, let's compute T(t) for t=1 to 5:T(t) = 4t³ - 6t² + 3tt=1: 4(1) - 6(1) + 3(1) = 4 - 6 + 3 = 1t=2: 4(8) - 6(4) + 3(2) = 32 - 24 + 6 = 14t=3: 4(27) - 6(9) + 3(3) = 108 - 54 + 9 = 63t=4: 4(64) - 6(16) + 3(4) = 256 - 96 + 12 = 172t=5: 4(125) - 6(25) + 3(5) = 500 - 150 + 15 = 365Now, multiply each by 10/7 (~1.4286):t=1: 1 * (10/7) ≈ 1.4286t=2: 14 * (10/7) = 20t=3: 63 * (10/7) = 90t=4: 172 * (10/7) ≈ 245.7143t=5: 365 * (10/7) ≈ 521.4286Now, sum these up:1.4286 + 20 = 21.428621.4286 + 90 = 111.4286111.4286 + 245.7143 ≈ 357.1429357.1429 + 521.4286 ≈ 878.5715So, total tasks by testers: approximately 878.5715Now, total tasks by both developers and testers: 300 + 878.5715 ≈ 1178.5715But wait, the problem says "the total number of tasks completed by developers and testers over the first 5 weeks". So, approximately 1178.57 tasks. Since tasks are whole numbers, we might need to round, but the problem doesn't specify, so perhaps we can leave it as a decimal.Alternatively, maybe we should keep it exact. Let's see:For testers, the total is 1*(10/7) + 14*(10/7) + 63*(10/7) + 172*(10/7) + 365*(10/7)Which is (1 + 14 + 63 + 172 + 365) * (10/7)Compute the sum inside:1 + 14 = 1515 + 63 = 7878 + 172 = 250250 + 365 = 615So, 615 * (10/7) = 6150 / 7 = 878.57142857...So, exactly 878 and 4/7 tasks.So, total tasks: 300 + 878 4/7 = 1178 4/7 ≈ 1178.5714So, Sub-problem 1 answer is approximately 1178.57 tasks.But let me double-check my calculations because sometimes when dealing with fractions, it's easy to make a mistake.Wait, for developers, the sum was 5 + 20 + 47.5 + 87.5 + 140 = 300. That seems correct.For testers, the sum was 1.4286 + 20 + 90 + 245.7143 + 521.4286 ≈ 878.5715. Yes, that's correct.So, total is 300 + 878.5715 ≈ 1178.5715.So, I think that's correct.Now, moving on to Sub-problem 2: Determine the minimum number of weeks required for the team to reach a goal of 1000 tasks, considering the time reductions.So, we need to find the smallest integer t such that the total tasks completed by developers and testers up to week t is at least 1000.But wait, in Sub-problem 1, over 5 weeks, they completed approximately 1178.57 tasks, which is more than 1000. So, perhaps the minimum number of weeks is less than 5? Wait, but let's check.Wait, actually, in Sub-problem 1, we summed up to week 5, but maybe the cumulative tasks reach 1000 before week 5. So, we need to find the smallest t where the cumulative tasks >= 1000.But wait, in Sub-problem 1, the total was 1178.57 at week 5, which is more than 1000. So, perhaps the minimum number of weeks is 5? But maybe it's less. Let's check the cumulative tasks week by week.Wait, let me compute the cumulative tasks for each week and see when it crosses 1000.But first, let's note that the functions D(t) and T(t) are increasing functions, so the cumulative tasks will also increase each week.So, let's compute the cumulative tasks for each week t=1,2,3,4,5.From Sub-problem 1, we have:Developers' tasks per week after adjustment:t=1: 5t=2: 20t=3: 47.5t=4: 87.5t=5: 140Cumulative developers:t=1: 5t=2: 5 + 20 = 25t=3: 25 + 47.5 = 72.5t=4: 72.5 + 87.5 = 160t=5: 160 + 140 = 300Testers' tasks per week after adjustment:t=1: 10/7 ≈ 1.4286t=2: 20t=3: 90t=4: 245.7143t=5: 521.4286Cumulative testers:t=1: ≈1.4286t=2: 1.4286 + 20 ≈21.4286t=3: 21.4286 + 90 ≈111.4286t=4: 111.4286 + 245.7143 ≈357.1429t=5: 357.1429 + 521.4286 ≈878.5715Total cumulative tasks:t=1: 5 + 1.4286 ≈6.4286t=2: 25 + 21.4286 ≈46.4286t=3: 72.5 + 111.4286 ≈183.9286t=4: 160 + 357.1429 ≈517.1429t=5: 300 + 878.5715 ≈1178.5715Wait, so at week 4, the total is approximately 517.14, which is less than 1000. At week 5, it's approximately 1178.57, which is more than 1000. So, the minimum number of weeks required is 5.But wait, let me check if perhaps the cumulative tasks cross 1000 partway through week 5. But since the problem is defined in whole weeks, and the rates are given per week, we can only consider whole weeks. Therefore, the team reaches the goal in week 5.But wait, let me think again. Maybe the functions are continuous, and we can model the cumulative tasks as a function of continuous time t, and find the exact t where the cumulative tasks reach 1000.But the problem says "the number of weeks since the project started", which is discrete, but perhaps the functions D(t) and T(t) are defined for continuous t, and the project managers' effect is a constant multiplier. So, maybe we can model the cumulative tasks as integrals from 0 to t of D(s)*1.25 + T(s)*(10/7) ds, and find t such that the integral equals 1000.Wait, that might be a more accurate approach, considering that the functions are given as D(t) and T(t) which could be continuous.Let me consider that possibility.So, if t is continuous, then the total tasks completed by developers up to time t is the integral from 0 to t of D(s)*1.25 ds, and similarly for testers.So, total tasks T_total(t) = 1.25 * ∫₀ᵗ (5s² - 3s + 2) ds + (10/7) * ∫₀ᵗ (4s³ - 6s² + 3s) dsCompute these integrals:First, compute ∫ (5s² - 3s + 2) ds:= (5/3)s³ - (3/2)s² + 2s + CSimilarly, ∫ (4s³ - 6s² + 3s) ds:= s⁴ - 2s³ + (3/2)s² + CSo, T_total(t) = 1.25 * [ (5/3)t³ - (3/2)t² + 2t ] + (10/7) * [ t⁴ - 2t³ + (3/2)t² ]Simplify:First term: 1.25*(5/3 t³ - 3/2 t² + 2t) = (1.25*5/3) t³ - (1.25*3/2) t² + (1.25*2) tCompute coefficients:1.25 * 5/3 = (5/4) * 5/3 = 25/12 ≈2.08331.25 * 3/2 = (5/4)*(3/2) = 15/8 = 1.8751.25 * 2 = 2.5So, first term: (25/12) t³ - (15/8) t² + (5/2) tSecond term: (10/7)*(t⁴ - 2t³ + (3/2)t²) = (10/7)t⁴ - (20/7)t³ + (15/7)t²Now, combine both terms:T_total(t) = (25/12) t³ - (15/8) t² + (5/2) t + (10/7)t⁴ - (20/7)t³ + (15/7)t²Combine like terms:t⁴ term: (10/7) t⁴t³ terms: (25/12 - 20/7) t³Compute 25/12 - 20/7:Find common denominator, which is 84.25/12 = 175/8420/7 = 240/84So, 175/84 - 240/84 = -65/84t³ term: (-65/84) t³t² terms: (-15/8 + 15/7) t²Compute -15/8 + 15/7:Common denominator 56.-15/8 = -105/5615/7 = 120/56So, -105/56 + 120/56 = 15/56t² term: (15/56) t²t term: (5/2) tSo, overall:T_total(t) = (10/7) t⁴ - (65/84) t³ + (15/56) t² + (5/2) tWe need to solve for t such that T_total(t) = 1000.This is a quartic equation, which is quite complex. Let me see if I can find an approximate solution.Alternatively, since in the discrete case, at t=5 weeks, the total is ~1178.57, which is more than 1000, and at t=4 weeks, it's ~517.14, which is less than 1000. So, the continuous case might have the total crossing 1000 somewhere between t=4 and t=5.But since the problem is about weeks, and the rates are given per week, perhaps we should stick to the discrete model, as in Sub-problem 1, and say that the minimum number of weeks is 5.But let me check the continuous case as well, just to be thorough.So, let's set up the equation:(10/7) t⁴ - (65/84) t³ + (15/56) t² + (5/2) t = 1000This is a quartic equation, which is difficult to solve analytically. So, we can use numerical methods to approximate the solution.Let me try plugging in t=4:Compute each term:(10/7)*(256) = (10/7)*256 ≈365.7143-(65/84)*(64) ≈-65/84*64 ≈-65*0.7619 ≈-49.5238(15/56)*(16) ≈15/56*16 ≈15*0.2857 ≈4.2857(5/2)*4 =10Sum: 365.7143 -49.5238 +4.2857 +10 ≈365.7143 -49.5238 = 316.1905 +4.2857 = 320.4762 +10 = 330.4762Which is much less than 1000.t=5:(10/7)*625 ≈892.8571-(65/84)*125 ≈-65/84*125 ≈-65*1.4881 ≈-96.7262(15/56)*25 ≈15/56*25 ≈15*0.4464 ≈6.6964(5/2)*5 =12.5Sum: 892.8571 -96.7262 ≈796.1309 +6.6964 ≈802.8273 +12.5 ≈815.3273Still less than 1000.t=6:(10/7)*1296 ≈1851.4286-(65/84)*216 ≈-65/84*216 ≈-65*2.5714 ≈-167.1429(15/56)*36 ≈15/56*36 ≈15*0.6429 ≈9.6429(5/2)*6 =15Sum: 1851.4286 -167.1429 ≈1684.2857 +9.6429 ≈1693.9286 +15 ≈1708.9286Which is more than 1000.So, between t=5 and t=6, the total crosses 1000.To find the exact t, we can use linear approximation or more precise methods.But since the problem is about weeks, and the rates are given per week, it's more appropriate to consider the discrete model, where the total at week 5 is ~1178.57, which is more than 1000, and at week 4 it's ~517.14, which is less than 1000. Therefore, the minimum number of weeks required is 5.But wait, in the continuous model, the total at t=5 is ~815.3273, which is still less than 1000, and at t=6, it's ~1708.9286. So, the continuous model suggests that the goal is reached between week 5 and 6. But since the problem is about weeks, and the rates are given per week, perhaps the answer is 6 weeks.But this is conflicting with the discrete model.Wait, perhaps I made a mistake in interpreting the problem. Let me re-examine the problem statement.The problem says: "Calculate the total number of tasks completed by developers and testers over the first 5 weeks of the project. Take into account the time reduction facilitated by the project managers."So, for Sub-problem 1, it's over the first 5 weeks, considering the time reduction.For Sub-problem 2: "Assume that the project has a set goal of 1000 tasks to be completed. Considering the rates provided and the time reductions, determine the minimum number of weeks required for the team to reach this goal."So, the rates are given as tasks per week, with the time reductions applied. So, perhaps the time reduction is applied per week, meaning that each week, the effective rate is increased.Therefore, the total tasks completed up to week t is the sum from s=1 to s=t of [D(s)*1.25 + T(s)*(10/7)].So, in that case, the total tasks at week 5 is ~1178.57, which is more than 1000, and at week 4, it's ~517.14, which is less than 1000. Therefore, the minimum number of weeks required is 5.But wait, in the continuous model, the total at t=5 is ~815, which is less than 1000, but in the discrete model, it's ~1178.57. So, which one is correct?I think the problem is intended to be discrete, as the functions D(t) and T(t) are given per week, and t is the number of weeks. Therefore, the time reduction is applied per week, so each week's tasks are multiplied by the respective factors.Therefore, the total tasks up to week t is the sum from s=1 to s=t of [D(s)*1.25 + T(s)*(10/7)].So, in that case, the minimum number of weeks is 5.But let me check the cumulative tasks week by week:At week 1: ~6.4286Week 2: ~46.4286Week 3: ~183.9286Week 4: ~517.1429Week 5: ~1178.5715So, the cumulative tasks reach 1000 between week 4 and week 5. But since we can't have a fraction of a week, we need to round up to the next whole week, which is week 5.Therefore, the minimum number of weeks required is 5.But wait, in the continuous model, the total at t=5 is ~815, which is less than 1000, but in the discrete model, it's ~1178.57. So, the answer depends on whether we model it as continuous or discrete.Given that the problem states "the number of weeks since the project started", and the rates are given per week, it's more appropriate to model it as discrete weeks, summing the tasks each week. Therefore, the answer is 5 weeks.But just to be thorough, let me check if the cumulative tasks in the discrete model cross 1000 before week 5.Wait, at week 4, the total is ~517.14, which is less than 1000. At week 5, it's ~1178.57, which is more than 1000. So, the minimum number of weeks is 5.Therefore, the answer to Sub-problem 2 is 5 weeks.But wait, let me think again. If the project managers reduce the time needed by 20% and 30%, does that mean that the effective rate is increased by those percentages? Or is it that the time per task is reduced, so the number of tasks per week is increased by 20% and 30% respectively?Wait, the problem says "facilitate the completion of tasks by reducing the time needed by 20% for developers and 30% for testers."So, if the time needed is reduced by 20%, that means that the time per task is 80% of the original. Therefore, the rate is increased by 1/0.8 = 1.25 times, which is a 25% increase. Similarly, for testers, 1/0.7 ≈1.4286, which is a ~42.86% increase.Therefore, the effective rates are D(t)*1.25 and T(t)*(10/7).So, that part is correct.Therefore, the total tasks completed up to week t is the sum from s=1 to s=t of [D(s)*1.25 + T(s)*(10/7)].So, as per the discrete model, the answer is 5 weeks.Therefore, the answers are:Sub-problem 1: Approximately 1178.57 tasks.Sub-problem 2: 5 weeks.But let me present the exact fractions instead of decimals for precision.For Sub-problem 1:Developers' total: 300Testers' total: 615*(10/7) = 6150/7 = 878 4/7Total: 300 + 878 4/7 = 1178 4/7 tasks.So, 1178 4/7 tasks.For Sub-problem 2: 5 weeks.Therefore, the final answers are:Sub-problem 1: 1178 4/7 tasks.Sub-problem 2: 5 weeks.</think>

🧠:<think>Alright, so I have this problem about calculating the growth rate of a recipient country's GDP based on the aid they receive. Let me try to break it down step by step. First, the problem gives me a function for the growth rate, which is ( r(G_0, A) = frac{k cdot A}{G_0 + c cdot A} ). They've also provided the values for ( k ) and ( c ): ( k = 0.3 ) and ( c = 0.1 ). The initial GDP ( G_0 ) is 50 billion, and the aid amount ( A ) is 10 billion. Okay, so for part 1, I need to plug these values into the function to find the growth rate ( r ). Let me write that out:( r = frac{0.3 times 10}{50 + 0.1 times 10} )Let me compute the numerator first: 0.3 multiplied by 10. That should be 3. Now the denominator: 50 plus 0.1 times 10. 0.1 times 10 is 1, so 50 plus 1 is 51. So now, the growth rate ( r ) is 3 divided by 51. Hmm, 3 divided by 51. Let me compute that. 3 divided by 51 is the same as 1 divided by 17, which is approximately 0.0588. Wait, let me check that division again. 51 goes into 3 zero times, so we add a decimal point. 51 goes into 30 zero times, 51 goes into 300 five times because 5 times 51 is 255. Subtract 255 from 300, we get 45. Bring down another zero: 450. 51 goes into 450 eight times because 8 times 51 is 408. Subtract 408 from 450, we get 42. Bring down another zero: 420. 51 goes into 420 eight times again because 8 times 51 is 408. Subtract 408 from 420, we get 12. Bring down another zero: 120. 51 goes into 120 twice because 2 times 51 is 102. Subtract 102 from 120, we get 18. Bring down another zero: 180. 51 goes into 180 three times because 3 times 51 is 153. Subtract 153 from 180, we get 27. Bring down another zero: 270. 51 goes into 270 five times because 5 times 51 is 255. Subtract 255 from 270, we get 15. Hmm, I see a pattern here. So, 3 divided by 51 is approximately 0.0588235294... So, rounding to a reasonable decimal place, maybe four decimal places: 0.0588. So, the growth rate ( r ) is approximately 5.88%. Wait, let me make sure I didn't make a calculation error. So, 0.3 times 10 is 3, correct. 0.1 times 10 is 1, so 50 plus 1 is 51, correct. 3 divided by 51 is indeed approximately 0.0588, which is about 5.88%. That seems reasonable. Okay, so part 1 is done. The growth rate is approximately 5.88%. Now, moving on to part 2. They want me to find the total GDP after one year, ( G_1 ), using the formula ( G_1 = G_0 cdot (1 + r) ). We already have ( G_0 = 50 ) billion and ( r ) is approximately 0.0588. So, let me compute ( 1 + r ) first. That would be 1 + 0.0588 = 1.0588. Now, multiply that by ( G_0 ): 50 billion times 1.0588. Let me compute that. 50 times 1 is 50, and 50 times 0.0588 is... Let me compute 50 times 0.05 is 2.5, and 50 times 0.0088 is 0.44. So, 2.5 + 0.44 is 2.94. Therefore, 50 times 1.0588 is 50 + 2.94 = 52.94 billion. So, ( G_1 ) is approximately 52.94 billion. Wait, let me verify that multiplication another way. 50 times 1.0588. Since 1.0588 is the same as 1 + 0.0588, as I did before. So, 50 times 1 is 50, and 50 times 0.0588 is 2.94. Adding them together gives 52.94. Alternatively, I can compute 50 times 1.0588 directly. 50 times 1 is 50, 50 times 0.05 is 2.5, 50 times 0.008 is 0.4, and 50 times 0.0008 is 0.04. Adding all those: 50 + 2.5 + 0.4 + 0.04 = 52.94. Yep, same result. So, that seems consistent. Therefore, the GDP after one year is approximately 52.94 billion. Wait, just to make sure, let me compute 50 multiplied by 1.0588 on a calculator. 50 times 1.0588. Let me do 50 times 1 is 50, 50 times 0.05 is 2.5, 50 times 0.008 is 0.4, and 50 times 0.0008 is 0.04. So, adding those up: 50 + 2.5 is 52.5, plus 0.4 is 52.9, plus 0.04 is 52.94. Yes, that's correct. So, 52.94 billion is the GDP after one year. Is there another way to compute this? Maybe using fractions instead of decimals? Let's see. We had ( r = frac{3}{51} ), which simplifies to ( frac{1}{17} ). So, ( r = frac{1}{17} ). Therefore, ( 1 + r = 1 + frac{1}{17} = frac{18}{17} ). So, ( G_1 = 50 times frac{18}{17} ). Let me compute that. 50 divided by 17 is approximately 2.9412, and then multiplied by 18. Let me compute 2.9412 times 18. 2.9412 times 10 is 29.412, 2.9412 times 8 is 23.5296. Adding those together: 29.412 + 23.5296 = 52.9416. So, that's approximately 52.9416 billion, which is about 52.94 billion when rounded to two decimal places. So, that's consistent with the previous calculation. Therefore, I can be confident that the growth rate is approximately 5.88%, and the GDP after one year is approximately 52.94 billion. Just to recap: 1. Calculated the growth rate using the given formula, substituted the values, did the division, got approximately 5.88%. 2. Then, used that growth rate to compute the new GDP by multiplying the initial GDP by (1 + growth rate), which gave me approximately 52.94 billion. I think that covers both parts of the problem. I don't see any mistakes in my calculations, and both methods (decimal division and fraction multiplication) gave me the same result. So, I think I'm good.Final Answer1. The growth rate ( r(G_0, A) ) is boxed{0.0588}.2. The total GDP after one year ( G_1 ) is boxed{52.94} billion dollars.</think>