🧠:<think>Alright, so I have this problem where I need to optimize workforce management for a company. The goal is to minimize costs while meeting new policy constraints. Let me try to break this down step by step.First, let me understand the problem. There are 10 employees, each with different skill levels across 5 projects. Each employee has an hourly rate as well. The new policy says each employee must work on at least two different projects each week, and no project can have more than 3 employees assigned to it. Additionally, after allocation, each project's average skill level must be at least 7. If not, I need to suggest a reallocation.So, part 1 is about formulating the optimization problem. Part 2 is about checking if the average skill levels meet the requirement and adjusting if necessary.Starting with part 1: Formulating the optimization problem.I need to define variables, objective function, and constraints.Let me denote the variables first. Let’s say x_ij is a binary variable where i is the employee index (from 1 to 10) and j is the project index (from 1 to 5). So, x_ij = 1 if employee i is assigned to project j, else 0.The objective is to minimize the total cost. The total cost would be the sum over all employees of their hourly rate multiplied by the number of hours they work. Wait, but the problem doesn't specify the number of hours each employee works on each project. Hmm, maybe I need to assume that each employee works the same number of hours on each project they are assigned to? Or perhaps, since the rates are given per hour, and the problem is about allocation, maybe the cost is just the sum of the rates multiplied by the number of projects each is assigned to? Wait, but each employee must work on at least two projects, but the exact number of hours isn't specified.Wait, maybe I need to consider that each employee is assigned to a certain number of projects, and each project has a certain number of employees. But the cost is per employee per hour, but without knowing the hours, maybe the cost is proportional to the number of projects they are assigned to? Hmm, that might not make sense because if an employee is assigned to more projects, they might be working more hours, but the rate is per hour. So, perhaps the total cost is the sum over all employees of (number of projects they are assigned to) multiplied by their hourly rate. But that might not be accurate because each project might require a different number of hours.Wait, the problem doesn't specify the hours each project requires, so maybe we can assume that each project requires the same amount of work, and the number of employees assigned to a project affects the total hours each employee works on that project. But without specific hours, it's tricky.Alternatively, maybe the cost is just the sum of the hourly rates for each employee multiplied by the number of projects they are assigned to, assuming each project requires one unit of time. But that might not be precise.Wait, perhaps the problem is intended to have each employee assigned to projects, and the cost is the sum of their rates multiplied by the number of projects they are assigned to. So, if an employee is assigned to two projects, their cost is 2 * R_i. That might make sense.Alternatively, maybe each project requires a certain number of hours, but since it's not given, perhaps we can assume that each project requires the same amount of work, so the number of employees assigned to a project inversely affects the hours each employee works on that project. For example, if a project has 3 employees, each works 1/3 of the total hours required for the project.But without knowing the total hours per project, it's hard to model the exact cost. Hmm, maybe the problem is intended to be a binary assignment problem where the cost is just the sum of the rates for each employee multiplied by the number of projects they are assigned to. That is, each project assignment adds their rate to the total cost.Wait, but that might not capture the fact that if an employee is assigned to more projects, they might be working more hours, thus increasing the cost. So, perhaps the cost is the sum over all employees of R_i multiplied by the number of projects they are assigned to. That is, for each employee, if they are assigned to k projects, their cost is k * R_i.But let me think again. The problem says "minimizing costs" with the given hourly rates. So, if an employee is assigned to multiple projects, the total cost would be their hourly rate multiplied by the total hours they work across all projects. But since the hours per project aren't specified, maybe we can assume that each project requires the same amount of work, say H hours, so each employee assigned to a project works H hours on that project. Therefore, the total cost would be the sum over all employees of R_i multiplied by the number of projects they are assigned to times H. Since H is a constant, it can be factored out, and we can minimize the sum of R_i multiplied by the number of projects each is assigned to.Alternatively, maybe the problem is intended to have each project requiring a certain number of hours, but since it's not given, perhaps we can treat the cost as the sum of R_i multiplied by the number of projects each is assigned to, as a proxy for total hours.Alternatively, maybe the problem is about assigning employees to projects such that the total cost is minimized, considering that each employee can work on multiple projects, but each project can have at most 3 employees. So, the cost is the sum over all projects of the sum of R_i for employees assigned to that project. But that would be the same as summing R_i multiplied by the number of projects they are assigned to.Wait, that makes sense. Because each employee's rate is added to each project they are assigned to. So, the total cost is the sum over all projects of the sum of R_i for employees assigned to project j. Which is equivalent to summing over all employees, R_i multiplied by the number of projects they are assigned to.Therefore, the objective function is to minimize the sum over i=1 to 10 of R_i * (sum over j=1 to 5 of x_ij). So, that's equivalent to minimizing the total cost.Now, the constraints:1. Each employee must be assigned to at least two projects: For each i, sum over j=1 to 5 of x_ij >= 2.2. Each project can have at most 3 employees: For each j, sum over i=1 to 10 of x_ij <= 3.Additionally, we have the binary constraints: x_ij is either 0 or 1.But wait, in part 2, we also have the average skill level per project must be at least 7. So, that would be another constraint: For each project j, (sum over i=1 to 10 of A_ij * x_ij) / (sum over i=1 to 10 of x_ij) >= 7. But since we are formulating the problem for part 1, maybe the average skill level constraint isn't part of the initial optimization, but only checked in part 2. So, perhaps in part 1, we only have the constraints about the number of projects per employee and the number of employees per project.Wait, the problem says in part 1: "formulate an optimization problem to minimize the total cost while satisfying the new policy constraints." The new policy constraints are: each employee must work on at least two projects, and no project can have more than 3 employees. So, the average skill level is a requirement that we check after solving part 1, and if it's not met, we adjust in part 2.Therefore, in part 1, the constraints are:- For each employee i: sum_j x_ij >= 2- For each project j: sum_i x_ij <= 3- x_ij is binary.So, the optimization problem is:Minimize sum_{i=1 to 10} R_i * sum_{j=1 to 5} x_ijSubject to:sum_{j=1 to 5} x_ij >= 2, for all i=1 to 10sum_{i=1 to 10} x_ij <= 3, for all j=1 to 5x_ij ∈ {0,1}That seems correct.Now, moving to part 2: After finding the optimal allocation, calculate the average skill level for each project and check if it's at least 7. If not, suggest a reallocation.But since I'm just formulating the problem in part 1, I don't need to solve it here. However, if I were to solve it, I would need to use integer programming, perhaps with a solver.But since I'm just formulating, I think I've covered part 1.Wait, but the problem also mentions the skill levels in matrix A. So, perhaps the objective function should consider the skill levels as well? But the problem says to minimize the total cost, which is based on the hourly rates. So, the skill levels might be a constraint or a factor in the objective, but the problem doesn't specify that. It only mentions that the average skill level must be at least 7 in part 2.So, in part 1, the optimization is purely about minimizing cost with the given constraints on the number of projects per employee and the number of employees per project.Therefore, I think the formulation is correct.Now, for part 2, assuming we have the optimal allocation, we need to compute the average skill level for each project. If any project has an average below 7, we need to reallocate employees to increase the average while still minimizing cost.But since I don't have the optimal solution, I can't compute the averages yet. However, I can think about how to approach it.If a project has an average skill level below 7, we need to replace some employees with higher-skilled ones. But we have to ensure that the new allocation still satisfies the constraints: each employee is on at least two projects, and no project has more than three employees.So, perhaps we can swap an employee with a lower skill in that project with an employee who has a higher skill and is currently on only two projects, freeing up their spot elsewhere.Alternatively, if an employee is on three projects, we might need to adjust their assignments to free up capacity.But this would require solving another optimization problem with the added constraints on the average skill levels.Alternatively, since the average skill is a linear constraint, we could include it in the optimization problem from the start, but the problem separates it into part 2.So, in summary, for part 1, the optimization problem is as formulated above, and for part 2, after solving, if any project's average skill is below 7, we need to adjust the assignments by replacing lower-skilled employees with higher-skilled ones, ensuring the constraints are still met.I think that's the approach.</think>

🧠:<think>Okay, so I'm trying to tackle this problem about optimizing a machine learning algorithm using linear algebra and differential geometry. It's a bit intimidating, but I'll take it step by step.First, part 1 is about PCA. I remember PCA is a technique used for dimensionality reduction. The idea is to find the directions (principal components) that explain the most variance in the data. So, given a matrix X with n data points in d dimensions, I need to perform PCA and find the first principal component.I think the steps for PCA involve centering the data, computing the covariance matrix, and then finding the eigenvectors of that covariance matrix. The first principal component would be the eigenvector corresponding to the largest eigenvalue. That makes sense because the largest eigenvalue represents the direction of maximum variance.Let me write down the steps:1. Center the data: subtract the mean of each feature from all data points. So, compute the mean vector μ = (1/n) * sum of each column of X, then X_centered = X - μ.2. Compute the covariance matrix: Cov = (1/(n-1)) * X_centered^T * X_centered. Alternatively, sometimes it's scaled by 1/n, depending on the convention.3. Find the eigenvalues and eigenvectors of Cov. The eigenvector with the largest eigenvalue is the first principal component.So, the first principal component is the eigenvector v corresponding to the maximum eigenvalue λ of the covariance matrix Cov.Now, the condition under which the first principal component maximizes the variance. I think this is inherent in PCA. The first principal component is the direction in which the data varies the most. So, the variance is maximized along this direction. Mathematically, it's the direction that maximizes the variance, which is equivalent to maximizing the quadratic form v^T Cov v, subject to the constraint that v is a unit vector.So, using Lagrange multipliers, we set up the optimization problem:Maximize v^T Cov vSubject to v^T v = 1The solution is given by the eigenvectors of Cov, with the maximum eigenvalue corresponding to the maximum variance. Therefore, the first principal component is the eigenvector that satisfies this condition.Moving on to part 2, it's about differential geometry and curvature. The dataset is embedded on a smooth manifold M with intrinsic dimensionality k < d. I need to derive the curvature κ at a point p ∈ M and relate it to the efficiency of the machine learning algorithm.Hmm, curvature in differential geometry can be a bit tricky. There are different types of curvature, like scalar curvature, Ricci curvature, and sectional curvature. I think for a point on a manifold, the curvature tensor encapsulates all the information about the curvature at that point.But maybe they're referring to the Gaussian curvature, which is a specific type of curvature for 2-dimensional surfaces. However, since the manifold has intrinsic dimensionality k, which could be higher than 2, perhaps we need a more general approach.Alternatively, in the context of machine learning, especially manifold learning, curvature might relate to how "twisted" or "bent" the manifold is, which could affect how well the algorithm can learn the structure.I recall that in differential geometry, the curvature tensor R is defined in terms of the Riemann curvature tensor. For a point p on the manifold, the curvature tensor can be expressed using the Christoffel symbols, which are derived from the metric tensor g.The Riemann curvature tensor R is given by:R^l_{ijk} = ∂_k Γ^l_{ij} - ∂_j Γ^l_{ik} + Γ^l_{mk} Γ^m_{ij} - Γ^l_{mj} Γ^m_{ik}Where Γ are the Christoffel symbols of the second kind, which depend on the metric tensor g.But maybe the question is asking for the scalar curvature, which is a contraction of the Ricci tensor. The Ricci tensor is a contraction of the Riemann tensor, and scalar curvature is a contraction of the Ricci tensor.Alternatively, if we're considering a 2-dimensional manifold, the Gaussian curvature K can be expressed in terms of the metric tensor components and their derivatives.But since the manifold has dimension k, perhaps the curvature is more complex. Maybe the question is referring to the sectional curvature, which generalizes Gaussian curvature to higher dimensions.In any case, curvature measures how much the manifold deviates from being flat. High curvature regions are more "curved," which might mean that the data is more concentrated or has more complex structure in those areas.Regarding the efficiency of the machine learning algorithm, I think that if the manifold has high curvature, it might be harder for the algorithm to learn the underlying structure because the data is more "twisted." Conversely, if the curvature is low, the manifold is closer to being flat, and the algorithm might perform better because the structure is simpler.So, theoretically, the efficiency of the algorithm could be inversely related to the curvature of the manifold. High curvature regions might require more complex models or more data to capture the structure accurately, leading to lower efficiency. On the other hand, low curvature might allow for simpler models and more efficient learning.But I'm not entirely sure about the exact relationship. Maybe in regions of high curvature, the local geometry is more variable, making it harder for the algorithm to generalize or find the optimal parameters. Alternatively, high curvature could indicate areas of high information content, which might require more careful handling.I think the key idea is that curvature affects the geometry of the data manifold, which in turn influences how machine learning algorithms, especially those that rely on the geometric properties of the data, perform. Algorithms that can adapt to the curvature might be more efficient, while those that assume flatness might struggle in highly curved regions.So, to summarize, the curvature κ at a point p on the manifold M can be derived using the Riemann curvature tensor, which involves the Christoffel symbols and their derivatives. The relationship between curvature and algorithm efficiency is that higher curvature regions may lead to lower efficiency due to the increased complexity of the manifold's structure, requiring more computational resources or more data to achieve the same performance as in regions of lower curvature.I think that's the gist of it. I might have missed some details, especially regarding the exact expression for curvature, but I believe this captures the main points.</think>

🧠:<think>First, I need to determine the maximum number of children the parent can invite without exceeding a budget of 750 for the clown. The clown charges a base fee of 150 plus 25 per child.I'll start by setting up the cost equation:Total Cost = Base Fee + (Cost per Child × Number of Children)750 = 150 + (25 × N)Next, I'll solve for N:750 - 150 = 25 × N600 = 25 × NN = 600 / 25N = 24So, the parent can invite up to 24 children within the budget.Now, I need to determine the maximum number of additional children (x) the parent can invite without exceeding a total party duration of 3 hours. The initial duration for 20 children is 2 hours, and the clown extends the duration by 10 minutes for every 5 additional children.First, I'll convert the total allowed duration into minutes:3 hours = 180 minutesThe initial duration is 2 hours, which is 120 minutes. The additional duration added by x children is (10 minutes / 5 children) × x = 2x minutes.Setting up the equation for total duration:120 + 2x ≤ 180Solving for x:2x ≤ 60x ≤ 30Therefore, the parent can invite up to 30 additional children without exceeding the time limit.</think>

🧠:<think>Alright, so I've got these two sub-problems to solve about the growth of two Japanese broadcasting networks. Let me start with Sub-problem 1.Sub-problem 1:The differential equation given is:[ frac{dV_A}{dt} = k_1 V_A (M - V_A) ]This looks like a logistic growth model. I remember that the general solution to the logistic equation is:[ V(t) = frac{M}{1 + left( frac{M - V_0}{V_0} right) e^{-k_1 M t}} ]But let me verify that. The logistic equation is:[ frac{dV}{dt} = r V left(1 - frac{V}{K}right) ]Comparing this to our equation, ( r = k_1 M ) and ( K = M ). So, the solution should be similar.Let me solve the differential equation step by step.First, rewrite the equation:[ frac{dV_A}{dt} = k_1 V_A (M - V_A) ]This is a separable equation. Let's separate variables:[ frac{dV_A}{V_A (M - V_A)} = k_1 dt ]I can use partial fractions to integrate the left side. Let me set:[ frac{1}{V_A (M - V_A)} = frac{A}{V_A} + frac{B}{M - V_A} ]Multiplying both sides by ( V_A (M - V_A) ):[ 1 = A (M - V_A) + B V_A ]To find A and B, let's plug in suitable values for ( V_A ).Let ( V_A = 0 ):[ 1 = A M Rightarrow A = frac{1}{M} ]Let ( V_A = M ):[ 1 = B M Rightarrow B = frac{1}{M} ]So, the partial fractions are:[ frac{1}{M V_A} + frac{1}{M (M - V_A)} ]Therefore, the integral becomes:[ int left( frac{1}{M V_A} + frac{1}{M (M - V_A)} right) dV_A = int k_1 dt ]Integrating both sides:Left side:[ frac{1}{M} ln |V_A| - frac{1}{M} ln |M - V_A| + C_1 ]Right side:[ k_1 t + C_2 ]Combine constants:[ frac{1}{M} ln left( frac{V_A}{M - V_A} right) = k_1 t + C ]Exponentiate both sides:[ frac{V_A}{M - V_A} = e^{M k_1 t + C} = e^C e^{M k_1 t} ]Let ( e^C = C' ), a constant:[ frac{V_A}{M - V_A} = C' e^{M k_1 t} ]Solve for ( V_A ):Multiply both sides by ( M - V_A ):[ V_A = C' e^{M k_1 t} (M - V_A) ]Expand:[ V_A = C' M e^{M k_1 t} - C' V_A e^{M k_1 t} ]Bring terms with ( V_A ) to the left:[ V_A + C' V_A e^{M k_1 t} = C' M e^{M k_1 t} ]Factor ( V_A ):[ V_A (1 + C' e^{M k_1 t}) = C' M e^{M k_1 t} ]Solve for ( V_A ):[ V_A = frac{C' M e^{M k_1 t}}{1 + C' e^{M k_1 t}} ]Let me rewrite this as:[ V_A(t) = frac{M}{frac{1}{C'} e^{-M k_1 t} + 1} ]Let ( frac{1}{C'} = C'' ), another constant:[ V_A(t) = frac{M}{C'' e^{-M k_1 t} + 1} ]Now, apply the initial condition ( V_A(0) = V_0 ):At ( t = 0 ):[ V_0 = frac{M}{C'' + 1} ]Solve for ( C'' ):[ C'' + 1 = frac{M}{V_0} Rightarrow C'' = frac{M}{V_0} - 1 ]So, the solution becomes:[ V_A(t) = frac{M}{left( frac{M}{V_0} - 1 right) e^{-M k_1 t} + 1} ]Alternatively, we can write it as:[ V_A(t) = frac{M}{1 + left( frac{M - V_0}{V_0} right) e^{-M k_1 t}} ]Which is the standard logistic growth solution.Now, we are told that Network A reached half the market capacity ( M/2 ) in ( T ) years. So, ( V_A(T) = M/2 ).Plugging into the equation:[ frac{M}{2} = frac{M}{1 + left( frac{M - V_0}{V_0} right) e^{-M k_1 T}} ]Divide both sides by M:[ frac{1}{2} = frac{1}{1 + left( frac{M - V_0}{V_0} right) e^{-M k_1 T}} ]Take reciprocals:[ 2 = 1 + left( frac{M - V_0}{V_0} right) e^{-M k_1 T} ]Subtract 1:[ 1 = left( frac{M - V_0}{V_0} right) e^{-M k_1 T} ]Solve for ( e^{-M k_1 T} ):[ e^{-M k_1 T} = frac{V_0}{M - V_0} ]Take natural logarithm of both sides:[ -M k_1 T = ln left( frac{V_0}{M - V_0} right) ]Multiply both sides by -1:[ M k_1 T = - ln left( frac{V_0}{M - V_0} right) ]Simplify the right side:[ M k_1 T = ln left( frac{M - V_0}{V_0} right) ]Solve for ( k_1 ):[ k_1 = frac{1}{M T} ln left( frac{M - V_0}{V_0} right) ]Alternatively, since ( ln(a/b) = -ln(b/a) ), we can write:[ k_1 = - frac{1}{M T} ln left( frac{V_0}{M - V_0} right) ]Either form is acceptable, but perhaps the first is more straightforward.So, summarizing:The expression for ( V_A(t) ) is:[ V_A(t) = frac{M}{1 + left( frac{M - V_0}{V_0} right) e^{-M k_1 t}} ]And ( k_1 ) is:[ k_1 = frac{1}{M T} ln left( frac{M - V_0}{V_0} right) ]Sub-problem 2:The viewership for Network B is given by:[ V_B(t) = frac{L}{1 + e^{-k_2 (t - t_0)}} ]This is a logistic growth function where ( t_0 ) is the midpoint, meaning that at ( t = t_0 ), ( V_B(t_0) = L/2 ).Given:- At ( t = 0 ), ( V_B(0) = V_1 )- At ( t = T_1 ), ( V_B(T_1) = 2 V_1 )We need to find ( k_2 ) and ( t_0 ) in terms of ( V_1 ), ( L ), and ( T_1 ).First, let's write the equations based on the given conditions.1. At ( t = 0 ):[ V_1 = frac{L}{1 + e^{-k_2 (-t_0)}} = frac{L}{1 + e^{k_2 t_0}} ]2. At ( t = T_1 ):[ 2 V_1 = frac{L}{1 + e^{-k_2 (T_1 - t_0)}} ]So, we have two equations:1. ( V_1 = frac{L}{1 + e^{k_2 t_0}} ) -- Equation (1)2. ( 2 V_1 = frac{L}{1 + e^{-k_2 (T_1 - t_0)}} ) -- Equation (2)Let me solve Equation (1) for ( e^{k_2 t_0} ):From Equation (1):[ V_1 (1 + e^{k_2 t_0}) = L ][ V_1 + V_1 e^{k_2 t_0} = L ][ V_1 e^{k_2 t_0} = L - V_1 ][ e^{k_2 t_0} = frac{L - V_1}{V_1} ]Take natural logarithm:[ k_2 t_0 = ln left( frac{L - V_1}{V_1} right) ]So,[ t_0 = frac{1}{k_2} ln left( frac{L - V_1}{V_1} right) ] -- Equation (3)Now, let's work on Equation (2):[ 2 V_1 = frac{L}{1 + e^{-k_2 (T_1 - t_0)}} ]Multiply both sides by denominator:[ 2 V_1 (1 + e^{-k_2 (T_1 - t_0)}) = L ]Expand:[ 2 V_1 + 2 V_1 e^{-k_2 (T_1 - t_0)} = L ]Subtract ( 2 V_1 ):[ 2 V_1 e^{-k_2 (T_1 - t_0)} = L - 2 V_1 ]Divide both sides by ( 2 V_1 ):[ e^{-k_2 (T_1 - t_0)} = frac{L - 2 V_1}{2 V_1} ]Take natural logarithm:[ -k_2 (T_1 - t_0) = ln left( frac{L - 2 V_1}{2 V_1} right) ]Multiply both sides by -1:[ k_2 (T_1 - t_0) = - ln left( frac{L - 2 V_1}{2 V_1} right) ]Simplify the right side:Note that ( ln(a/b) = - ln(b/a) ), so:[ k_2 (T_1 - t_0) = ln left( frac{2 V_1}{L - 2 V_1} right) ]From Equation (3), we have ( t_0 ) in terms of ( k_2 ). Let's substitute that into this equation.So,[ k_2 left( T_1 - frac{1}{k_2} ln left( frac{L - V_1}{V_1} right) right) = ln left( frac{2 V_1}{L - 2 V_1} right) ]Multiply through:[ k_2 T_1 - ln left( frac{L - V_1}{V_1} right) = ln left( frac{2 V_1}{L - 2 V_1} right) ]Bring the logarithm term to the right:[ k_2 T_1 = ln left( frac{2 V_1}{L - 2 V_1} right) + ln left( frac{L - V_1}{V_1} right) ]Combine the logarithms:[ k_2 T_1 = ln left( frac{2 V_1}{L - 2 V_1} cdot frac{L - V_1}{V_1} right) ]Simplify the argument:[ frac{2 V_1}{L - 2 V_1} cdot frac{L - V_1}{V_1} = frac{2 (L - V_1)}{L - 2 V_1} ]So,[ k_2 T_1 = ln left( frac{2 (L - V_1)}{L - 2 V_1} right) ]Therefore,[ k_2 = frac{1}{T_1} ln left( frac{2 (L - V_1)}{L - 2 V_1} right) ]Now, let's find ( t_0 ) using Equation (3):[ t_0 = frac{1}{k_2} ln left( frac{L - V_1}{V_1} right) ]Substitute ( k_2 ):[ t_0 = frac{T_1}{ln left( frac{2 (L - V_1)}{L - 2 V_1} right)} ln left( frac{L - V_1}{V_1} right) ]So, that's ( t_0 ) expressed in terms of ( V_1 ), ( L ), and ( T_1 ).Let me just double-check the algebra to make sure I didn't make any mistakes.Starting from Equation (2):We had:[ 2 V_1 = frac{L}{1 + e^{-k_2 (T_1 - t_0)}} ]Which led to:[ e^{-k_2 (T_1 - t_0)} = frac{L - 2 V_1}{2 V_1} ]Taking ln:[ -k_2 (T_1 - t_0) = ln left( frac{L - 2 V_1}{2 V_1} right) ]Which is correct.Then, substituting ( t_0 ) from Equation (3):Yes, that seems right.So, the final expressions are:For ( k_2 ):[ k_2 = frac{1}{T_1} ln left( frac{2 (L - V_1)}{L - 2 V_1} right) ]And for ( t_0 ):[ t_0 = frac{T_1 ln left( frac{L - V_1}{V_1} right)}{ln left( frac{2 (L - V_1)}{L - 2 V_1} right)} ]I think that's correct.Final AnswerSub-problem 1: The expression for ( V_A(t) ) is ( boxed{V_A(t) = dfrac{M}{1 + left( dfrac{M - V_0}{V_0} right) e^{-M k_1 t}}} ) and ( k_1 = boxed{dfrac{1}{M T} ln left( dfrac{M - V_0}{V_0} right)} ).Sub-problem 2: The parameters are ( k_2 = boxed{dfrac{1}{T_1} ln left( dfrac{2(L - V_1)}{L - 2V_1} right)} ) and ( t_0 = boxed{dfrac{T_1 ln left( dfrac{L - V_1}{V_1} right)}{ln left( dfrac{2(L - V_1)}{L - 2V_1} right)}} ).</think>