🧠:<think>Okay, so I'm trying to figure out these two questions about Voronoi diagrams and coffee shops. Let me start with the first one.1. Voronoi Cell Condition: I remember that a Voronoi diagram divides the plane into regions based on proximity to each point. Each region, or Voronoi cell, consists of all points closer to its associated point than to any other. So, for a point P to be in the Voronoi cell of coffee shop A, it must be closer to A than to any other coffee shop. Let me denote the location of coffee shop A as point A, and the other coffee shops as points B, C, D, etc. The condition for P being in A's Voronoi cell is that the distance from P to A is less than the distance from P to any other coffee shop. Mathematically, for all other points B_i, the distance PA must be less than PB_i. So, in terms of Euclidean distance, this would be: For all i, ||P - A|| < ||P - B_i|| That seems straightforward. So, the condition is that P is closer to A than to any other coffee shop.2. Expected Area of Voronoi Cell: Now, this seems trickier. The city is a square of side length L, centered at the origin. So, the square goes from (-L/2, -L/2) to (L/2, L/2). There's a coffee shop at (x0, y0), and m other coffee shops randomly distributed within the square. I need to find the expected area of the Voronoi cell for the coffee shop at (x0, y0). Since the distribution is uniform, I can assume that the other points are randomly placed, so the Voronoi cell's area depends on the average distance to the nearest neighbors. I recall that in a Poisson point process, the expected area of a Voronoi cell is inversely proportional to the density of points. But here, it's a finite square with m points, so maybe it's similar. Wait, actually, in a uniform distribution over a square, the expected area of a Voronoi cell can be approximated by the area of the square divided by the number of points. But that might be too simplistic because edge effects come into play, especially if the coffee shop is near the boundary. Hmm, so maybe the expected area isn't just L²/(m+1). Because if the coffee shop is near the edge, its Voronoi cell might be smaller on average. Alternatively, maybe we can model this using the concept of the average nearest neighbor distance. But I'm not sure. Let me think about it differently. The Voronoi cell around a point is determined by the perpendicular bisectors between that point and all others. The area depends on how close the nearest neighbors are. If the points are uniformly distributed, the expected number of points in a region around (x0, y0) can be modeled using probability. The probability that another point is within a distance r from (x0, y0) is proportional to the area of the circle of radius r, but adjusted for the square boundaries. Wait, maybe I can use the concept of the expected value of the area. The area of the Voronoi cell is the integral over all points P where P is closer to (x0, y0) than to any other point. So, the expected area would involve integrating over the square, the probability that a random point P is closer to (x0, y0) than to any of the m other points. But that seems complicated. Maybe there's a known formula for the expected area of a Voronoi cell in a uniform distribution. I think in a Poisson process, the expected area is 1/λ, where λ is the intensity. But here, it's a finite square with m points, so the intensity λ would be (m+1)/L². Wait, so the expected area would be L²/(m+1). But does that hold when considering edge effects? Actually, in the case of a finite square, especially when the point is near the edge, the Voronoi cell might extend beyond the square, but since we're only considering the city as the square, the Voronoi cell is clipped at the boundaries. So, perhaps the expected area is approximately L²/(m+1), but adjusted for the position of (x0, y0). If (x0, y0) is near the center, the area is roughly L²/(m+1), but if it's near the edge, it's smaller. However, the question says to assume a uniform distribution of coffee shops. So maybe we can average over all possible positions of (x0, y0). But the problem specifies that the coffee shop is at (x0, y0), so perhaps we need to consider its position. Alternatively, maybe the expected area is the same regardless of (x0, y0) due to uniformity. But that doesn't seem right because edge effects would cause variance. Wait, maybe the question is assuming that the coffee shop is randomly placed as well, so the expectation would be over all possible positions of (x0, y0) and all other points. If that's the case, then the expected area would be L²/(m+1). Because each point contributes equally on average. But I'm not entirely sure. Let me check my reasoning. If you have n points uniformly distributed in a square, the expected area of each Voronoi cell is roughly the total area divided by n. So here, n = m+1, so the expected area is L²/(m+1). But wait, in reality, the Voronoi cells near the edges have smaller areas on average because they are cut off by the square boundaries. However, if we average over all possible positions, including the edges, the expected area might still be L²/(m+1) because the distribution is uniform. Alternatively, maybe it's more complex. I think in the case of a toroidal space (where edges wrap around), the expected area is exactly L²/(m+1). But in a square with edges, it's slightly different. However, for large L and m, the edge effects become negligible, so the expected area approaches L²/(m+1). Since the problem says the city is a perfect square, but doesn't specify anything about the size relative to m, maybe we can assume that the edge effects are negligible, or that the coffee shop is not near the edge. Alternatively, if we consider the entire square and average over all possible positions, including edges, the expected area might still be L²/(m+1) because of symmetry. Hmm, I'm a bit torn here. On one hand, edge effects do reduce the expected area for points near the boundary, but when averaging over all positions, including the center, it might balance out. Wait, actually, no. The Voronoi cells near the edges have smaller areas, but the cells near the center have larger areas. So, the average might still be L²/(m+1) because the total area is fixed. Let me think about it this way: the sum of all Voronoi cell areas is equal to the area of the square, L². So, if there are m+1 points, the average area per cell is L²/(m+1). Therefore, the expected area of a Voronoi cell is L²/(m+1). That makes sense because expectation is linear, and the sum of the areas is fixed. So, regardless of the distribution, the average must be L²/(m+1). Therefore, the expected area is L²/(m+1). Wait, but does this hold when considering that the coffee shop is at a specific point (x0, y0)? If (x0, y0) is fixed, say at the center, then its Voronoi cell might be larger on average, but if it's fixed near the edge, it's smaller. But the question says to assume the distribution of coffee shops is uniform, so maybe (x0, y0) is also uniformly distributed. If (x0, y0) is uniformly distributed over the square, then the expected area would still be L²/(m+1) because of the linearity of expectation. So, I think the expected area is L² divided by (m+1), so the formula is E[Area] = L² / (m+1). Let me double-check. Suppose m=0, then there's only one coffee shop, so the Voronoi cell is the entire square, which has area L². Plugging m=0 into the formula gives L²/(0+1) = L², which is correct. If m=1, then there are two points. The expected area for each Voronoi cell would be L²/2, which also makes sense because each cell would be roughly half the square on average. So, yes, that seems to hold. Therefore, the expected area is L²/(m+1).Final Answer1. A point ( P ) belongs to the Voronoi cell of coffee shop ( A ) if and only if the Euclidean distance from ( P ) to ( A ) is less than the distance from ( P ) to any other coffee shop. This can be expressed as: [ boxed{|P - A| < |P - B_i| text{ for all other coffee shops } B_i} ]2. The expected area of the Voronoi cell corresponding to the coffee shop at ( (x_0, y_0) ) is: [ boxed{dfrac{L^2}{m + 1}} ]</think>

🧠:<think>Alright, so I have this problem about life expectancy and healthcare costs. It's divided into three parts: finding the growth rate ( k ) for life expectancy, finding the annual increase ( m ) for healthcare costs, and then using those to calculate the total healthcare cost over the next 10 years for someone currently 65. Let me take it step by step.Starting with part 1: The life expectancy model is given by ( L(t) = L_0 e^{kt} ). They told me ( L_0 ) is 75 years, and after 20 years, it's 85 years. I need to find ( k ). Hmm, okay, so I can plug in the known values into the equation.So, ( L(20) = 85 = 75 e^{20k} ). To solve for ( k ), I can divide both sides by 75 first. That gives me ( 85 / 75 = e^{20k} ). Simplifying ( 85/75 ) is the same as ( 17/15 ), which is approximately 1.1333. So, ( 1.1333 = e^{20k} ).To solve for ( k ), I need to take the natural logarithm of both sides. Remember, ( ln(e^{20k}) = 20k ). So, ( ln(1.1333) = 20k ). Calculating ( ln(1.1333) ), I can use a calculator for that. Let me see, ( ln(1.1333) ) is approximately 0.124. So, 0.124 = 20k. Then, dividing both sides by 20, ( k = 0.124 / 20 ), which is 0.0062. So, ( k ) is approximately 0.0062 per year.Wait, let me double-check that. If I plug ( k = 0.0062 ) back into the equation, does it give me 85 after 20 years? Let's see: ( 75 e^{0.0062 * 20} ). 0.0062 * 20 is 0.124. ( e^{0.124} ) is approximately 1.132, and 75 * 1.132 is about 84.9, which is roughly 85. Okay, that seems correct. So, ( k ) is approximately 0.0062.Moving on to part 2: The healthcare cost model is linear, ( C(t) = C_0 + mt ). They gave me ( C_0 = 10,000 ) dollars, and after 20 years, it's 20,000. So, I need to find ( m ).So, plugging in the values: ( C(20) = 20,000 = 10,000 + m * 20 ). Subtracting 10,000 from both sides gives ( 10,000 = 20m ). So, ( m = 10,000 / 20 = 500 ). So, the annual increase ( m ) is 500 dollars per year.That seems straightforward. Let me verify: Starting at 10,000, adding 500 each year for 20 years would be 10,000 + 500*20 = 10,000 + 10,000 = 20,000. Perfect, that checks out.Now, the third part is to calculate the total healthcare cost for an older adult who is currently 65 years old over the next 10 years. Hmm, okay. So, currently, the person is 65. The models are based on time since the start of the study, which I assume is when the person was, say, 65? Or is it a general model for all older adults? Wait, the problem says "for a sample of older adults over the past 20 years." So, maybe the models are general, not specific to a person's age.Wait, but the person is currently 65. So, if we're predicting the next 10 years, we need to model their healthcare costs from now until they are 75. But how does their age relate to the models? The models are in terms of ( t ), which is the time since the start of the study. Hmm, I need to clarify.Wait, the models are for older adults in general, so ( t ) is the time since the study started, not since the person was born or something. So, if the study has been going on for 20 years, and the person is currently 65, how does that tie in?Wait, maybe I need to think differently. The models are functions of ( t ), which is time since the start of the study. So, if the study started 20 years ago, and now we're looking at the next 10 years, ( t ) would go from 20 to 30.But the person is currently 65. So, their age is 65, and we need to model their healthcare costs over the next 10 years. But how does their age relate to the models? The models are about older adults in general, so maybe the healthcare cost is per year, regardless of age? Or is the healthcare cost dependent on age?Wait, the problem says the healthcare cost is modeled by ( C(t) = C_0 + mt ), which is a linear function of time since the study started. So, it's not directly dependent on the person's age, but on the time since the study began. So, if the person is currently 65, and the study started 20 years ago, then ( t ) is 20 now, and in 10 years, it will be 30.So, to calculate the total healthcare cost over the next 10 years, we need to compute the integral of ( C(t) ) from ( t = 20 ) to ( t = 30 ). Because ( C(t) ) is the annual cost, integrating over the next 10 years will give the total cost.Alternatively, since it's a linear function, the total cost can be calculated as the average cost over the 10 years multiplied by 10. The average cost would be the average of ( C(20) ) and ( C(30) ).Wait, let's think about it. The healthcare cost each year is ( C(t) = 10,000 + 500t ). So, in year 20, it's 20,000, as given. In year 30, it would be ( 10,000 + 500*30 = 10,000 + 15,000 = 25,000 ).So, the cost increases from 20,000 to 25,000 over the next 10 years. The average cost per year would be ( (20,000 + 25,000)/2 = 22,500 ). Therefore, the total cost over 10 years would be 22,500 * 10 = 225,000 dollars.Alternatively, integrating ( C(t) ) from 20 to 30: ( int_{20}^{30} (10,000 + 500t) dt ). The integral of 10,000 is 10,000t, and the integral of 500t is 250t². Evaluating from 20 to 30:At 30: 10,000*30 + 250*(30)^2 = 300,000 + 250*900 = 300,000 + 225,000 = 525,000.At 20: 10,000*20 + 250*(20)^2 = 200,000 + 250*400 = 200,000 + 100,000 = 300,000.Subtracting, 525,000 - 300,000 = 225,000. So, same result.Therefore, the total healthcare cost over the next 10 years is 225,000 dollars.Wait, but hold on. The person is currently 65, and we're predicting their healthcare costs over the next 10 years. But the models are based on time since the study started, not on the person's age. So, if the study started 20 years ago, and the person is currently 65, that would mean they were 45 when the study started? Or is the study about older adults, so maybe the participants were already older when the study began?Wait, the problem says it's a sample of older adults over the past 20 years. So, perhaps the study has been tracking older adults for 20 years, so the participants were older adults at the start, say, 65 or older. So, if the person is currently 65, and the study has been going on for 20 years, then ( t = 20 ) corresponds to their current age? Hmm, maybe not necessarily, because the study could have started with people of various ages.Wait, this is getting a bit confusing. Let me reread the problem."The advocate has collected data on the life expectancy (in years) and annual healthcare costs (in thousands of dollars) for a sample of older adults over the past 20 years. The data fits an exponential growth model for life expectancy and a linear growth model for healthcare costs."So, the data is over the past 20 years, so ( t = 0 ) is 20 years ago, and ( t = 20 ) is now. So, the person is currently 65, so 20 years ago, they were 45. But the study is on older adults, so maybe the participants were older than 65 when the study started. Hmm, this is unclear.Wait, perhaps the models are independent of the individual's age, just based on the passage of time. So, regardless of when the person was born, the healthcare cost and life expectancy are modeled based on the time since the study started.So, if the study started 20 years ago, and now we're at ( t = 20 ), and we want to predict the next 10 years, ( t = 20 ) to ( t = 30 ). So, the person's age is 65 now, but the models don't directly depend on their age, just on ( t ).Therefore, the healthcare cost each year is given by ( C(t) = 10,000 + 500t ). So, in the next 10 years, from ( t = 20 ) to ( t = 30 ), the cost per year increases from 20,000 to 25,000, as I calculated before. So, the total cost is 225,000.Alternatively, if the person is 65 now, and we need to model their healthcare costs until they reach a certain age, but since the models are based on time since the study started, not on the person's age, it's still about the next 10 years in the study's timeline.Wait, maybe another approach: The person is 65 now, and we need to model their healthcare costs for the next 10 years, so until they are 75. But the models are in terms of ( t ), which is the time since the study started. So, if the study started 20 years ago, and the person is 65 now, that means 20 years ago, they were 45. But the study is on older adults, so maybe they weren't part of the study back then. Hmm, this is getting complicated.Wait, perhaps the models are general, so regardless of when the person was born, the healthcare cost and life expectancy are functions of the time since the study started. So, if the person is currently 65, and the study started 20 years ago, then ( t = 20 ) corresponds to their current age. But that would mean when ( t = 0 ), they were 45, which is not an older adult. So, maybe the models are applicable to older adults, so the participants were older than a certain age when the study started.Wait, this is confusing. Maybe I need to make an assumption here. Let's assume that the person is part of the study, so when the study started 20 years ago, they were already an older adult. So, if they're 65 now, 20 years ago they were 45, which is not older. Hmm, conflicting.Alternatively, maybe the models are for the entire population of older adults, so regardless of when the person was born, their healthcare cost is modeled by ( C(t) ), where ( t ) is the time since they became an older adult? But the problem says "time in years since the start of the study," so ( t ) is fixed for everyone.Wait, maybe the key is that the person is currently 65, so their remaining life expectancy is modeled by ( L(t) ), but ( t ) is the time since the study started. So, if the study started 20 years ago, and they are 65 now, their remaining life expectancy is ( L(20) = 85 ). Wait, no, ( L(t) ) is the life expectancy at time ( t ), not the remaining life.Wait, the problem says "life expectancy (in years) and annual healthcare costs (in thousands of dollars) for a sample of older adults over the past 20 years." So, perhaps the life expectancy is the expected remaining life at each time point. So, at ( t = 0 ), life expectancy was 75, meaning that older adults were expected to live 75 more years? That doesn't make sense because 75 is already a long lifespan.Wait, maybe life expectancy is the average age at death. So, if life expectancy increased from 75 to 85 over 20 years, that's an increase in lifespan. So, the models are about the general population's life expectancy and healthcare costs over time.So, if someone is currently 65, their life expectancy now is ( L(20) = 85 ). So, their remaining life expectancy is 85 - 65 = 20 years? Wait, no, because ( L(t) ) is the life expectancy at time ( t ). So, if ( t = 20 ), life expectancy is 85. So, if someone is 65 now, their life expectancy is 85, meaning they are expected to live 20 more years? That seems plausible.But then, the healthcare cost is ( C(t) = 10,000 + 500t ). So, at ( t = 20 ), it's 20,000 per year. So, if the person is expected to live 20 more years, their total healthcare cost would be the integral of ( C(t) ) from ( t = 20 ) to ( t = 40 ). Wait, but that would be 20 years into the future, but the problem says to predict over the next 10 years.Wait, the problem says: "predict the total healthcare cost for an older adult who is currently 65 years old over the next 10 years." So, regardless of their life expectancy, just calculate the cost for the next 10 years.So, if the person is 65 now, and we need to calculate the healthcare cost for the next 10 years, regardless of whether they might die before that. So, we just need to compute the cost from now until 10 years later.But since the models are based on ( t ), which is the time since the study started, and the study has been going on for 20 years, now ( t = 20 ). So, the next 10 years would be ( t = 20 ) to ( t = 30 ). Therefore, the healthcare cost each year is ( C(t) = 10,000 + 500t ). So, from ( t = 20 ) to ( t = 30 ), the cost per year goes from 20,000 to 25,000.Therefore, the total cost is the integral of ( C(t) ) from 20 to 30, which we calculated as 225,000. So, 225,000 dollars over the next 10 years.Alternatively, if we model it as the person's age, but since the models are based on ( t ), which is study time, not age, I think the correct approach is to use ( t ) from 20 to 30.Wait, but let me think again. If the person is 65 now, and the study started 20 years ago, does that mean they were 45 when the study started? But the study is about older adults, so maybe they weren't part of the study back then. So, perhaps their healthcare costs are modeled starting from now, ( t = 20 ), as if they are entering the study now. So, their healthcare cost from now on would follow ( C(t) ) where ( t ) is measured from the start of the study, which is 20 years ago.Therefore, their healthcare cost in the next 10 years would be from ( t = 20 ) to ( t = 30 ). So, same as before, the total cost is 225,000.Alternatively, if the models were based on the person's age, then we'd have to adjust, but since ( t ) is the time since the study started, I think it's correct to use ( t = 20 ) to ( t = 30 ).Therefore, the total healthcare cost is 225,000 dollars.Wait, but let me make sure I didn't make a mistake in interpreting the models. The life expectancy model is ( L(t) = 75 e^{0.0062 t} ). So, at ( t = 20 ), ( L(20) = 75 e^{0.124} ≈ 85 ). So, that's correct.But does this life expectancy model affect the healthcare cost calculation? The problem says the advocate wants to use these models to predict the total healthcare cost. So, maybe the life expectancy affects how long the person will need healthcare. But the problem specifically says "over the next 10 years," so regardless of how long they live, we just need to calculate the cost for the next 10 years.Therefore, even if the person's life expectancy is 85, meaning they might live 20 more years, the problem only asks for the next 10 years. So, we don't need to factor in their life expectancy beyond that.Therefore, the total healthcare cost is 225,000 dollars.Wait, but just to be thorough, let me recast the problem. If the person is 65 now, and the study started 20 years ago, so ( t = 20 ) now. The life expectancy now is 85, meaning they are expected to live 20 more years. But the problem asks for the next 10 years, so regardless of their life expectancy, we just calculate the cost for the next 10 years.So, yes, 225,000 is the answer.But let me also think about units. The healthcare cost is in thousands of dollars. Wait, in the problem statement, it says "annual healthcare costs (in thousands of dollars)". So, ( C(t) ) is in thousands. So, when I calculated 225,000, that's in thousands of dollars? Wait, no.Wait, no, hold on. Let me check the problem again."annual healthcare costs (in thousands of dollars)" So, ( C(t) ) is in thousands. So, when I calculated ( C(t) = 10,000 + 500t ), that's in thousands. So, 10,000 is 10,000 thousand dollars, which is 10 million dollars? That can't be right.Wait, hold on, maybe I misread. Let me check:"annual healthcare costs (in thousands of dollars)" So, ( C(t) ) is in thousands of dollars. So, ( C_0 = 10,000 ) is 10,000 thousand dollars, which is 10 million dollars? That seems way too high.Wait, no, that can't be. Maybe ( C(t) ) is in thousands, so ( C_0 = 10 ) (thousand dollars), and ( m = 0.5 ) (thousand dollars per year). So, the equation would be ( C(t) = 10 + 0.5 t ), with ( C(t) ) in thousands.Wait, the problem says: "annual healthcare costs (in thousands of dollars)" and "initial annual healthcare cost ( C_0 ) was 10,000". Wait, that's conflicting. If ( C(t) ) is in thousands, then ( C_0 = 10 ) (meaning 10,000). But the problem says ( C_0 ) was 10,000. So, maybe ( C(t) ) is in dollars, not thousands.Wait, let me reread the problem."annual healthcare costs (in thousands of dollars) for a sample of older adults over the past 20 years. The data fits ... a linear growth model for healthcare costs.2. The annual healthcare cost ( C(t) ) per older adult is modeled by the linear function ( C(t) = C_0 + mt ), where ( C_0 ) is the initial annual healthcare cost, ( m ) is the annual increase in healthcare cost, and ( t ) is the time in years since the start of the study. If the initial annual healthcare cost ( C_0 ) was 10,000 and it has increased to 20,000 after 20 years, determine the annual increase in healthcare cost ( m )."So, ( C(t) ) is in dollars, because ( C_0 ) is 10,000. The problem says "annual healthcare costs (in thousands of dollars)", but then defines ( C(t) ) with ( C_0 = 10,000 ) dollars. So, maybe it's a typo, or maybe ( C(t) ) is in thousands, but the numbers are given in dollars.Wait, this is confusing. Let me parse it again.The problem says: "annual healthcare costs (in thousands of dollars)" So, the unit is thousands of dollars. Then, in part 2, it says: "the initial annual healthcare cost ( C_0 ) was 10,000". So, if the unit is thousands, then 10,000 is 10 thousand dollars, so ( C_0 = 10 ). Similarly, after 20 years, it's 20,000, which is 20 thousand dollars, so ( C(20) = 20 ).Therefore, in the equation ( C(t) = C_0 + mt ), ( C_0 = 10 ) (thousand dollars), and ( C(20) = 20 ). So, ( 20 = 10 + 20m ). Therefore, ( m = (20 - 10)/20 = 0.5 ). So, ( m = 0.5 ) thousand dollars per year, which is 500 per year.So, in that case, ( C(t) = 10 + 0.5 t ), where ( C(t) ) is in thousands of dollars. So, to get the actual dollars, we have to multiply by 1,000. But since the problem says "annual healthcare costs (in thousands of dollars)", and then gives ( C_0 ) as 10,000, which is 10 thousand dollars, so it's consistent.Therefore, my earlier calculation was correct in terms of units. ( C(t) = 10 + 0.5 t ), with ( C(t) ) in thousands. So, at ( t = 20 ), ( C(20) = 10 + 0.5*20 = 20 ) thousand dollars, which is 20,000. So, that's correct.Therefore, when I calculated the total healthcare cost over the next 10 years as 225,000, that was in thousands of dollars? Wait, no. Wait, no, hold on. Wait, ( C(t) ) is in thousands, so the integral would be in thousands of dollars per year times years, so thousands of dollars.Wait, no, the integral of ( C(t) ) from 20 to 30 would be in thousands of dollars, because ( C(t) ) is in thousands per year. So, 225,000 would be in thousands, meaning 225,000 thousand dollars, which is 225 million dollars. That can't be right.Wait, no, wait. Let's clarify.If ( C(t) ) is in thousands of dollars per year, then the integral ( int_{20}^{30} C(t) dt ) would be in thousands of dollars times years, which is thousands of dollars per year multiplied by years, resulting in thousands of dollars.Wait, no, actually, no. The integral of cost over time is total cost. If ( C(t) ) is in thousands of dollars per year, then integrating over 10 years gives total cost in thousands of dollars.Wait, no, that's not right. Let's think carefully.If ( C(t) ) is in thousands of dollars per year, then each year, the cost is, say, 20 thousand dollars. So, over 10 years, it's 20,000 * 10 = 200,000 thousand dollars, which is 200 million dollars. But that seems too high.Wait, no, no, no. Wait, if ( C(t) ) is in thousands of dollars, then ( C(t) ) is, for example, 20 (thousand dollars) per year. So, each year, the cost is 20,000 dollars. So, over 10 years, it's 20,000 * 10 = 200,000 dollars.But wait, in our case, the cost is increasing each year. So, the average cost is 22.5 thousand dollars per year, so total cost is 22.5 thousand * 10 = 225 thousand dollars.Wait, so 225,000 dollars in total. So, that's the correct interpretation.Wait, so ( C(t) ) is in thousands, so when we integrate, we get total cost in thousands. But no, actually, no. Wait, the integral of ( C(t) ) over time is total cost. If ( C(t) ) is in thousands per year, then the integral is in thousands.Wait, no, let's use units:( C(t) ) is in [thousands of dollars]/year.Integrate over t (years):Total cost = ( int C(t) dt ) = [thousands of dollars]/year * year = thousands of dollars.So, total cost is in thousands of dollars.But in our calculation, we had:Integral from 20 to 30 of (10 + 0.5 t) dt = 225.But since ( C(t) ) is in thousands, 225 is in thousands of dollars. So, total cost is 225 thousand dollars, which is 225,000.So, that's correct.Therefore, the total healthcare cost over the next 10 years is 225,000.Wait, so my initial calculation was correct, but I was confused about the units. So, 225,000 is in thousands, so 225,000 thousand dollars? No, wait, no.Wait, no, no, no. Wait, no, the integral result is 225, which is in thousands of dollars. So, 225 thousand dollars, which is 225,000.Wait, let me clarify:- ( C(t) ) is in thousands of dollars per year. So, if ( C(t) = 20 ), that's 20,000 per year.- The integral ( int_{20}^{30} C(t) dt ) is the total cost in thousands of dollars.- The result of the integral was 225, which is in thousands of dollars. So, 225 thousand dollars is 225,000.Yes, that makes sense.Therefore, the total healthcare cost over the next 10 years is 225,000.So, summarizing:1. Growth rate ( k ) is approximately 0.0062 per year.2. Annual increase ( m ) is 0.5 thousand dollars per year, which is 500.3. Total healthcare cost over the next 10 years is 225,000.I think that's solid.Final AnswerThe total healthcare cost for the older adult over the next 10 years is boxed{225000} dollars.</think>

🧠:<think>Alright, so I have this problem about engagement rate for a mental health blog and forum. It's split into two parts. Let me tackle them one by one.Starting with part 1. I need to find the engagement rate E(t) as a function of time t. Engagement rate is defined as the ratio of total interactions to total unique visitors. So, E(t) = I(t) / V(t). Given:- V(t) = 2000 + 300t- I(t) = 500t² + 1500t + 2000So, plugging these into the formula, E(t) should be (500t² + 1500t + 2000) / (2000 + 300t). That seems straightforward.Now, I need to calculate the average engagement rate over the first year. Since it's a year, t goes from 0 to 12 months. To find the average, I think I need to integrate E(t) from t=0 to t=12 and then divide by the interval length, which is 12.So, average E = (1/12) * ∫₀¹² E(t) dt = (1/12) * ∫₀¹² [500t² + 1500t + 2000] / [2000 + 300t] dt.Hmm, integrating this might be a bit tricky. Let me see if I can simplify the integrand first. Maybe perform polynomial division or partial fractions.Let me denote the numerator as N(t) = 500t² + 1500t + 2000 and the denominator as D(t) = 300t + 2000.I can factor out 100 from D(t) to make it simpler: D(t) = 100(3t + 20). Similarly, N(t) can be factored as 100(5t² + 15t + 20). So, E(t) simplifies to (5t² + 15t + 20)/(3t + 20).Now, let me perform polynomial division on 5t² + 15t + 20 divided by 3t + 20.Divide 5t² by 3t to get (5/3)t. Multiply (3t + 20) by (5/3)t to get 5t² + (100/3)t. Subtract this from N(t):(5t² + 15t + 20) - (5t² + (100/3)t) = (15t - 100/3 t) + 20 = (45/3 t - 100/3 t) + 20 = (-55/3 t) + 20.Now, divide (-55/3 t) by 3t to get (-55/9). Multiply (3t + 20) by (-55/9) to get (-55/3 t) - (1100/9). Subtract this from the previous remainder:(-55/3 t + 20) - (-55/3 t - 1100/9) = 20 + 1100/9 = (180/9 + 1100/9) = 1280/9.So, the division gives us:E(t) = (5/3)t - 55/9 + (1280/9)/(3t + 20)Therefore, E(t) can be rewritten as:E(t) = (5/3)t - 55/9 + (1280/9)/(3t + 20)Now, integrating E(t) from 0 to 12:∫₀¹² E(t) dt = ∫₀¹² [(5/3)t - 55/9 + (1280/9)/(3t + 20)] dtLet's break this into three separate integrals:1. ∫ (5/3)t dt = (5/6)t²2. ∫ (-55/9) dt = (-55/9)t3. ∫ (1280/9)/(3t + 20) dtFor the third integral, let me make a substitution. Let u = 3t + 20, then du = 3 dt, so dt = du/3.So, ∫ (1280/9)/u * (du/3) = (1280/27) ∫ (1/u) du = (1280/27) ln|u| + C = (1280/27) ln(3t + 20) + CPutting it all together:∫ E(t) dt = (5/6)t² - (55/9)t + (1280/27) ln(3t + 20) + CNow, evaluate from 0 to 12:At t=12:(5/6)(144) - (55/9)(12) + (1280/27) ln(3*12 + 20)= (5/6)(144) = 120= (55/9)(12) = (55*12)/9 = 660/9 = 73.333...= (1280/27) ln(56) ≈ (1280/27)*4.025 ≈ 188.481 * 4.025 ≈ 760.0 (approx)Wait, let me calculate more accurately:First term: 5/6 * 144 = 5 * 24 = 120Second term: 55/9 * 12 = (55*12)/9 = 660/9 = 73.333...Third term: (1280/27) * ln(56). Let's compute ln(56):ln(56) ≈ 4.02536So, (1280/27) * 4.02536 ≈ (47.4074) * 4.02536 ≈ 190.93So, total at t=12: 120 - 73.333 + 190.93 ≈ 120 - 73.333 = 46.667 + 190.93 ≈ 237.597At t=0:(5/6)(0) - (55/9)(0) + (1280/27) ln(20) ≈ 0 - 0 + (1280/27)*2.9957 ≈ (47.4074)*2.9957 ≈ 142.0So, the definite integral is 237.597 - 142.0 ≈ 95.597Therefore, average E = (1/12)*95.597 ≈ 7.966So, approximately 7.97. Let me check my calculations again because the numbers seem a bit rough.Wait, let me recalculate the third term at t=12:(1280/27) * ln(56) ≈ 47.407 * 4.025 ≈ 47.407 * 4 = 189.628, plus 47.407 * 0.025 ≈ 1.185, total ≈ 190.813So, total at t=12: 120 - 73.333 + 190.813 ≈ 120 - 73.333 = 46.667 + 190.813 ≈ 237.48At t=0: (1280/27)*ln(20) ≈ 47.407 * 2.9957 ≈ 142.0So, integral ≈ 237.48 - 142.0 ≈ 95.48Average ≈ 95.48 / 12 ≈ 7.956, which is approximately 7.96.So, the average engagement rate over the first year is roughly 7.96.Wait, but let me think about units. Engagement rate is interactions per visitor, so it's a ratio. So, it's unitless, just a number.But let me double-check if I did the integral correctly.Alternatively, maybe I can use substitution or another method.Alternatively, perhaps I made a mistake in the polynomial division. Let me verify that.Original E(t) = (500t² + 1500t + 2000)/(300t + 2000)Factor numerator and denominator:Numerator: 500t² + 1500t + 2000 = 100*(5t² + 15t + 20)Denominator: 300t + 2000 = 100*(3t + 20)So, E(t) = (5t² + 15t + 20)/(3t + 20)Divide 5t² +15t +20 by 3t +20.First term: (5t²)/(3t) = (5/3)tMultiply (3t +20) by (5/3)t: 5t² + (100/3)tSubtract from numerator: (5t² +15t +20) - (5t² +100/3 t) = (15t -100/3 t) +20 = (45/3 t -100/3 t) +20 = (-55/3 t) +20Now, divide (-55/3 t) by 3t: (-55/9)Multiply (3t +20) by (-55/9): (-55/3 t) - (1100/9)Subtract: (-55/3 t +20) - (-55/3 t -1100/9) = 20 +1100/9 = (180/9 +1100/9)=1280/9So, the division is correct: E(t)= (5/3)t -55/9 + (1280/9)/(3t +20)So, integrating term by term:∫(5/3)t dt = (5/6)t²∫(-55/9) dt = (-55/9)t∫(1280/9)/(3t +20) dt = (1280/27) ln|3t +20|So, the integral is correct.Evaluated from 0 to12:At t=12:(5/6)(144) = 120(-55/9)(12) = -73.333(1280/27) ln(56) ≈ 47.407 *4.025 ≈190.813Total: 120 -73.333 +190.813 ≈237.48At t=0:(5/6)(0)=0(-55/9)(0)=0(1280/27) ln(20)≈47.407*2.9957≈142.0So, integral≈237.48 -142=95.48Average≈95.48/12≈7.956≈7.96So, the average engagement rate is approximately 7.96.Wait, but let me check if I should have kept more decimal places for accuracy.Alternatively, maybe I can compute it more precisely.Compute ln(56):ln(56)=ln(7*8)=ln7 + ln8≈1.9459 +2.0794≈4.0253Similarly, ln(20)=ln(4*5)=ln4 + ln5≈1.3863 +1.6094≈2.9957So, (1280/27)*ln(56)= (1280/27)*4.0253≈47.4074*4.0253≈47.4074*4=189.6296 +47.4074*0.0253≈1.197≈190.8266Similarly, (1280/27)*ln(20)=47.4074*2.9957≈47.4074*3=142.2222 -47.4074*0.0043≈≈142.2222 -0.203≈142.0192So, integral≈237.48 -142.0192≈95.4608Average≈95.4608/12≈7.955≈7.96So, approximately 7.96.So, the average engagement rate over the first year is approximately 7.96.Now, moving to part 2.They introduce a new feature that increases interactions exponentially: I_new(t)=I(t)*e^{0.1t}So, I_new(t)= (500t² +1500t +2000)*e^{0.1t}The number of visitors remains the same: V(t)=2000 +300tSo, the new engagement rate E_new(t)=I_new(t)/V(t)= [ (500t² +1500t +2000)*e^{0.1t} ] / (2000 +300t)We need to find the percentage increase in engagement rate at the end of the first year, i.e., at t=12, compared to the original engagement rate.So, first, compute E(12) and E_new(12), then find ((E_new(12) - E(12))/E(12))*100%First, compute E(12):E(12)=I(12)/V(12)I(12)=500*(144)+1500*12 +2000=72000 +18000 +2000=92000V(12)=2000 +300*12=2000 +3600=5600So, E(12)=92000/5600≈16.4286Wait, that's different from the average we calculated earlier. Because E(t) varies each month, but at t=12, it's 92000/5600≈16.4286.Wait, but earlier when we integrated, we found that the average was around 7.96, but at t=12, E(t)=16.4286.So, now, E_new(12)= [I(12)*e^{0.1*12}]/V(12)= [92000 * e^{1.2}]/5600Compute e^{1.2}≈3.3201So, E_new(12)=92000*3.3201 /5600≈(92000/5600)*3.3201≈16.4286*3.3201≈54.545So, E_new(12)≈54.545Original E(12)=16.4286Percentage increase= ((54.545 -16.4286)/16.4286)*100≈(38.1164/16.4286)*100≈231.4%So, approximately 231.4% increase.Wait, let me verify the calculations step by step.First, I(12)=500*(12)^2 +1500*12 +2000=500*144=72000 +1500*12=18000 +2000=92000V(12)=2000 +300*12=2000+3600=5600E(12)=92000/5600=16.4285714I_new(12)=92000*e^{0.1*12}=92000*e^{1.2}Compute e^{1.2}≈3.32011692So, I_new(12)=92000*3.32011692≈92000*3.3201≈306, let's compute 92000*3=276000, 92000*0.3201≈92000*0.3=27600, 92000*0.0201≈1849.2, so total≈276000 +27600 +1849.2≈305,449.2So, I_new(12)=305,449.2E_new(12)=305,449.2 /5600≈54.5445So, E_new(12)=54.5445Original E(12)=16.4285714Increase=54.5445 -16.4285714≈38.1159Percentage increase= (38.1159 /16.4285714)*100≈231.4%So, approximately 231.4% increase.Alternatively, to be precise, let's compute 38.1159 /16.4285714:16.4285714 *2=32.857142838.1159 -32.8571428≈5.2587572So, 2 + (5.2587572 /16.4285714)≈2 +0.320≈2.320, which is 232%. So, approximately 232%.But earlier I had 231.4%, so maybe 231.4% is more accurate.Alternatively, let's compute 38.1159 /16.4285714:16.4285714 *2.314≈16.4285714*2=32.8571428 +16.4285714*0.314≈16.4285714*0.3=4.92857142 +16.4285714*0.014≈0.229999999≈≈4.92857142 +0.23≈5.15857142So, total≈32.8571428 +5.15857142≈38.0157142, which is close to 38.1159, so 2.314 is accurate.So, approximately 231.4% increase.So, the percentage increase in engagement rate at the end of the first year is approximately 231.4%.But let me check if I made any mistake in the calculation.Wait, when I computed E_new(12)=I_new(12)/V(12)= (92000*e^{1.2})/5600But e^{1.2}=3.32011692So, 92000*3.32011692=92000*3 +92000*0.32011692=276000 +92000*0.32011692Compute 92000*0.32011692:0.3*92000=276000.02011692*92000≈92000*0.02=1840, plus 92000*0.00011692≈10.76So, total≈27600 +1840 +10.76≈29450.76So, total I_new(12)=276000 +29450.76≈305,450.76Thus, E_new(12)=305,450.76 /5600≈54.54478So, E_new(12)=54.54478E(12)=16.4285714Difference=54.54478 -16.4285714≈38.1162Percentage increase= (38.1162 /16.4285714)*100≈231.4%Yes, that's correct.So, the percentage increase is approximately 231.4%.So, summarizing:1. E(t)= (500t² +1500t +2000)/(2000 +300t). The average engagement rate over the first year is approximately 7.96.2. E_new(t)= [ (500t² +1500t +2000)*e^{0.1t} ] / (2000 +300t). The percentage increase in engagement rate at t=12 is approximately 231.4%.I think that's it.</think>

🧠:<think>Alright, so I've got this problem about electoral fraud in a local election in Queensland. There are five candidates: A, B, C, D, and E. The voting system is preferential, which means voters rank the candidates in order of preference. There are 1000 voters in total.The resident is looking into potential fraud, and there are two sub-problems to solve. Let me tackle them one by one.Sub-problem 1: The first thing I notice is the distribution of first-preference votes. Candidate A got 35%, B got 25%, C got 20%, D got 15%, and E got 5%. So, converting percentages to actual votes, that would be:- A: 350 votes- B: 250 votes- C: 200 votes- D: 150 votes- E: 50 votesNow, the question is about the expected number of second-preference votes Candidate B would receive, assuming that the second-preference votes are distributed uniformly at random among the remaining candidates.Hmm, okay. So, for each voter, their second preference is equally likely to be any of the other four candidates, right? But wait, actually, no. Because if a voter has already given their first preference to a candidate, their second preference can't be the same candidate. So, for each voter, the second preference is among the remaining four candidates, each with equal probability.But wait, is that the case? Or does the second preference depend on who the first preference was? Hmm, the problem says the second-preference votes are distributed uniformly at random among the remaining candidates. So, regardless of who the first preference was, each of the other four candidates has an equal chance of being the second preference.So, for each voter, the probability that their second preference is B is 1/4, since there are four other candidates.But wait, hold on. For each voter, the second preference is among the remaining four candidates, so if the first preference was, say, A, then the second preference could be B, C, D, or E, each with equal probability. Similarly, if the first preference was B, the second preference could be A, C, D, or E, each with equal probability.But in this case, the question is about the expected number of second-preference votes for B. So, for each voter, the probability that their second preference is B is 1/4, regardless of their first preference.Therefore, the expected number of second-preference votes for B would be the total number of voters multiplied by 1/4.But wait, hold on. The total number of voters is 1000, so 1000 * 1/4 = 250. But that seems too high because B already has 250 first-preference votes. Wait, but in preferential voting, the second preferences are counted only after the first preferences are exhausted. So, actually, the second preferences are only counted if the first preference is eliminated.Wait, no, hold on. The question is just about the expected number of second-preference votes, not about the actual count in the election process. So, perhaps it's just the number of voters who have B as their second preference, regardless of whether their first preference is still in the race.So, in that case, if each voter has a 1/4 chance of having B as their second preference, then the expected number is 1000 * 1/4 = 250.But that seems counterintuitive because B already has 250 first-preference votes. But in reality, the second-preference votes are separate from the first-preference votes. So, actually, each voter can have a different second preference, regardless of their first.Wait, but if a voter's first preference is B, can their second preference also be B? No, because you can't rank the same candidate twice. So, for voters who have B as their first preference, their second preference must be someone else. So, for those 250 voters, their second preference is among A, C, D, E, each with equal probability.Similarly, for voters who have A as their first preference, their second preference is among B, C, D, E, each with equal probability. Same for the others.So, to calculate the expected number of second-preference votes for B, we need to consider each group of voters based on their first preference.Let me break it down:1. Voters who gave their first preference to A: 350 voters. For each of these, the probability that their second preference is B is 1/4. So, expected second-preference votes from A's first-preference voters: 350 * 1/4 = 87.5.2. Voters who gave their first preference to B: 250 voters. Since their second preference can't be B, the probability of B being their second preference is 0. So, expected second-preference votes from B's first-preference voters: 250 * 0 = 0.3. Voters who gave their first preference to C: 200 voters. Probability their second preference is B: 1/4. So, expected votes: 200 * 1/4 = 50.4. Voters who gave their first preference to D: 150 voters. Probability their second preference is B: 1/4. So, expected votes: 150 * 1/4 = 37.5.5. Voters who gave their first preference to E: 50 voters. Probability their second preference is B: 1/4. So, expected votes: 50 * 1/4 = 12.5.Now, adding all these up: 87.5 + 0 + 50 + 37.5 + 12.5 = 187.5.So, the expected number of second-preference votes for B is 187.5.Wait, but earlier I thought it was 250, but that was a mistake because I didn't consider that voters who have B as their first preference can't have B as their second preference. So, the correct calculation is 187.5.But let me verify this. The total number of second-preference votes for B is the sum over all voters of the probability that their second preference is B.For each voter, the probability that their second preference is B is:- If their first preference is not B, then 1/4.- If their first preference is B, then 0.So, the total expected number is the sum over all voters: for each voter, if their first preference is not B, add 1/4; else, add 0.So, the number of voters who didn't choose B as their first preference is 1000 - 250 = 750. Each of these has a 1/4 chance of having B as their second preference. So, 750 * 1/4 = 187.5. That's consistent with the previous calculation.So, the answer to Sub-problem 1 is 187.5.Sub-problem 2: Now, the distribution of preferences might not be uniform. The second-preference votes for B follow a different probability distribution depending on the first preference.Specifically:- From voters who gave their first preference to A: probability of second preference B is 0.3.- From voters who gave their first preference to C: probability is 0.4.- From voters who gave their first preference to D: probability is 0.2.- From voters who gave their first preference to E: probability is 0.1.So, we need to calculate the new expected number of second-preference votes for B.Again, we'll break it down by each group of first-preference voters.1. Voters who gave their first preference to A: 350 voters. Probability of second preference B: 0.3. So, expected votes: 350 * 0.3 = 105.2. Voters who gave their first preference to B: 250 voters. Probability of second preference B: 0. So, expected votes: 0.3. Voters who gave their first preference to C: 200 voters. Probability of second preference B: 0.4. So, expected votes: 200 * 0.4 = 80.4. Voters who gave their first preference to D: 150 voters. Probability of second preference B: 0.2. So, expected votes: 150 * 0.2 = 30.5. Voters who gave their first preference to E: 50 voters. Probability of second preference B: 0.1. So, expected votes: 50 * 0.1 = 5.Now, adding all these up: 105 + 0 + 80 + 30 + 5 = 220.So, the new expected number of second-preference votes for B is 220.Wait, let me double-check:- A: 350 * 0.3 = 105- C: 200 * 0.4 = 80- D: 150 * 0.2 = 30- E: 50 * 0.1 = 5Total: 105 + 80 = 185; 185 + 30 = 215; 215 + 5 = 220. Yep, that's correct.So, comparing Sub-problem 1 and Sub-problem 2, the expected number of second-preference votes for B increased from 187.5 to 220 when the distribution is not uniform. This suggests that if the second preferences are distributed as per the given probabilities, B is expected to receive more second-preference votes than under a uniform distribution. This could indicate potential tampering or strategic voting, which might be a red flag for fraud.But the resident needs to determine whether this indicates fraud. Well, if the second-preference distribution is significantly different from what would be expected under uniform distribution, it could suggest manipulation. However, without more information, it's hard to conclusively prove fraud, but it's a point of concern.So, in summary:- Sub-problem 1: Expected second-preference votes for B: 187.5- Sub-problem 2: Expected second-preference votes for B: 220These are the calculations.</think>