🧠:<think>Alright, so I need to help this bank manager optimize their loan portfolio. Let me try to break down the problem step by step. First, the bank offers two types of loans: personal and mortgage. The interest rates are 5% and 3% per annum respectively. The goal is to maximize the annual interest income. There are three constraints given:1. Total funds allocated shouldn't exceed 100 million.2. Personal loans can't be more than 40% of the total.3. The interest from mortgage loans needs to be at least twice that from personal loans.Okay, so for Sub-problem 1, I need to set up a linear programming model. Let me define the variables first.Let’s say:- Let ( x ) be the amount allocated to personal loans (in millions of dollars).- Let ( y ) be the amount allocated to mortgage loans (in millions of dollars).Our objective is to maximize the annual interest income. The interest from personal loans would be 5% of ( x ), which is ( 0.05x ). Similarly, the interest from mortgage loans would be 3% of ( y ), which is ( 0.03y ). So, the total interest income is ( 0.05x + 0.03y ). Therefore, the objective function is:Maximize ( Z = 0.05x + 0.03y )Now, let's look at the constraints.1. The total amount allocated should not exceed 100 million. So, ( x + y leq 100 ).2. The amount allocated for personal loans shouldn't exceed 40% of the total. So, ( x leq 0.4(x + y) ). Let me rearrange this inequality to make it simpler. Starting with ( x leq 0.4x + 0.4y ). Subtract ( 0.4x ) from both sides: ( 0.6x leq 0.4y ). Then, divide both sides by 0.2 to simplify: ( 3x leq 2y ) or ( 3x - 2y leq 0 ).3. The total annual interest from mortgage loans should be at least twice that from personal loans. So, ( 0.03y geq 2 times 0.05x ). Simplifying this: ( 0.03y geq 0.10x ) or ( 0.03y - 0.10x geq 0 ). To make it cleaner, I can multiply both sides by 100 to eliminate decimals: ( 3y - 10x geq 0 ).Also, we have the non-negativity constraints: ( x geq 0 ) and ( y geq 0 ).So, putting it all together, the linear programming problem is:Maximize ( Z = 0.05x + 0.03y )Subject to:1. ( x + y leq 100 )2. ( 3x - 2y leq 0 )3. ( -10x + 3y geq 0 )4. ( x geq 0 )5. ( y geq 0 )Wait, let me double-check constraint 3. The interest from mortgage is at least twice that from personal. So, ( 0.03y geq 2 times 0.05x ) simplifies to ( 0.03y geq 0.10x ). Multiplying both sides by 100 gives ( 3y geq 10x ), which is ( -10x + 3y geq 0 ). Yeah, that seems correct.Now, for Sub-problem 2, I need to determine the maximum annual interest income. To do this, I can solve the linear programming problem either graphically or using the simplex method. Since it's a two-variable problem, maybe graphing would be straightforward.Let me try to sketch the feasible region.First, plot the constraints:1. ( x + y = 100 ): This is a straight line from (100,0) to (0,100).2. ( 3x - 2y = 0 ): This can be rewritten as ( y = (3/2)x ). It's a line passing through the origin with a slope of 1.5.3. ( -10x + 3y = 0 ): Rewriting, ( y = (10/3)x ). This is a steeper line, slope about 3.333, passing through the origin.Now, the feasible region is where all constraints are satisfied.Let me find the intersection points of these constraints to identify the vertices of the feasible region.First, find where ( x + y = 100 ) intersects with ( y = (3/2)x ).Substitute ( y = (3/2)x ) into ( x + y = 100 ):( x + (3/2)x = 100 )( (5/2)x = 100 )( x = 40 )Then, ( y = (3/2)*40 = 60 )So, intersection at (40,60)Next, find where ( x + y = 100 ) intersects with ( y = (10/3)x ).Substitute ( y = (10/3)x ) into ( x + y = 100 ):( x + (10/3)x = 100 )( (13/3)x = 100 )( x = (100 * 3)/13 ≈ 23.08 )Then, ( y = (10/3)*23.08 ≈ 76.92 )So, intersection at approximately (23.08, 76.92)Now, check where ( y = (3/2)x ) intersects with ( y = (10/3)x ).Set ( (3/2)x = (10/3)x )Multiply both sides by 6: 9x = 20xWhich gives 11x = 0 => x=0, y=0. So, they intersect at the origin.So, the feasible region is a polygon with vertices at:1. (0,0)2. (0,100) – but wait, is this point feasible? Let me check constraints.Wait, if x=0, then from constraint 3: ( 3y geq 0 ), which is always true. From constraint 2: ( 3x - 2y leq 0 ) => 0 - 2y ≤ 0 => y ≥ 0, which is also satisfied. So, (0,100) is a vertex, but we need to check if it's part of the feasible region.But wait, the intersection points we found are (40,60) and (23.08,76.92). So, the feasible region is bounded by (0,0), (0,100), (23.08,76.92), (40,60), and back to (0,0). Wait, no, because the constraint ( y = (10/3)x ) is steeper than ( y = (3/2)x ), so the feasible region is actually between these lines.Wait, maybe I need to plot these lines properly.Alternatively, perhaps the feasible region is a triangle with vertices at (0,0), (40,60), and (23.08,76.92). Hmm, but (0,100) is another point, but does it satisfy all constraints?Wait, let's evaluate all constraints at (0,100):1. ( x + y = 100 ): yes.2. ( 3x - 2y = 0 - 200 = -200 ≤ 0 ): yes.3. ( -10x + 3y = 0 + 300 = 300 ≥ 0 ): yes.So, (0,100) is a feasible point. Similarly, (0,0) is trivially feasible.Wait, but the intersection of ( x + y = 100 ) and ( y = (10/3)x ) is (23.08,76.92). So, the feasible region is a quadrilateral with vertices at (0,0), (0,100), (23.08,76.92), and (40,60). Hmm, but actually, when you plot these lines, the feasible region is bounded by:- From (0,0) to (0,100) along y-axis.- From (0,100) to (23.08,76.92) along ( x + y = 100 ).- From (23.08,76.92) to (40,60) along ( y = (10/3)x ).- From (40,60) back to (0,0) along ( y = (3/2)x ).Wait, no, because ( y = (3/2)x ) is another boundary. So, actually, the feasible region is a polygon with vertices at (0,0), (0,100), (23.08,76.92), (40,60), and back to (0,0). So, it's a quadrilateral.But to find the maximum of Z, we need to evaluate Z at each vertex.So, let's compute Z at each vertex:1. At (0,0): Z = 0 + 0 = 02. At (0,100): Z = 0 + 0.03*100 = 33. At (23.08,76.92): Z = 0.05*23.08 + 0.03*76.92 ≈ 1.154 + 2.3076 ≈ 3.46164. At (40,60): Z = 0.05*40 + 0.03*60 = 2 + 1.8 = 3.8So, the maximum Z is at (40,60) with Z=3.8 million dollars.Wait, but let me confirm if (40,60) satisfies all constraints:1. ( x + y = 100 ): 40+60=100, yes.2. ( 3x - 2y = 120 - 120 = 0 ≤ 0 ): yes.3. ( -10x + 3y = -400 + 180 = -220 ≥ 0 ): Wait, that's -220, which is not ≥ 0. So, this point doesn't satisfy constraint 3.Oh no, that's a problem. So, (40,60) is not feasible because it violates the third constraint.Wait, that means my earlier assumption was wrong. So, I need to check which points are actually feasible.Let me re-examine the constraints.At (40,60):- Constraint 1: 40 + 60 = 100 ≤ 100: okay.- Constraint 2: 3*40 - 2*60 = 120 - 120 = 0 ≤ 0: okay.- Constraint 3: -10*40 + 3*60 = -400 + 180 = -220 ≥ 0: Not okay.So, (40,60) is not feasible. That means the feasible region is actually different.Wait, so perhaps the feasible region is bounded by (0,0), (0,100), (23.08,76.92), and another point where ( y = (3/2)x ) intersects with ( -10x + 3y = 0 ).Wait, let's find the intersection of ( y = (3/2)x ) and ( -10x + 3y = 0 ).Substitute ( y = (3/2)x ) into ( -10x + 3*(3/2)x = 0 ):( -10x + (9/2)x = 0 )( (-20/2 + 9/2)x = 0 )( (-11/2)x = 0 )So, x=0, y=0. So, they only intersect at the origin.Therefore, the feasible region is actually a triangle with vertices at (0,0), (0,100), and (23.08,76.92). Because beyond (23.08,76.92), the point (40,60) is not feasible.Wait, but let me check if (23.08,76.92) satisfies all constraints:1. ( x + y ≈ 23.08 + 76.92 = 100 ): yes.2. ( 3x - 2y ≈ 69.24 - 153.84 ≈ -84.6 ≤ 0 ): yes.3. ( -10x + 3y ≈ -230.8 + 230.76 ≈ -0.04 ≥ 0 ): Hmm, approximately zero, but slightly negative. So, it's almost on the boundary.Wait, maybe I should calculate it more precisely.Let me compute x and y exactly.From ( x + y = 100 ) and ( y = (10/3)x ):Substitute y into the first equation:( x + (10/3)x = 100 )( (13/3)x = 100 )( x = 100 * 3 /13 ≈ 23.076923 )So, x ≈ 23.076923Then, y = 100 - x ≈ 76.923077Now, check constraint 3 at this point:( -10x + 3y = -10*(23.076923) + 3*(76.923077) )= -230.76923 + 230.76923= 0Ah, so exactly at this point, constraint 3 is satisfied as equality. So, (23.076923, 76.923077) is a feasible point.Therefore, the feasible region has vertices at:1. (0,0)2. (0,100)3. (23.076923,76.923077)Because beyond that point, the other intersection (40,60) is not feasible.So, now, let's evaluate Z at these vertices:1. At (0,0): Z=02. At (0,100): Z=0 + 0.03*100=33. At (23.076923,76.923077): Z=0.05*23.076923 + 0.03*76.923077Let me compute this:0.05*23.076923 ≈ 1.1538460.03*76.923077 ≈ 2.307692Total Z ≈ 1.153846 + 2.307692 ≈ 3.461538 million dollars.So, the maximum Z is approximately 3.4615 million dollars at the point (23.076923,76.923077).Wait, but let me check if there's another vertex where constraint 2 and 3 intersect, but not on the total funds constraint.Wait, constraint 2 is ( 3x - 2y ≤ 0 ) and constraint 3 is ( -10x + 3y ≥ 0 ). Let's find their intersection.Set ( 3x - 2y = 0 ) and ( -10x + 3y = 0 ).From the first equation: ( y = (3/2)x )Substitute into the second equation: ( -10x + 3*(3/2)x = 0 )Simplify: ( -10x + (9/2)x = 0 )Convert to common denominator: ( (-20/2 + 9/2)x = 0 )So, ( (-11/2)x = 0 ) => x=0, y=0.So, they only intersect at the origin.Therefore, the only other vertex is where constraint 3 intersects the total funds constraint, which is at (23.076923,76.923077).So, the maximum Z is indeed at that point.Therefore, the optimal allocation is approximately 23.08 million to personal loans and 76.92 million to mortgage loans, yielding a maximum annual interest income of approximately 3.4615 million.But let me express this more precisely. Since 23.076923 is 23 and 1/13 million, and 76.923077 is 76 and 12/13 million.So, x = 23 + 1/13 ≈ 23.0769 milliony = 76 + 12/13 ≈ 76.9231 millionThus, the maximum Z is 0.05*(23 + 1/13) + 0.03*(76 + 12/13)Calculating:0.05*(23 + 1/13) = 1.15 + 0.05/13 ≈ 1.15 + 0.003846 ≈ 1.1538460.03*(76 + 12/13) = 2.28 + (0.03*12)/13 ≈ 2.28 + 0.02769 ≈ 2.30769Adding them: 1.153846 + 2.30769 ≈ 3.461536 million.So, approximately 3.4615 million.But let me see if there's a way to express this exactly.Since x = 300/13 ≈23.0769 and y=900/13≈76.9231.So, Z = 0.05*(300/13) + 0.03*(900/13)= (15/13) + (27/13)= 42/13 ≈3.2308 million.Wait, wait, that can't be. Wait, 0.05*(300/13) is (15/1300)*300? Wait, no.Wait, 0.05 is 5/100=1/20. So, 0.05*(300/13)= (1/20)*(300/13)= (300)/(260)= 30/26=15/13≈1.1538.Similarly, 0.03=3/100. So, 0.03*(900/13)= (3/100)*(900/13)= (2700)/(1300)=27/13≈2.0769.Wait, wait, that's different from earlier. Wait, 0.03*(900/13)= (3/100)*(900/13)= (2700)/(1300)=27/13≈2.0769.Wait, so total Z=15/13 +27/13=42/13≈3.2308 million.Wait, that contradicts my earlier calculation. So, which is correct?Wait, 0.05*(300/13)= (0.05*300)/13=15/13≈1.15380.03*(900/13)= (0.03*900)/13=27/13≈2.0769Total Z=15/13 +27/13=42/13≈3.2308 million.Wait, but earlier I thought it was approximately 3.4615. So, which is correct?Wait, I think I made a mistake in the earlier decimal calculation. Let me recalculate:0.05*23.076923=1.1538460.03*76.923077=2.307692Total=1.153846+2.307692=3.461538But according to fractions:x=300/13≈23.0769y=900/13≈76.9231Z=0.05x +0.03y= (0.05*300)/13 + (0.03*900)/13=15/13 +27/13=42/13≈3.2308Wait, this is inconsistent. There must be a miscalculation.Wait, 0.05*300=15, so 15/13≈1.15380.03*900=27, so 27/13≈2.0769Total=1.1538+2.0769≈3.2307But earlier, when I calculated 0.05*23.0769≈1.1538 and 0.03*76.9231≈2.3077, adding to≈3.4615.Wait, but 0.03*76.9231=2.3077, which is more than 2.0769.Wait, so which is correct? Let me compute 0.03*(900/13):900/13≈76.92310.03*76.9231≈2.3077But 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769Wait, that can't be. Because 0.03*76.9231 is 2.3077, which is not equal to 27/13≈2.0769.Wait, so I think I made a mistake in the fraction calculation.Wait, 0.03*(900/13)= (3/100)*(900/13)= (2700)/1300=27/13≈2.0769But 0.03*76.9231=2.3077Wait, so which is correct? It seems like a discrepancy.Wait, 900/13≈76.9231, so 0.03*76.9231≈2.3077But 27/13≈2.0769So, 2.3077≠2.0769Therefore, my earlier fraction approach was wrong.Wait, no. Wait, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, that can't be. There's a miscalculation here.Wait, 900/13≈76.9231, correct.0.03*76.9231=2.3077, correct.But 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769Wait, so which is correct? It seems like a conflict.Wait, no, actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, that can't be both. There must be a mistake in the way I'm calculating.Wait, 900/13 is approximately 76.9231, correct.0.03*76.9231=2.3077But 0.03*(900/13)=27/13≈2.0769Wait, so which is correct? It seems like the decimal calculation is correct because 0.03*76.9231≈2.3077But why does 0.03*(900/13)=27/13≈2.0769?Wait, 900/13 is 76.9231, so 0.03*(900/13)=0.03*76.9231≈2.3077But 27/13≈2.0769So, 27/13 is approximately 2.0769, but 0.03*(900/13)=2.3077Therefore, my earlier fraction approach was wrong because I incorrectly calculated 0.03*(900/13) as 27/13, but actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769, but that's not correct because 0.03*900=27, so 27/13≈2.0769, but 0.03*(900/13)=0.03*(76.9231)=2.3077Wait, so which is correct? It seems like a conflict in calculations.Wait, no, actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, that can't be both. There must be a mistake in the way I'm calculating.Wait, no, actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, this is confusing. Let me compute 0.03*(900/13):First, 900 divided by 13 is approximately 76.9231.Then, 0.03*76.9231≈2.3077Alternatively, 0.03*900=27, then 27 divided by 13≈2.0769Wait, so which is correct? It seems like both can't be correct.Wait, actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, so which is correct? It must be that I'm making a mistake in the order of operations.Wait, no, actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, so which is correct? It seems like a conflict in the calculation.Wait, perhaps I'm misunderstanding the expression. Let me clarify:0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, so which is correct? It seems like both can't be correct.Wait, no, actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, so which is correct? It must be that I'm making a mistake in the way I'm calculating.Wait, perhaps I should compute 0.03*(900/13) as (0.03*900)/13=27/13≈2.0769But when I compute 0.03*76.9231, I get≈2.3077Wait, so which is correct? It seems like a conflict.Wait, no, actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, so which is correct? It must be that I'm making a mistake in the way I'm calculating.Wait, perhaps I should compute 0.03*(900/13) as (0.03*900)/13=27/13≈2.0769But when I compute 0.03*76.9231, I get≈2.3077Wait, so which is correct? It seems like a conflict.Wait, no, actually, 0.03*(900/13)= (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, so which is correct? It must be that I'm making a mistake in the way I'm calculating.Wait, perhaps I should compute 0.03*(900/13) as (0.03*900)/13=27/13≈2.0769But 0.03*(900/13)=0.03*76.9231≈2.3077Wait, so which is correct? It seems like a conflict.Wait, I think the confusion arises because 0.03*(900/13) can be calculated in two ways:1. First compute 900/13≈76.9231, then multiply by 0.03:≈2.30772. First compute 0.03*900=27, then divide by 13:≈2.0769But these two methods should give the same result, but they don't. Therefore, I must have made a mistake in one of the calculations.Wait, let me compute 0.03*(900/13):First, 900 divided by 13 is exactly 69.230769... (Wait, no, 13*76=988, which is more than 900. Wait, 13*69=897, so 900-897=3, so 900/13=69 + 3/13≈69.230769Wait, so 900/13≈69.230769Then, 0.03*69.230769≈2.076923Ah! So, I made a mistake earlier when I thought 900/13≈76.9231. That was incorrect.Wait, no, 13*76=988, which is more than 900. So, 900/13 is actually≈69.230769, not 76.9231.Wait, that's a big mistake. So, I think I confused x and y earlier.Wait, let's go back.We had x=300/13≈23.0769 and y=900/13≈69.2308Wait, no, wait, from earlier:From ( x + y = 100 ) and ( y = (10/3)x ), substituting:x + (10/3)x =100 => (13/3)x=100 =>x=300/13≈23.0769Then, y=100 -x≈76.9231But wait, if y=(10/3)x, then y=(10/3)*(300/13)=1000/39≈25.641, which contradicts y≈76.9231Wait, no, that can't be. Wait, y=(10/3)x, so if x=300/13, then y=(10/3)*(300/13)=1000/13≈76.9231Ah, okay, so y=1000/13≈76.9231So, 900/13≈69.2308 is incorrect. It's actually 1000/13≈76.9231So, my earlier mistake was in calculating y as 900/13, but it's actually 1000/13.Therefore, y=1000/13≈76.9231So, 0.03*y=0.03*(1000/13)=30/13≈2.3077Similarly, 0.05*x=0.05*(300/13)=15/13≈1.1538So, total Z=15/13 +30/13=45/13≈3.4615 million dollars.Ah, that makes sense now. So, the earlier confusion was because I incorrectly calculated y as 900/13 instead of 1000/13.So, to summarize:x=300/13≈23.0769 milliony=1000/13≈76.9231 millionZ=0.05x +0.03y=15/13 +30/13=45/13≈3.4615 million dollars.Therefore, the maximum annual interest income is 45/13 million dollars, which is approximately 3.4615 million.So, the optimal allocation is approximately 23.08 million to personal loans and 76.92 million to mortgage loans, yielding a maximum annual interest income of 3.4615 million.But let me express this exactly. 45/13 is approximately 3.4615, but as a fraction, it's 45/13.So, the exact maximum interest income is 45/13 million dollars.Therefore, the answers are:Sub-problem 1: The linear programming model as formulated above.Sub-problem 2: The maximum annual interest income is 45/13 million dollars, approximately 3.4615 million.</think>

🧠:<think>Okay, so I have this problem where my nephew is trying to help his retired aunt and uncle track down some scammers. The scammers are using a rotating series of phone numbers, and the nephew has figured out that these numbers can be modeled by the function ( P(t) = A cdot sin(Bt + C) + D ). My job is to find the constants ( A, B, C, ) and ( D ) using three data points and then predict the next phone number at ( t = 4 ) hours.Let me start by writing down the given data points:- At ( t = 1 ) hour, ( P(1) = 5000 ).- At ( t = 2 ) hours, ( P(2) = 5500 ).- At ( t = 3 ) hours, ( P(3) = 5000 ).So, I have three equations here:1. ( A cdot sin(B cdot 1 + C) + D = 5000 )2. ( A cdot sin(B cdot 2 + C) + D = 5500 )3. ( A cdot sin(B cdot 3 + C) + D = 5000 )Hmm, three equations with four unknowns. That might be tricky, but maybe there's some pattern or symmetry here that can help me figure it out.Looking at the data points, I notice that at ( t = 1 ) and ( t = 3 ), the phone number is the same, 5000. That suggests that the function might have a maximum or minimum at ( t = 2 ). Since ( P(2) = 5500 ) is higher than the other two, it's likely that ( t = 2 ) is the maximum point of the sine wave.If that's the case, then the sine function reaches its maximum at ( t = 2 ). The sine function ( sin(theta) ) reaches its maximum of 1 at ( theta = frac{pi}{2} + 2pi k ) for integer ( k ). So, let's set up the equation for ( t = 2 ):( B cdot 2 + C = frac{pi}{2} + 2pi k )But since we're dealing with a function that's likely periodic and we don't have information about multiple cycles, I can probably assume ( k = 0 ) for simplicity. So,( 2B + C = frac{pi}{2} ) --- Equation (4)Now, looking at the other two points, ( t = 1 ) and ( t = 3 ), both give 5000. Let's write those equations:For ( t = 1 ):( A cdot sin(B + C) + D = 5000 ) --- Equation (1)For ( t = 3 ):( A cdot sin(3B + C) + D = 5000 ) --- Equation (3)Since both equal 5000, we can set them equal to each other:( A cdot sin(B + C) + D = A cdot sin(3B + C) + D )Subtracting ( D ) from both sides:( A cdot sin(B + C) = A cdot sin(3B + C) )Assuming ( A neq 0 ) (otherwise, the function would be constant, which doesn't fit the data), we can divide both sides by ( A ):( sin(B + C) = sin(3B + C) )Now, when is ( sin(alpha) = sin(beta) )? This happens when:1. ( alpha = beta + 2pi n ) for some integer ( n ), or2. ( alpha = pi - beta + 2pi n ) for some integer ( n ).Let's consider both cases.Case 1: ( B + C = 3B + C + 2pi n )Simplify:( B + C - 3B - C = 2pi n )( -2B = 2pi n )( B = -pi n )Since ( B ) is a constant in the function, it can be positive or negative, but the sine function is periodic, so a negative ( B ) would just shift the phase. However, for simplicity, let's consider ( n = 0 ), which gives ( B = 0 ). But if ( B = 0 ), the function becomes ( P(t) = A cdot sin(C) + D ), which is a constant, contradicting the data where ( P(2) ) is different. So, Case 1 with ( n = 0 ) is invalid. If ( n = 1 ), then ( B = -pi ). Let's see if that works.Case 2: ( B + C = pi - (3B + C) + 2pi n )Simplify:( B + C = pi - 3B - C + 2pi n )Bring like terms to one side:( B + C + 3B + C = pi + 2pi n )( 4B + 2C = pi + 2pi n )Divide both sides by 2:( 2B + C = frac{pi}{2} + pi n )Wait a minute, this is similar to Equation (4), which was ( 2B + C = frac{pi}{2} ). So, if we set ( n = 0 ), we get the same equation. If ( n = 1 ), then ( 2B + C = frac{3pi}{2} ), but that would mean the sine function is at a minimum at ( t = 2 ), which contradicts ( P(2) = 5500 ) being a maximum. So, ( n = 0 ) is the correct choice here.So, from Case 2, we have:( 2B + C = frac{pi}{2} ) --- Equation (4)Which is consistent with our earlier conclusion.Now, let's go back to Equation (1) and Equation (3):Equation (1): ( A cdot sin(B + C) + D = 5000 )Equation (3): ( A cdot sin(3B + C) + D = 5000 )But since ( sin(B + C) = sin(3B + C) ), as we found earlier, and both equal 5000, we can't get more information from these two equations directly. So, we need to use Equation (2):Equation (2): ( A cdot sin(2B + C) + D = 5500 )But from Equation (4), ( 2B + C = frac{pi}{2} ), so ( sin(2B + C) = sin(frac{pi}{2}) = 1 ). Therefore, Equation (2) becomes:( A cdot 1 + D = 5500 )So,( A + D = 5500 ) --- Equation (5)Now, let's go back to Equation (1):( A cdot sin(B + C) + D = 5000 )But from Equation (4), ( 2B + C = frac{pi}{2} ), so ( C = frac{pi}{2} - 2B ). Let's substitute this into Equation (1):( A cdot sin(B + (frac{pi}{2} - 2B)) + D = 5000 )Simplify the argument of sine:( B + frac{pi}{2} - 2B = -B + frac{pi}{2} )So,( A cdot sin(-B + frac{pi}{2}) + D = 5000 )Recall that ( sin(-x + frac{pi}{2}) = sin(frac{pi}{2} - x) = cos(x) ). So,( A cdot cos(B) + D = 5000 ) --- Equation (6)Similarly, let's look at Equation (3):( A cdot sin(3B + C) + D = 5000 )Again, substitute ( C = frac{pi}{2} - 2B ):( A cdot sin(3B + frac{pi}{2} - 2B) + D = 5000 )Simplify the argument:( 3B + frac{pi}{2} - 2B = B + frac{pi}{2} )So,( A cdot sin(B + frac{pi}{2}) + D = 5000 )Again, ( sin(B + frac{pi}{2}) = cos(B) ), so:( A cdot cos(B) + D = 5000 ) --- Equation (7)Wait, that's the same as Equation (6). So, both Equation (1) and Equation (3) reduce to the same equation, which is ( A cdot cos(B) + D = 5000 ).So, now we have:Equation (5): ( A + D = 5500 )Equation (6): ( A cdot cos(B) + D = 5000 )Let's subtract Equation (6) from Equation (5):( (A + D) - (A cdot cos(B) + D) = 5500 - 5000 )Simplify:( A - A cdot cos(B) = 500 )Factor out ( A ):( A (1 - cos(B)) = 500 ) --- Equation (8)Now, we need another equation to solve for ( A ) and ( B ). Let's recall that the sine function has a period of ( frac{2pi}{B} ). Since the function repeats every period, and we have data points at ( t = 1, 2, 3 ), it's possible that the period is 2 hours because the function goes from 5000 to 5500 to 5000, which is a full cycle (peak to peak). Wait, actually, from 5000 to 5500 to 5000 is a half cycle, so the period would be 4 hours? Hmm, let me think.Wait, the function goes from 5000 at ( t = 1 ), peaks at 5500 at ( t = 2 ), and goes back to 5000 at ( t = 3 ). So, that's a half-period. Therefore, the full period would be 4 hours. So, the period ( T = 4 ) hours.The period of the sine function is ( T = frac{2pi}{B} ). So,( frac{2pi}{B} = 4 )Solving for ( B ):( B = frac{2pi}{4} = frac{pi}{2} )So, ( B = frac{pi}{2} ).Now, let's plug ( B = frac{pi}{2} ) into Equation (8):( A (1 - cos(frac{pi}{2})) = 500 )We know that ( cos(frac{pi}{2}) = 0 ), so:( A (1 - 0) = 500 )( A = 500 )Now, from Equation (5):( A + D = 5500 )( 500 + D = 5500 )( D = 5500 - 500 = 5000 )So, ( D = 5000 ).Now, let's find ( C ) using Equation (4):( 2B + C = frac{pi}{2} )We know ( B = frac{pi}{2} ), so:( 2 cdot frac{pi}{2} + C = frac{pi}{2} )Simplify:( pi + C = frac{pi}{2} )( C = frac{pi}{2} - pi = -frac{pi}{2} )So, ( C = -frac{pi}{2} ).Let me recap the constants we've found:- ( A = 500 )- ( B = frac{pi}{2} )- ( C = -frac{pi}{2} )- ( D = 5000 )Let me verify these values with the original equations to make sure they fit.First, Equation (1):( P(1) = 500 cdot sin(frac{pi}{2} cdot 1 - frac{pi}{2}) + 5000 )Simplify the argument:( frac{pi}{2} - frac{pi}{2} = 0 )So,( 500 cdot sin(0) + 5000 = 0 + 5000 = 5000 ) ✔️Equation (2):( P(2) = 500 cdot sin(frac{pi}{2} cdot 2 - frac{pi}{2}) + 5000 )Simplify the argument:( pi - frac{pi}{2} = frac{pi}{2} )So,( 500 cdot sin(frac{pi}{2}) + 5000 = 500 cdot 1 + 5000 = 5500 ) ✔️Equation (3):( P(3) = 500 cdot sin(frac{pi}{2} cdot 3 - frac{pi}{2}) + 5000 )Simplify the argument:( frac{3pi}{2} - frac{pi}{2} = pi )So,( 500 cdot sin(pi) + 5000 = 0 + 5000 = 5000 ) ✔️Great, all three data points check out.Now, moving on to Sub-problem 2: Determine the next phone number at ( t = 4 ) hours.Using the function ( P(t) = 500 cdot sin(frac{pi}{2} t - frac{pi}{2}) + 5000 ).Let's compute ( P(4) ):( P(4) = 500 cdot sin(frac{pi}{2} cdot 4 - frac{pi}{2}) + 5000 )Simplify the argument:( 2pi - frac{pi}{2} = frac{4pi}{2} - frac{pi}{2} = frac{3pi}{2} )So,( 500 cdot sin(frac{3pi}{2}) + 5000 )We know that ( sin(frac{3pi}{2}) = -1 ), so:( 500 cdot (-1) + 5000 = -500 + 5000 = 4500 )Therefore, the phone number at ( t = 4 ) hours is 4500.Wait, let me double-check the calculation for ( P(4) ):( frac{pi}{2} cdot 4 = 2pi )So,( 2pi - frac{pi}{2} = frac{4pi}{2} - frac{pi}{2} = frac{3pi}{2} )Yes, that's correct. And ( sin(frac{3pi}{2}) = -1 ), so the calculation is correct.So, the next phone number is 4500.</think>

🧠:<think>Alright, so I have this problem about calculating the diversity index and the gender balance ratio for a set of films. Let me try to break it down step by step.First, part 1 is about the diversity index. The formula given is Simpson's Diversity Index, which is ( D = 1 - sum_{i=1}^{n} p_i^2 ). I remember Simpson's index is a measure of biodiversity, but here it's applied to cultural groups in films. The film in question has 5 different cultural groups with proportions 0.20, 0.25, 0.15, 0.30, and 0.10.Okay, so I need to calculate each ( p_i^2 ) and then sum them up. Let me write them down:1. ( p_1 = 0.20 ), so ( p_1^2 = 0.20^2 = 0.04 )2. ( p_2 = 0.25 ), so ( p_2^2 = 0.25^2 = 0.0625 )3. ( p_3 = 0.15 ), so ( p_3^2 = 0.15^2 = 0.0225 )4. ( p_4 = 0.30 ), so ( p_4^2 = 0.30^2 = 0.09 )5. ( p_5 = 0.10 ), so ( p_5^2 = 0.10^2 = 0.01 )Now, adding all these up: 0.04 + 0.0625 + 0.0225 + 0.09 + 0.01.Let me compute this step by step:- 0.04 + 0.0625 = 0.1025- 0.1025 + 0.0225 = 0.125- 0.125 + 0.09 = 0.215- 0.215 + 0.01 = 0.225So the sum of ( p_i^2 ) is 0.225. Then, the diversity index ( D ) is 1 minus this sum. So:( D = 1 - 0.225 = 0.775 )Hmm, that seems straightforward. Let me double-check my calculations. Each squared proportion seems correct. Adding them up: 0.04 + 0.0625 is indeed 0.1025, plus 0.0225 is 0.125, plus 0.09 is 0.215, plus 0.01 is 0.225. So yes, 1 - 0.225 is 0.775. Okay, that seems solid.Now moving on to part 2. The gender balance ratio ( G ) is given by ( G = frac{M}{F} ), where ( M ) is the number of male lead roles and ( F ) is the number of female lead roles. The critic notes that 10% of the films have a perfect gender balance, meaning ( G = 1 ). The average number of male lead roles per film is 3, and the average number of female lead roles per film is 2. We need to find the overall gender balance ratio ( G_{text{total}} ) for the entire dataset of 1000 films.Wait, so let me parse this. The average number of male lead roles is 3 per film, and female is 2 per film. So, across all films, the total number of male leads is 1000 films * 3 = 3000 male leads. Similarly, total female leads are 1000 * 2 = 2000 female leads.But hold on, the gender balance ratio is given per film as ( G = frac{M}{F} ). However, the overall ratio is not just the total male divided by total female, because each film contributes a ratio, and we have to consider how these ratios combine across all films.But wait, the problem says "determine the overall gender balance ratio ( G_{text{total}} ) for the entire dataset." It doesn't specify whether it's the average ratio or the total male over total female. Hmm.Wait, the gender balance ratio is defined as ( G = frac{M}{F} ) for each film. So, if we have 1000 films, each with their own ( G ), then ( G_{text{total}} ) could be interpreted in different ways. It could be the average of all ( G )s, or it could be the total male leads divided by total female leads.But the problem says "overall gender balance ratio ( G_{text{total}} )", so I think it's probably referring to the total male leads divided by total female leads for the entire dataset. Let me check.Given that the average number of male lead roles per film is 3 and female is 2, so total male leads would be 1000 * 3 = 3000, and total female leads would be 1000 * 2 = 2000. So, the overall ratio would be 3000 / 2000 = 1.5.But wait, the problem also mentions that 10% of the films have a perfect gender balance, i.e., ( G = 1 ). So, does this affect the overall ratio? Or is the overall ratio just the total male over total female regardless of individual film ratios?Wait, the average number of male and female leads is given, so regardless of how individual films are balanced, the total across all films is 3000 male and 2000 female. So, the overall ratio would be 3000 / 2000 = 1.5.But let me think again. If 10% of the films have G=1, meaning M=F for those films. So, for those 100 films, M=F. So, the number of male and female leads in those films are equal. Let's denote:Let’s denote:Total films: 1000Number of films with G=1: 10% of 1000 = 100 films.For these 100 films, M=F. Let’s denote the number of male leads in these films as M1 and female leads as F1. So, M1 = F1.For the remaining 900 films, G ≠ 1, so M ≠ F. Let’s denote their male leads as M2 and female leads as F2.Given that the average number of male leads per film is 3, total male leads = 1000 * 3 = 3000.Similarly, total female leads = 1000 * 2 = 2000.So, total male leads = M1 + M2 = 3000Total female leads = F1 + F2 = 2000But for the 100 films with G=1, M1 = F1.Let me denote M1 = F1 = x.So, M1 = x, F1 = x.Then, M2 = 3000 - xF2 = 2000 - xBut for the remaining 900 films, each film has some G ≠ 1, but we don't know the distribution. However, the problem doesn't give us more information about these films, so perhaps we can't compute the exact overall ratio based on individual film ratios. But the question is about the overall gender balance ratio for the entire dataset.Wait, if we consider the entire dataset, regardless of individual films, the total male leads are 3000 and total female leads are 2000. So, the overall ratio is 3000 / 2000 = 1.5.But the problem mentions that 10% of the films have G=1, but since we're looking for the overall ratio, which is total male over total female, it's 1.5 regardless of how individual films are balanced.Alternatively, if the question was asking for the average gender balance ratio across all films, that would be different. Because for 100 films, G=1, and for the other 900, we don't know G. But since the question says "overall gender balance ratio ( G_{text{total}} )", I think it refers to total male / total female.So, 3000 / 2000 = 1.5.But let me think again. If we have 100 films with G=1, meaning M=F for each of those films. So, if each of those 100 films has, say, 2 male and 2 female leads, then M1=200, F1=200. Then, the remaining 900 films would have M2=3000 - 200 = 2800, and F2=2000 - 200 = 1800. So, the average for the remaining 900 films would be M=2800/900 ≈ 3.111 and F=1800/900=2. So, their average G would be 3.111 / 2 ≈ 1.555. Then, the overall average G would be (100*1 + 900*1.555)/1000 ≈ (100 + 1400)/1000 = 1500/1000 = 1.5. So, same result.Alternatively, if the 100 films with G=1 have different numbers of leads, say, some have 1 male and 1 female, some have 2 and 2, etc., but as long as M1 = F1, the total male and female leads from those films are equal, so M1 = F1. Then, the remaining films have M2 = 3000 - M1 and F2 = 2000 - M1. So, the overall ratio is (3000)/(2000) = 1.5 regardless.Therefore, I think the overall gender balance ratio is 1.5.But let me make sure. The problem says "the number of films with a perfect gender balance (i.e., G = 1) is 10% of the total films." It doesn't specify anything about the number of leads in those films, just that G=1, which is M/F=1, so M=F for each of those films. However, the average number of male and female leads per film is given as 3 and 2, respectively. So, regardless of how the leads are distributed among the films, the total male leads are 3000, total female leads are 2000, so the overall ratio is 1.5.Therefore, I think the answer is 1.5.Final Answer1. The diversity index is boxed{0.775}.2. The overall gender balance ratio is boxed{1.5}.</think>

🧠:<think>Alright, so I have this problem where a financial literacy educator is looking at two investment opportunities for their clients. Both involve complex instruments, but with different risk and return profiles. The first is a structured product linked to a basket of stocks, and the second is a REIT with fluctuating dividends. I need to analyze both and figure out which might be better for a client seeking stable long-term growth.Starting with the first investment, the structured product. The return is given by the function R(t) = 0.02 + 0.01 sin(πt/6). So, this is an annual return that varies over time. It's compounded continuously, which means the growth is exponential based on the instantaneous rate. I need to find the expected return over a 10-year period.Hmm, okay. So, for continuous compounding, the formula for the amount after time t is A = P * e^(rt), where r is the continuous growth rate. But in this case, the growth rate r isn't constant; it's changing over time as R(t). So, the total return would be the integral of R(t) from 0 to 10, right? Because for continuous compounding, the overall growth factor is e^(∫R(t) dt from 0 to 10). So, the expected return would be the integral of R(t) over 10 years.Let me write that down. The expected return factor is e^(∫₀¹⁰ R(t) dt). So, first, I need to compute the integral of R(t) from 0 to 10. R(t) is 0.02 + 0.01 sin(πt/6). So, integrating term by term.The integral of 0.02 dt from 0 to 10 is straightforward: 0.02 * 10 = 0.2.The integral of 0.01 sin(πt/6) dt. Let's compute that. The integral of sin(ax) dx is (-1/a) cos(ax) + C. So, here, a = π/6. So, the integral becomes 0.01 * (-6/π) cos(πt/6) evaluated from 0 to 10.Calculating that: 0.01 * (-6/π) [cos(10π/6) - cos(0)]. Let's compute cos(10π/6). 10π/6 is 5π/3, which is equivalent to 300 degrees. Cos(300°) is 0.5. Cos(0) is 1. So, [0.5 - 1] = -0.5.So, plugging in: 0.01 * (-6/π) * (-0.5) = 0.01 * (6/π) * 0.5 = 0.01 * 3/π ≈ 0.01 * 0.9549 ≈ 0.009549.So, the integral of R(t) from 0 to 10 is 0.2 + 0.009549 ≈ 0.209549.Therefore, the expected return factor is e^0.209549. Let me compute that. e^0.2 is approximately 1.2214, and e^0.209549 is a bit more. Let me compute it more accurately.Using a calculator: e^0.209549 ≈ 1.233. So, the expected return is approximately 23.3%. But wait, that's the total growth factor. So, the expected return is 23.3% over 10 years, compounded continuously.Wait, but is that the expected return? Or is that the total growth? Because continuous compounding means that the overall growth is e^(∫R(t) dt). So, if I invest 1, after 10 years, it would be approximately 1.233, so a 23.3% increase. So, the expected return is about 23.3% over 10 years.But wait, is that the effective annual return? Or is it just the total return? Because the question says "expected return over a 10-year period, assuming the returns are compounded continuously." So, I think it's the total return, not an annualized return. So, 23.3% total return over 10 years.But let me double-check. The integral of R(t) is 0.209549, so e^0.209549 ≈ 1.233, so yes, that's the total growth factor. So, the expected return is 23.3%.Now, moving on to the second investment, the REIT. It pays quarterly dividends that follow a geometric progression. The first quarter's dividend is 100, and each subsequent quarter's dividend increases by 2%. So, it's a geometric series where each term is 1.02 times the previous term.We need to calculate the total dividend income over a 10-year period. Since it's quarterly, there are 40 payments (10 years * 4 quarters). So, the total dividend income is the sum of a geometric series with first term a = 100, common ratio r = 1.02, and number of terms n = 40.The formula for the sum of a geometric series is S_n = a*(r^n - 1)/(r - 1). Plugging in the numbers: S_40 = 100*(1.02^40 - 1)/(1.02 - 1).First, compute 1.02^40. Let me calculate that. 1.02^40 is approximately... I know that (1 + 0.02)^40. Using the rule of 72, 72/2 = 36, so doubling time is about 36 years, so in 40 years, it's a bit more than double. Let me compute it more accurately.Using logarithms: ln(1.02) ≈ 0.0198026. So, ln(1.02^40) = 40*0.0198026 ≈ 0.7921. So, e^0.7921 ≈ 2.208. So, 1.02^40 ≈ 2.208.Therefore, S_40 ≈ 100*(2.208 - 1)/(0.02) = 100*(1.208)/0.02 = 100*60.4 = 6040.So, the total dividend income over 10 years is approximately 6,040.Wait, but let me verify that calculation. 1.02^40 is actually approximately 2.208, yes. So, 2.208 - 1 = 1.208. Divided by 0.02 is 60.4. Multiply by 100 gives 6040. So, yes, 6,040 total dividends.But wait, is that the total income? Yes, because each quarter, the dividend increases by 2%, so it's a growing annuity. So, the sum is indeed 6040.Now, comparing the two investments. The structured product gives a total return of about 23.3% over 10 years, compounded continuously. The REIT gives a total dividend income of 6,040 over 10 years. But wait, we need to consider the initial investment for both.Wait, the problem doesn't specify the initial investment amount. Hmm. Maybe we can assume the same initial investment for both? Or perhaps we need to compute the total return in terms of percentage or in absolute terms.Wait, for the structured product, the return is 23.3% over 10 years. For the REIT, the total dividend income is 6,040. But to compare them, we need to know the initial investment. Since it's not given, perhaps we can assume the same principal, say 10,000, and compute the total return.Wait, but the problem doesn't specify. Hmm. Alternatively, maybe we can express the REIT's total return as a percentage of the initial investment. But since we don't have the initial investment, perhaps we need to consider the dividend yield.Wait, but the REIT pays 6,040 over 10 years. If the initial investment is, say, 10,000, then the total return would be 6,040, which is a 60.4% return over 10 years. But without knowing the initial investment, it's hard to compare.Wait, perhaps the structured product's return is 23.3% over 10 years, while the REIT's dividend income is 6,040. If we assume that the REIT's principal remains the same, then the total return is just the dividends. But if the REIT's value also appreciates, then the total return would be more. But the problem doesn't mention the appreciation of the REIT, only the dividends.So, perhaps the REIT's total return is just the sum of dividends, which is 6,040. While the structured product's total return is 23.3% of the initial investment. So, if the initial investment is the same, say 10,000, then structured product would give 2,330 return, while REIT gives 6,040. So, REIT is better in terms of total return.But wait, the structured product's return is compounded continuously, so it's a growth in the principal, while the REIT's return is in dividends, which are separate from the principal. So, the structured product's principal grows to 1.233 times the initial investment, while the REIT's principal remains the same, but you get 6,040 in dividends.So, if the initial investment is P, then structured product gives P*1.233, while REIT gives P + 6040. So, to compare, we need to see which is larger: 1.233P or P + 6040.So, 1.233P - P = 0.233P. So, 0.233P vs 6040. So, if 0.233P > 6040, then structured product is better. If 0.233P < 6040, then REIT is better.But without knowing P, we can't say. However, perhaps the REIT's dividend is based on the initial investment? Wait, the first dividend is 100, which is presumably based on the initial investment. So, if the initial investment is P, then the first dividend is 100, which is 100/P of the principal. So, if P is 10,000, then the dividend yield is 1%, which seems low. But maybe the REIT's dividend is fixed at 100 per quarter, regardless of the principal. That would mean that the dividend is fixed, not based on the principal.Wait, the problem says "the first quarter's dividend is 100, and each subsequent quarter's dividend is expected to increase by 2%." So, it's a fixed amount, starting at 100, growing by 2% each quarter. So, the total dividends are 6,040 over 10 years, regardless of the principal. So, if the principal is, say, 10,000, then the dividend yield is 1% per quarter, which is 4% annually, but growing by 2% each quarter.Wait, but the structured product's return is 23.3% over 10 years, which is about 2.33% per year, compounded continuously. So, if the initial investment is the same, say 10,000, then structured product would grow to 12,330, while the REIT would give 6,040 in dividends, keeping the principal at 10,000.So, in terms of total value, structured product gives 12,330, while REIT gives 10,000 + 6,040 = 16,040. So, REIT is better in this case.But wait, that can't be right, because the structured product's return is 23.3%, which is less than the REIT's total return of 60.4% (6040/10000). So, REIT is better.But wait, the structured product's return is 23.3% over 10 years, while the REIT's total return is 60.4% over 10 years. So, REIT is better in terms of total return.But wait, the structured product's return is compounded continuously, so it's a growth in the principal, while the REIT's return is in dividends, which are separate. So, if the client is looking for stable long-term growth, they might prefer the REIT because it provides predictable income (dividends) that grow over time, while the structured product's return is volatile, as it's linked to the stock market.But wait, the structured product's return is modeled as R(t) = 0.02 + 0.01 sin(πt/6). So, it's oscillating between 1% and 3% annually. So, it's not extremely volatile, but it does vary. The REIT's dividends are predictable, growing by 2% each quarter, so they are more stable.Therefore, considering that the client seeks stable long-term growth, the REIT might be more suitable because it provides predictable, growing income, whereas the structured product's returns, although they have a lower volatility (since it's a basket of stocks), still vary over time, and the total return over 10 years is lower than the REIT's.Wait, but in my earlier calculation, the REIT's total return was higher. So, even though the structured product's returns are modeled as varying, the total return over 10 years is lower than the REIT's. So, the REIT is better in terms of total return and also provides more stable income.Therefore, the REIT might be more suitable for a client seeking stable long-term growth.</think>