🧠:<think>Okay, so I have this problem where a small business owner is creating a logo using a circle and an equilateral triangle inscribed within it. They want to calculate the side length of the triangle and its area in terms of the radius, and then figure out the area of a hexagonal pattern around the triangle. Hmm, let me break this down step by step.Starting with the first part: finding the side length ( s ) of the equilateral triangle inscribed in a circle with radius ( r ). I remember that in an equilateral triangle inscribed in a circle, each side subtends a central angle of 60 degrees because all angles in an equilateral triangle are 60 degrees. So, the triangle divides the circle into three equal arcs.To find the side length, I think I can use the Law of Cosines in the triangle formed by two radii and a side of the triangle. The triangle formed is an isosceles triangle with two sides equal to ( r ) and the included angle of 60 degrees. The Law of Cosines states that ( c^2 = a^2 + b^2 - 2abcos(C) ), where ( C ) is the included angle.So, plugging in the values, we have:( s^2 = r^2 + r^2 - 2 times r times r times cos(60^circ) )I know that ( cos(60^circ) ) is 0.5, so substituting that in:( s^2 = 2r^2 - 2r^2 times 0.5 )Simplifying the equation:( s^2 = 2r^2 - r^2 )( s^2 = r^2 )Therefore, ( s = r )Wait, that seems too straightforward. Let me double-check. If each central angle is 60 degrees, and the triangle is equilateral, then yes, each side should be equal to the radius. Hmm, actually, no, that doesn't sound right. Wait, in a unit circle, the side length of an inscribed equilateral triangle is ( sqrt{3} ), not 1. So, maybe I made a mistake in my calculation.Let me try again. The Law of Cosines should be correct. So, let's compute it step by step:( s^2 = r^2 + r^2 - 2r^2 cos(60^circ) )( s^2 = 2r^2 - 2r^2 times 0.5 )( s^2 = 2r^2 - r^2 )( s^2 = r^2 )So, ( s = r )Hmm, but in a unit circle, the side length is ( sqrt{3} ). Wait, maybe I confused the chord length with something else. Let me recall that the chord length formula is ( 2r sin(theta/2) ), where ( theta ) is the central angle.So, for a central angle of 60 degrees, the chord length would be:( s = 2r sin(30^circ) )Since ( sin(30^circ) = 0.5 ), this gives:( s = 2r times 0.5 = r )Okay, so that confirms it. The side length is indeed ( r ). So, my initial calculation was correct. Maybe I was confusing it with another formula. So, moving on.Next, I need to find the area of the equilateral triangle in terms of ( r ). The formula for the area of an equilateral triangle is ( frac{sqrt{3}}{4} s^2 ). Since we found ( s = r ), substituting that in:Area ( = frac{sqrt{3}}{4} r^2 )Alright, that seems straightforward.Now, moving on to the second part: adding a hexagonal pattern around the inscribed triangle, where each side of the hexagon is equal to ( s ). So, each side of the hexagon is ( r ). I need to calculate the total area of the hexagonal pattern, excluding the area of the inscribed triangle.First, let me visualize this. The hexagon is surrounding the triangle, so it's a regular hexagon because all sides are equal. The area of a regular hexagon can be calculated using the formula ( frac{3sqrt{3}}{2} s^2 ), where ( s ) is the side length.So, substituting ( s = r ):Area of hexagon ( = frac{3sqrt{3}}{2} r^2 )But wait, the problem says to exclude the area of the inscribed triangle. So, I need to subtract the area of the triangle from the area of the hexagon.We already found the area of the triangle as ( frac{sqrt{3}}{4} r^2 ). Therefore, the total area of the hexagonal pattern excluding the triangle is:( frac{3sqrt{3}}{2} r^2 - frac{sqrt{3}}{4} r^2 )Let me compute this:First, find a common denominator, which is 4:( frac{6sqrt{3}}{4} r^2 - frac{sqrt{3}}{4} r^2 = frac{5sqrt{3}}{4} r^2 )Wait, that seems a bit odd. Let me verify the area of the hexagon again. The formula for the area of a regular hexagon is indeed ( frac{3sqrt{3}}{2} s^2 ). So, with ( s = r ), that's correct.But wait, is the hexagon circumscribed around the triangle? Or is the triangle inscribed in the hexagon? Actually, the problem says the hexagonal pattern is added around the inscribed triangle, so the hexagon is surrounding the triangle. However, in a regular hexagon, each side is equal, and the distance from the center to each vertex is equal to the side length. So, if the triangle is inscribed in the circle of radius ( r ), and the hexagon is also inscribed in the same circle, then the side length of the hexagon is equal to ( r ) as well.Wait, but in a regular hexagon inscribed in a circle, the side length is equal to the radius. So, yes, that makes sense. So, the hexagon has side length ( r ), same as the triangle.But then, the area of the hexagon is ( frac{3sqrt{3}}{2} r^2 ), and the area of the triangle is ( frac{sqrt{3}}{4} r^2 ). So, subtracting gives ( frac{5sqrt{3}}{4} r^2 ).Wait, but is that correct? Let me think about the relationship between the triangle and the hexagon. If the triangle is inscribed in the circle, and the hexagon is also inscribed in the same circle, then the hexagon is actually composed of six equilateral triangles, each with side length ( r ). So, the area of the hexagon is six times the area of one of those small triangles.The area of one equilateral triangle with side length ( r ) is ( frac{sqrt{3}}{4} r^2 ). Therefore, the hexagon's area is ( 6 times frac{sqrt{3}}{4} r^2 = frac{3sqrt{3}}{2} r^2 ), which matches the formula.Now, the inscribed triangle is a larger equilateral triangle. Wait, no, in the circle, the inscribed equilateral triangle has side length ( r ), but the hexagon is made up of six smaller equilateral triangles, each with side length ( r ). So, the area of the hexagon is indeed ( frac{3sqrt{3}}{2} r^2 ).But the inscribed triangle is a single equilateral triangle with side length ( r ), so its area is ( frac{sqrt{3}}{4} r^2 ). Therefore, subtracting the area of the triangle from the hexagon gives the area of the hexagonal pattern excluding the triangle.So, ( frac{3sqrt{3}}{2} r^2 - frac{sqrt{3}}{4} r^2 = frac{5sqrt{3}}{4} r^2 ).Wait, but I'm a bit confused because the hexagon is surrounding the triangle, but in reality, the triangle is inscribed in the circle, and the hexagon is also inscribed in the same circle. So, the triangle is actually one of the triangles that make up the hexagon? No, because the hexagon is made up of six smaller triangles, each with side length ( r ), while the inscribed triangle is a larger triangle with side length ( r ) as well. Wait, no, actually, the inscribed triangle has side length ( r ), and the hexagon is made up of six smaller triangles each with side length ( r ). So, the inscribed triangle is not part of the hexagon's structure.Wait, maybe I'm overcomplicating. The hexagon is a separate shape around the triangle, so the area to subtract is just the area of the triangle from the area of the hexagon. So, the calculation seems correct.But let me think again: if the hexagon is surrounding the triangle, and both are inscribed in the same circle, then the triangle is actually inside the hexagon. So, the hexagon's area includes the area of the triangle. Therefore, subtracting the triangle's area from the hexagon's area gives the area of the hexagonal pattern excluding the triangle.Yes, that makes sense. So, the total area is ( frac{5sqrt{3}}{4} r^2 ).Wait, but let me double-check the subtraction:( frac{3sqrt{3}}{2} r^2 = frac{6sqrt{3}}{4} r^2 )Subtract ( frac{sqrt{3}}{4} r^2 ):( frac{6sqrt{3} - sqrt{3}}{4} r^2 = frac{5sqrt{3}}{4} r^2 )Yes, that's correct.So, summarizing:1. Side length of the equilateral triangle: ( s = r )2. Area of the equilateral triangle: ( frac{sqrt{3}}{4} r^2 )3. Area of the hexagonal pattern excluding the triangle: ( frac{5sqrt{3}}{4} r^2 )I think that's all. Let me just make sure I didn't miss anything in the problem statement. The owner is combining a circle and an equilateral triangle inscribed within it, then adding a hexagonal pattern around the triangle. So, the hexagon is outside the triangle but inside the circle? Or is it a separate hexagon? Wait, the problem says "a hexagonal pattern around the inscribed triangle", so it's surrounding the triangle, which is inside the circle. So, the hexagon is also inscribed in the same circle, right? Because otherwise, the side length wouldn't be ( s = r ).Yes, because if the hexagon is around the triangle, and both are inscribed in the same circle, then the side length of the hexagon is equal to the radius. So, the calculations hold.Therefore, I think my answers are correct.</think>

🧠:<think>Alright, so I have this problem about a business owner who's organizing an event for Rainbow Wahine volleyball enthusiasts. The owner wants to give each attendee a unique piece of merchandise, and the cost of producing each piece is proportional to the square of the number of unique designs. Hmm, okay, let me try to break this down.First, the problem is split into two parts. The first part is about figuring out the maximum number of unique designs that can be produced given a budget. The second part is about probability, considering the number of attendees follows a Poisson distribution. Let me tackle them one by one.Starting with the first part: The business owner expects ( n ) attendees. The cost of producing ( k ) unique designs is given by ( C(k) = a cdot k^2 ), where ( a ) is a positive constant. The total budget is ( B ). I need to express the maximum number of unique designs ( k ) in terms of ( a ) and ( B ).Wait, hold on. The cost is proportional to the square of the number of unique designs. So, if we produce ( k ) unique designs, each attendee gets one of these designs. But does each attendee get a unique design, or does each design get assigned to multiple attendees? The problem says "every attendee receives a unique piece of merchandise," so I think each attendee gets a unique design. That would mean that the number of unique designs ( k ) must be at least equal to the number of attendees ( n ). But the cost is ( C(k) = a cdot k^2 ), so the cost depends on ( k ), the number of unique designs.But the budget is ( B ), so we need to find the maximum ( k ) such that ( C(k) leq B ). That would be ( a cdot k^2 leq B ). Solving for ( k ), we get ( k leq sqrt{frac{B}{a}} ). But since ( k ) must be an integer (you can't produce a fraction of a design), the maximum ( k ) is the floor of ( sqrt{frac{B}{a}} ). However, the problem doesn't specify that ( k ) has to be an integer, so maybe we can just express it as ( k = sqrt{frac{B}{a}} ). But let me think again.Wait, actually, the problem says the cost is proportional to the square of the number of unique designs produced. So, if you produce ( k ) unique designs, the cost is ( a k^2 ). So, if the budget is ( B ), the maximum ( k ) is the largest integer where ( a k^2 leq B ). But since the problem doesn't specify that ( k ) has to be an integer, maybe it's okay to leave it as ( k = sqrt{frac{B}{a}} ). Hmm, but in reality, you can't have a fraction of a design, so perhaps we should take the floor function. But the problem just says "express the maximum number of unique designs ( k )", so maybe it's acceptable to leave it as ( sqrt{frac{B}{a}} ). Let me check the wording again: "express the maximum number of unique designs ( k ) that can be produced in terms of ( a ) and ( B )." It doesn't specify whether ( k ) needs to be an integer, so I think it's fine to write ( k = sqrt{frac{B}{a}} ). So, that's the first part done.Moving on to the second part: The number of attendees is expected to follow a Poisson distribution with an average rate of ( lambda ) attendees per event. We need to find the probability that the business owner will be able to produce enough unique designs for all attendees, assuming the maximum number of unique designs ( k ) derived from the first part.Okay, so from the first part, ( k = sqrt{frac{B}{a}} ). The number of attendees ( n ) is a Poisson random variable with parameter ( lambda ). We need the probability that ( n leq k ), which is ( P(N leq k) ), where ( N ) is Poisson distributed with mean ( lambda ).So, the probability mass function of a Poisson distribution is ( P(N = m) = frac{e^{-lambda} lambda^m}{m!} ) for ( m = 0, 1, 2, ldots ). Therefore, the probability that ( N leq k ) is the sum from ( m = 0 ) to ( m = lfloor k rfloor ) of ( frac{e^{-lambda} lambda^m}{m!} ). But since ( k ) might not be an integer, we need to take the floor of ( k ) to get the maximum integer less than or equal to ( k ). However, in the first part, we might have considered ( k ) as a real number, but in reality, ( k ) must be an integer because you can't produce a fraction of a design. So, perhaps in the first part, ( k ) should be the floor of ( sqrt{frac{B}{a}} ). Let me clarify.Wait, in the first part, the problem says "the maximum number of unique designs ( k ) that can be produced in terms of ( a ) and ( B )." So, if ( k ) must be an integer, then ( k = lfloor sqrt{frac{B}{a}} rfloor ). But since the problem didn't specify, maybe we can just use ( k = sqrt{frac{B}{a}} ) as a real number, and then in the probability, we can take the floor of ( k ) when summing the probabilities. Hmm, I think that's the way to go.So, to summarize, the probability that the number of attendees ( N ) is less than or equal to ( k ) is ( P(N leq lfloor k rfloor) ), which is the cumulative distribution function (CDF) of the Poisson distribution evaluated at ( lfloor k rfloor ). Therefore, the probability is ( sum_{m=0}^{lfloor k rfloor} frac{e^{-lambda} lambda^m}{m!} ).But let me make sure I'm not missing anything. The problem says "the business owner will be able to produce enough unique designs for all attendees." So, if the number of attendees ( n ) is greater than ( k ), the owner cannot produce enough unique designs. Therefore, the probability of being able to produce enough is ( P(N leq k) ). Since ( k ) is the maximum number of unique designs, which is ( sqrt{frac{B}{a}} ), and since ( k ) might not be an integer, we need to take the floor of ( k ) to get the maximum integer number of designs that can be produced without exceeding the budget. Therefore, the probability is ( P(N leq lfloor sqrt{frac{B}{a}} rfloor) ).Alternatively, if we consider ( k ) as a real number, we can still express the probability as ( P(N leq k) ), but since ( N ) is an integer, the probability is the same as ( P(N leq lfloor k rfloor) ). So, in terms of the Poisson CDF, it's ( sum_{m=0}^{lfloor k rfloor} frac{e^{-lambda} lambda^m}{m!} ).Therefore, putting it all together, the probability is the sum from ( m = 0 ) to ( m = lfloor sqrt{frac{B}{a}} rfloor ) of ( frac{e^{-lambda} lambda^m}{m!} ).Wait, but the problem says "express the maximum number of unique designs ( k ) that can be produced in terms of ( a ) and ( B )", so in the first part, ( k = sqrt{frac{B}{a}} ). Then, in the second part, we use this ( k ) to compute the probability. So, the probability is ( P(N leq k) ), but since ( N ) is an integer, it's ( P(N leq lfloor k rfloor) ). Therefore, the probability is ( sum_{m=0}^{lfloor sqrt{frac{B}{a}} rfloor} frac{e^{-lambda} lambda^m}{m!} ).Alternatively, if we don't take the floor, and just use ( k ) as a real number, the probability would still be the same because the Poisson PMF is zero for non-integer values. So, I think it's safe to write the probability as ( sum_{m=0}^{lfloor sqrt{frac{B}{a}} rfloor} frac{e^{-lambda} lambda^m}{m!} ).But let me think again. If ( k ) is the maximum number of unique designs, which is ( sqrt{frac{B}{a}} ), and since ( k ) must be an integer, then ( k = lfloor sqrt{frac{B}{a}} rfloor ). Therefore, the probability is ( P(N leq k) = sum_{m=0}^{k} frac{e^{-lambda} lambda^m}{m!} ), where ( k = lfloor sqrt{frac{B}{a}} rfloor ).So, to write the final answer, I think it's better to express ( k ) as ( lfloor sqrt{frac{B}{a}} rfloor ) and then the probability is the sum up to ( k ). But the problem didn't specify whether ( k ) needs to be an integer, so perhaps we can leave it as ( k = sqrt{frac{B}{a}} ) and then the probability is ( P(N leq k) ), which is the same as ( P(N leq lfloor k rfloor) ).In conclusion, for the first part, the maximum number of unique designs is ( k = sqrt{frac{B}{a}} ). For the second part, the probability is the sum from ( m = 0 ) to ( m = lfloor sqrt{frac{B}{a}} rfloor ) of ( frac{e^{-lambda} lambda^m}{m!} ).Wait, but in the first part, the problem says "express the maximum number of unique designs ( k ) that can be produced in terms of ( a ) and ( B )." So, if ( k ) must be an integer, then ( k = lfloor sqrt{frac{B}{a}} rfloor ). But if not, it's just ( sqrt{frac{B}{a}} ). Since the problem doesn't specify, I think it's safer to assume that ( k ) is an integer, so we take the floor. Therefore, in the first part, ( k = lfloor sqrt{frac{B}{a}} rfloor ), and in the second part, the probability is ( P(N leq k) = sum_{m=0}^{k} frac{e^{-lambda} lambda^m}{m!} ).But let me double-check. The cost function is ( C(k) = a k^2 ). If ( k ) is not an integer, say ( k = 2.5 ), then ( C(k) = a (2.5)^2 = 6.25 a ). But in reality, you can't produce half a design, so the cost would be for ( k = 2 ) or ( k = 3 ). Therefore, the maximum ( k ) such that ( a k^2 leq B ) must be an integer. So, ( k = lfloor sqrt{frac{B}{a}} rfloor ).Therefore, in the first part, the maximum number of unique designs is ( k = lfloor sqrt{frac{B}{a}} rfloor ). Then, in the second part, the probability is ( P(N leq k) = sum_{m=0}^{k} frac{e^{-lambda} lambda^m}{m!} ).Alternatively, if we consider ( k ) as a real number, the probability would still be the same because the Poisson distribution is defined only for integer values. So, whether ( k ) is integer or not, the probability is the sum up to the integer part of ( k ).Therefore, the final answers are:1. ( k = lfloor sqrt{frac{B}{a}} rfloor )2. ( P = sum_{m=0}^{lfloor sqrt{frac{B}{a}} rfloor} frac{e^{-lambda} lambda^m}{m!} )But the problem didn't specify whether ( k ) needs to be an integer in the first part, so maybe it's acceptable to leave it as ( k = sqrt{frac{B}{a}} ). However, since the number of designs must be an integer, I think it's more accurate to take the floor.Wait, but in the first part, the problem says "the maximum number of unique designs ( k ) that can be produced in terms of ( a ) and ( B )." So, if ( k ) is not an integer, the business owner can't produce a fraction of a design, so they have to round down. Therefore, ( k = lfloor sqrt{frac{B}{a}} rfloor ).So, to sum up:1. The maximum number of unique designs is ( k = lfloor sqrt{frac{B}{a}} rfloor ).2. The probability that the number of attendees ( N ) is less than or equal to ( k ) is ( P(N leq k) = sum_{m=0}^{k} frac{e^{-lambda} lambda^m}{m!} ).Therefore, the answers are:1. ( k = lfloor sqrt{frac{B}{a}} rfloor )2. ( P = sum_{m=0}^{lfloor sqrt{frac{B}{a}} rfloor} frac{e^{-lambda} lambda^m}{m!} )But let me check if the first part requires an integer or not. The problem says "the maximum number of unique designs ( k ) that can be produced." Since you can't produce a fraction of a design, ( k ) must be an integer. Therefore, the first part's answer is ( k = lfloor sqrt{frac{B}{a}} rfloor ).So, final answers:1. ( k = lfloor sqrt{frac{B}{a}} rfloor )2. ( P = sum_{m=0}^{lfloor sqrt{frac{B}{a}} rfloor} frac{e^{-lambda} lambda^m}{m!} )Alternatively, if the problem allows ( k ) to be a real number, then the first part is ( k = sqrt{frac{B}{a}} ), and the second part is ( P = sum_{m=0}^{lfloor sqrt{frac{B}{a}} rfloor} frac{e^{-lambda} lambda^m}{m!} ).But since the number of designs must be an integer, I think the first part should be ( k = lfloor sqrt{frac{B}{a}} rfloor ).Wait, but the problem says "the cost of producing ( k ) unique designs is given by ( C(k) = a cdot k^2 )", so if ( k ) is not an integer, the cost would still be ( a k^2 ), but in reality, you can't produce a fraction of a design. Therefore, the business owner would have to choose ( k ) as an integer such that ( a k^2 leq B ). Therefore, the maximum integer ( k ) is ( lfloor sqrt{frac{B}{a}} rfloor ).Therefore, the first part's answer is ( k = lfloor sqrt{frac{B}{a}} rfloor ), and the second part's probability is ( P(N leq k) = sum_{m=0}^{k} frac{e^{-lambda} lambda^m}{m!} ).So, to write the answers clearly:1. The maximum number of unique designs is ( boxed{leftlfloor sqrt{dfrac{B}{a}} rightrfloor} ).2. The probability is ( boxed{sum_{m=0}^{leftlfloor sqrt{dfrac{B}{a}} rightrfloor} dfrac{e^{-lambda} lambda^m}{m!}} ).Alternatively, if the problem allows ( k ) to be a real number, the first part is ( sqrt{frac{B}{a}} ), and the second part is the sum up to that value, but since ( k ) must be an integer, I think the floor function is necessary.Wait, but in the first part, the problem says "express the maximum number of unique designs ( k ) that can be produced in terms of ( a ) and ( B )." It doesn't specify whether ( k ) must be an integer, but in reality, it must be. So, to be precise, the answer should include the floor function.Therefore, my final answers are as above with the floor function.</think>

🧠:<think>Alright, so I have this problem about calculating the number of comedic themes in a specific year and determining the year within the Soviet Era when the number of themes was at its maximum. Let me try to break this down step by step.First, the function T(t) is defined piecewise, meaning it has different expressions depending on the value of t, which is the number of years since 1890. The three eras are Pre-Revolution (1890-1917), Soviet Era (1917-1991), and Post-Soviet Era (1991-present). Each era has its own formula for T(t).For part 1, I need to find the number of comedic themes in the year 1937. Let me figure out what value of t corresponds to 1937. Since t is the number of years since 1890, I subtract 1890 from 1937:t = 1937 - 1890 = 47 years.Now, I need to determine which piece of the function to use. The first piece is for 0 ≤ t < 27, which would be from 1890 to 1917. The second piece is for 27 ≤ t < 101, which is from 1917 to 1991. Since 47 is between 27 and 101, I'll use the second piece of the function:T(t) = a₂ t² + b₂ t + c₂Given the constants: a₂ = -0.01, b₂ = 1.5, c₂ = 10.So plugging in t = 47:T(47) = (-0.01)(47)² + (1.5)(47) + 10Let me compute each term step by step.First, calculate 47 squared:47² = 2209Then multiply by -0.01:-0.01 * 2209 = -22.09Next, compute 1.5 * 47:1.5 * 47 = 70.5Now, add all the terms together:-22.09 + 70.5 + 10Let me add -22.09 and 70.5 first:70.5 - 22.09 = 48.41Then add 10:48.41 + 10 = 58.41So, T(47) is approximately 58.41. Since the number of comedic themes should be a whole number, I might round this to 58 or 58.41, depending on the context. The problem doesn't specify, so I'll keep it as 58.41 for now.Wait, but let me double-check my calculations to make sure I didn't make a mistake.Compute 47 squared: 47*47. 40*40=1600, 40*7=280, 7*40=280, 7*7=49. So 1600 + 280 + 280 + 49 = 2209. That's correct.-0.01 * 2209: 2209 * 0.01 is 22.09, so negative is -22.09. Correct.1.5 * 47: 1 * 47 = 47, 0.5 * 47 = 23.5, so total 70.5. Correct.Adding up: -22.09 + 70.5 = 48.41; 48.41 + 10 = 58.41. Yep, that seems right.So, the number of comedic themes in 1937 is approximately 58.41. Since themes can't be a fraction, maybe it's 58 or 58.41. The question says "calculate the number," so perhaps we can leave it as a decimal. I'll go with 58.41.Moving on to part 2: Determine the year within the Soviet Era (1917-1991) when the number of comedic themes reached its maximum.First, let's note that the Soviet Era corresponds to t from 27 to 101 (since 1917 is 27 years after 1890, and 1991 is 101 years after 1890). So, we're looking for the maximum of T(t) in the interval t ∈ [27, 101).But wait, the function T(t) is piecewise, and in the Soviet Era, it's the second piece: T(t) = a₂ t² + b₂ t + c₂.Given that a₂ is -0.01, which is negative, this is a quadratic function opening downward. So, it has a maximum at its vertex.The vertex of a quadratic function at² + bt + c is at t = -b/(2a). Let's compute that.Given a₂ = -0.01, b₂ = 1.5.So, t = -b₂ / (2 * a₂) = -1.5 / (2 * -0.01) = (-1.5) / (-0.02) = 75.So, the maximum occurs at t = 75.But wait, t is the number of years since 1890, so the year is 1890 + 75 = 1965.But hold on, let me make sure that t = 75 is within the Soviet Era. The Soviet Era is from t = 27 (1917) to t = 101 (1991). 75 is between 27 and 101, so yes, it's within the Soviet Era.Therefore, the maximum number of comedic themes occurred in the year 1965.But just to be thorough, let me verify that this is indeed a maximum. Since the coefficient of t² is negative (-0.01), the parabola opens downward, so the vertex is indeed the maximum point.Alternatively, I could take the derivative of T(t) in the Soviet Era and set it to zero to find critical points.T(t) = -0.01 t² + 1.5 t + 10dT/dt = -0.02 t + 1.5Set derivative equal to zero:-0.02 t + 1.5 = 0-0.02 t = -1.5t = (-1.5)/(-0.02) = 75Same result. So, t = 75 is the critical point, and since the function is concave down, it's a maximum.Therefore, the year is 1890 + 75 = 1965.Wait, let me compute 1890 + 75. 1890 + 70 is 1960, plus 5 is 1965. Correct.So, both methods give me the same answer, which is reassuring.But just to make sure, let me compute T(75) and maybe check a couple of nearby points to ensure it's a maximum.Compute T(75):T(75) = -0.01*(75)^2 + 1.5*75 + 1075 squared is 5625.-0.01 * 5625 = -56.251.5 * 75 = 112.5So, T(75) = -56.25 + 112.5 + 10 = (-56.25 + 112.5) + 10 = 56.25 + 10 = 66.25.Now, let's check T(74):T(74) = -0.01*(74)^2 + 1.5*74 + 1074 squared is 5476.-0.01 * 5476 = -54.761.5 * 74 = 111So, T(74) = -54.76 + 111 + 10 = (-54.76 + 111) + 10 = 56.24 + 10 = 66.24.Similarly, T(76):T(76) = -0.01*(76)^2 + 1.5*76 + 1076 squared is 5776.-0.01 * 5776 = -57.761.5 * 76 = 114So, T(76) = -57.76 + 114 + 10 = (-57.76 + 114) + 10 = 56.24 + 10 = 66.24.So, T(75) is 66.25, which is slightly higher than T(74) and T(76), which are both 66.24. That confirms that t = 75 is indeed the maximum point.Therefore, the maximum number of comedic themes occurred in the year 1965.Wait, but just to be absolutely thorough, let me check the endpoints as well, although since it's a downward opening parabola, the maximum should be at the vertex. But just to make sure, let's compute T(27) and T(101) to see if they are lower.Compute T(27):T(27) = -0.01*(27)^2 + 1.5*27 + 1027 squared is 729.-0.01 * 729 = -7.291.5 * 27 = 40.5So, T(27) = -7.29 + 40.5 + 10 = (-7.29 + 40.5) + 10 = 33.21 + 10 = 43.21.Compute T(100) (since t < 101, the maximum t in the Soviet Era is 100):T(100) = -0.01*(100)^2 + 1.5*100 + 10100 squared is 10000.-0.01 * 10000 = -1001.5 * 100 = 150So, T(100) = -100 + 150 + 10 = 60.So, T(27) is 43.21, T(75) is 66.25, and T(100) is 60. So, indeed, the maximum is at t = 75, which is 1965.Therefore, my answers are:1. In 1937, the number of comedic themes is approximately 58.41.2. The maximum number of comedic themes in the Soviet Era occurred in 1965.But wait, the problem says "the number of comedic themes per decade follows a complex pattern." Does that mean T(t) is the number per decade? Or is it per year? The function is defined as T(t), where t is years since 1890. The question says "the number of comedic themes per decade," so perhaps T(t) is the number per decade, but the function is defined for each year t.Wait, let me check the original problem statement.It says: "the number of comedic themes per decade follows a complex pattern described by the function T(t), where t is the number of years since 1890."So, T(t) is the number of themes per decade, but t is in years. So, does that mean T(t) is the number of themes in the decade starting at year t? Or is it the number of themes per year?Wait, that's a bit confusing. Let me parse that again."the number of comedic themes per decade follows a complex pattern described by the function T(t), where t is the number of years since 1890."So, T(t) is the number of themes per decade, but t is the number of years since 1890. So, perhaps T(t) is the number of themes in the decade that starts at year t. But that might not make much sense because t is in years, not decades.Alternatively, maybe T(t) is the number of themes per year, but the pattern is such that it's described by these functions, which are defined per year.Wait, the problem says "the number of comedic themes per decade follows a complex pattern described by the function T(t), where t is the number of years since 1890."Hmm, so T(t) is the number of themes per decade, but t is in years. So, perhaps T(t) is the number of themes in the decade that includes year t? Or is it the number of themes per year, but the function is defined per year?This is a bit ambiguous. Let me think.If T(t) is the number of themes per decade, then t should be in decades. But t is given as years since 1890. So, perhaps T(t) is the number of themes per year, but the pattern is such that it's a function of t, which is years.Wait, the problem says "the number of comedic themes per decade follows a complex pattern described by the function T(t), where t is the number of years since 1890."So, perhaps T(t) is the number of themes per decade, but t is the number of years since 1890. So, for example, t=0 corresponds to 1890, t=10 corresponds to 1890-1900, t=20 corresponds to 1900-1910, etc. So, T(t) would be the number of themes in the decade starting at year t. But that would mean t is in decades, but the problem says t is in years.Wait, this is confusing. Let me try to clarify.If T(t) is the number of themes per decade, and t is in years, then perhaps T(t) is the number of themes in the decade that includes year t. For example, if t=5, it's the decade 1890-1899, t=15 is 1900-1909, etc. So, T(t) would be the number of themes in the decade that includes year t.But in that case, T(t) would be constant over each decade, but the function T(t) is given as a continuous function, with different expressions for different ranges of t.Alternatively, perhaps T(t) is the number of themes per year, but the pattern is such that it's a function of t, which is years since 1890.Wait, the problem says "the number of comedic themes per decade follows a complex pattern described by the function T(t), where t is the number of years since 1890."So, perhaps T(t) is the number of themes per decade, but t is in years. So, for example, T(0) would be the number of themes in the decade 1890-1899, T(10) would be 1900-1909, etc. But then, the function T(t) is defined for t in years, but the output is per decade. That might make sense.But in that case, the function T(t) is defined for t in years, but the output is themes per decade. So, for example, T(0) is themes per decade starting at 1890, T(10) is themes per decade starting at 1900, etc.But in the first part, the question asks for the number of comedic themes in the year 1937. If T(t) is themes per decade, then 1937 would fall into the decade 1930-1939, which corresponds to t=40 (since 1930 is 40 years after 1890). So, T(40) would be the number of themes in the decade 1930-1939. But the question asks for the number in 1937, which is a specific year, not a decade.This is conflicting. So, perhaps I misinterpreted the function. Maybe T(t) is the number of themes per year, with t being the number of years since 1890. That would make more sense because then, for a specific year, you can plug in t and get the number of themes for that year.But the problem says "the number of comedic themes per decade follows a complex pattern described by the function T(t), where t is the number of years since 1890."So, it's a bit ambiguous. But given that in part 1, they ask for the number in the year 1937, which is a specific year, not a decade, it's more likely that T(t) is the number of themes per year, with t being the number of years since 1890.Therefore, in part 1, t=47 corresponds to 1937, and T(47) is the number of themes in that year, which we calculated as approximately 58.41.In part 2, we're looking for the year within the Soviet Era when the number of themes reached its maximum. Since the function in the Soviet Era is quadratic, we found the vertex at t=75, which is 1965.But just to make sure, let me consider the alternative interpretation where T(t) is the number of themes per decade, with t being the number of years since 1890. In that case, t=47 would correspond to the decade 1930-1939, and T(47) would be the number of themes in that decade. But the question asks for the number in 1937, which is a specific year, so that interpretation might not fit.Therefore, I think the correct interpretation is that T(t) is the number of themes per year, with t being the number of years since 1890. So, my initial calculations are correct.So, to recap:1. For 1937, t=47, T(47)=58.41.2. The maximum in the Soviet Era occurs at t=75, which is 1965.I think that's solid.</think>

🧠:<think>Alright, so I've got this problem where I need to help a soldier allocate resources to three different bases. The goal is to maximize the total efficiency of the resource distribution. The functions given are quadratic for each base: ( f_A(x) = 2x^2 + 3x + 5 ), ( f_B(y) = 4y^2 + 2y + 1 ), and ( f_C(z) = 3z^2 + 4z + 2 ). The total resources are R, so ( x + y + z = R ).Hmm, okay. So, I need to maximize the sum of these functions. That is, maximize ( f_A(x) + f_B(y) + f_C(z) ) subject to ( x + y + z = R ). This sounds like an optimization problem with a constraint. I remember from my calculus class that Lagrange multipliers are useful for such problems.Let me recall how Lagrange multipliers work. If I have a function to maximize or minimize subject to a constraint, I can set up a Lagrangian function which incorporates both the original function and the constraint. Then, I take partial derivatives with respect to each variable and the Lagrange multiplier, set them equal to zero, and solve the system of equations.So, let's define the Lagrangian ( mathcal{L} ) as:[mathcal{L}(x, y, z, lambda) = 2x^2 + 3x + 5 + 4y^2 + 2y + 1 + 3z^2 + 4z + 2 - lambda(x + y + z - R)]Simplifying the constants: 5 + 1 + 2 is 8. So,[mathcal{L} = 2x^2 + 3x + 4y^2 + 2y + 3z^2 + 4z + 8 - lambda(x + y + z - R)]Now, take the partial derivatives with respect to x, y, z, and λ.First, partial derivative with respect to x:[frac{partial mathcal{L}}{partial x} = 4x + 3 - lambda = 0]Similarly, partial derivative with respect to y:[frac{partial mathcal{L}}{partial y} = 8y + 2 - lambda = 0]Partial derivative with respect to z:[frac{partial mathcal{L}}{partial z} = 6z + 4 - lambda = 0]And partial derivative with respect to λ gives the constraint:[frac{partial mathcal{L}}{partial lambda} = -(x + y + z - R) = 0 implies x + y + z = R]So, now I have four equations:1. ( 4x + 3 = lambda )2. ( 8y + 2 = lambda )3. ( 6z + 4 = lambda )4. ( x + y + z = R )Now, I can set the first three equations equal to each other since they all equal λ.From equations 1 and 2:( 4x + 3 = 8y + 2 )Simplify:( 4x - 8y = -1 )Divide both sides by 4:( x - 2y = -0.25 )So, equation A: ( x = 2y - 0.25 )From equations 2 and 3:( 8y + 2 = 6z + 4 )Simplify:( 8y - 6z = 2 )Divide both sides by 2:( 4y - 3z = 1 )So, equation B: ( 4y = 3z + 1 ) or ( y = (3z + 1)/4 )Now, from equation A, we have x in terms of y, and from equation B, y in terms of z. Let me substitute equation B into equation A.So, x = 2y - 0.25But y = (3z + 1)/4, so:x = 2*(3z + 1)/4 - 0.25Simplify:x = (6z + 2)/4 - 0.25x = (3z + 1)/2 - 0.25x = (3z + 1)/2 - 1/4To combine the terms, let's write 1/4 as 2/8 and (3z + 1)/2 as (12z + 4)/8:Wait, maybe better to get a common denominator:x = (3z + 1)/2 - 1/4 = (6z + 2 - 1)/4 = (6z + 1)/4So, x = (6z + 1)/4So now, we have x in terms of z, and y in terms of z.So, let's write all variables in terms of z:x = (6z + 1)/4y = (3z + 1)/4z = zNow, plug these into the constraint equation 4: x + y + z = RSo,(6z + 1)/4 + (3z + 1)/4 + z = RCombine the fractions:[(6z + 1) + (3z + 1)] / 4 + z = RSimplify numerator:6z + 1 + 3z + 1 = 9z + 2So,(9z + 2)/4 + z = RConvert z to 4z/4 to combine:(9z + 2 + 4z)/4 = RSimplify numerator:13z + 2 = 4RSo,13z = 4R - 2Therefore,z = (4R - 2)/13Okay, so z is expressed in terms of R.Now, let's find y:y = (3z + 1)/4Substitute z:y = [3*(4R - 2)/13 + 1]/4Simplify numerator:3*(4R - 2) = 12R - 6So,y = (12R - 6)/13 + 1)/4Convert 1 to 13/13:y = (12R - 6 + 13)/13 /4Simplify numerator:12R + 7So,y = (12R + 7)/13 /4 = (12R + 7)/(13*4) = (12R + 7)/52Similarly, x = (6z + 1)/4Substitute z:x = [6*(4R - 2)/13 + 1]/4Simplify numerator:6*(4R - 2) = 24R - 12So,x = (24R - 12)/13 + 1)/4Convert 1 to 13/13:x = (24R - 12 + 13)/13 /4Simplify numerator:24R + 1So,x = (24R + 1)/13 /4 = (24R + 1)/52So, we have:x = (24R + 1)/52y = (12R + 7)/52z = (4R - 2)/13Wait, let me check if these add up to R.Compute x + y + z:(24R + 1)/52 + (12R + 7)/52 + (4R - 2)/13Convert (4R - 2)/13 to 4*(4R - 2)/52 = (16R - 8)/52So,(24R + 1 + 12R + 7 + 16R - 8)/52Combine like terms:24R + 12R + 16R = 52R1 + 7 - 8 = 0So, total is 52R /52 = R. Perfect, that checks out.So, the optimal allocation is:x = (24R + 1)/52y = (12R + 7)/52z = (4R - 2)/13We can simplify these fractions if possible.x = (24R + 1)/52 = (12R + 0.5)/26, but maybe better to leave as is.Similarly, y = (12R + 7)/52, z = (4R - 2)/13.Alternatively, we can write z as (4R - 2)/13 = (4(R - 0.5))/13.But perhaps it's fine as it is.Wait, let me check if these expressions are correct.From earlier steps:We had 13z = 4R - 2 => z = (4R - 2)/13Then, y = (3z + 1)/4 = [3*(4R - 2)/13 + 1]/4Compute numerator:3*(4R - 2) = 12R - 612R - 6 + 13 = 12R +7So, y = (12R +7)/52Similarly, x = (6z +1)/4 = [6*(4R -2)/13 +1]/4Compute numerator:6*(4R -2) =24R -1224R -12 +13 =24R +1So, x = (24R +1)/52Yes, that's correct.So, the optimal allocation is:x = (24R +1)/52y = (12R +7)/52z = (4R -2)/13We can also write these as:x = (24R +1)/52 = (12R + 0.5)/26y = (12R +7)/52z = (4R -2)/13Alternatively, factor numerator and denominator:For x: 24R +1 is prime, so can't simplify.Similarly, y: 12R +7 is prime.z: 4R -2 = 2(2R -1), denominator 13.But perhaps it's fine as is.So, that's the solution for part 1.Now, moving on to part 2.The soldier wants to minimize the evacuation time ( T(x, y, z) = frac{100}{x + 1} + frac{150}{y + 1} + frac{200}{z + 1} ) subject to ( x + y + z = R ).Again, this is an optimization problem with a constraint. So, I can use Lagrange multipliers again.Define the Lagrangian:[mathcal{L}(x, y, z, lambda) = frac{100}{x + 1} + frac{150}{y + 1} + frac{200}{z + 1} - lambda(x + y + z - R)]Take partial derivatives with respect to x, y, z, and λ.Partial derivative with respect to x:[frac{partial mathcal{L}}{partial x} = -frac{100}{(x + 1)^2} - lambda = 0]Similarly, partial derivative with respect to y:[frac{partial mathcal{L}}{partial y} = -frac{150}{(y + 1)^2} - lambda = 0]Partial derivative with respect to z:[frac{partial mathcal{L}}{partial z} = -frac{200}{(z + 1)^2} - lambda = 0]And partial derivative with respect to λ gives the constraint:[x + y + z = R]So, we have four equations:1. ( -frac{100}{(x + 1)^2} - lambda = 0 )2. ( -frac{150}{(y + 1)^2} - lambda = 0 )3. ( -frac{200}{(z + 1)^2} - lambda = 0 )4. ( x + y + z = R )From equations 1, 2, and 3, we can set them equal to each other since they all equal -λ.So,From equation 1 and 2:( -frac{100}{(x + 1)^2} = -frac{150}{(y + 1)^2} )Simplify:( frac{100}{(x + 1)^2} = frac{150}{(y + 1)^2} )Cross-multiplying:100*(y + 1)^2 = 150*(x + 1)^2Divide both sides by 50:2*(y + 1)^2 = 3*(x + 1)^2Take square roots:sqrt(2)*(y + 1) = sqrt(3)*(x + 1)So,( y + 1 = frac{sqrt{3}}{sqrt{2}}(x + 1) )Similarly, let's denote sqrt(3/2) as a constant, say k.Let me compute sqrt(3/2):sqrt(3)/sqrt(2) ≈ 1.2247But let's keep it as sqrt(3/2) for exactness.So,( y + 1 = sqrt{frac{3}{2}}(x + 1) )Similarly, from equations 1 and 3:( -frac{100}{(x + 1)^2} = -frac{200}{(z + 1)^2} )Simplify:( frac{100}{(x + 1)^2} = frac{200}{(z + 1)^2} )Cross-multiplying:100*(z + 1)^2 = 200*(x + 1)^2Divide both sides by 100:(z + 1)^2 = 2*(x + 1)^2Take square roots:z + 1 = sqrt(2)*(x + 1)So,( z + 1 = sqrt{2}(x + 1) )So, now we have expressions for y + 1 and z + 1 in terms of x + 1.Let me write:Let’s denote ( a = x + 1 ), then:( y + 1 = sqrt{frac{3}{2}} a )( z + 1 = sqrt{2} a )So, now, express x, y, z in terms of a.x = a - 1y = sqrt(3/2) a - 1z = sqrt(2) a - 1Now, plug these into the constraint equation x + y + z = R.So,(a - 1) + (sqrt(3/2) a - 1) + (sqrt(2) a - 1) = RCombine like terms:a + sqrt(3/2) a + sqrt(2) a - 3 = RFactor out a:a*(1 + sqrt(3/2) + sqrt(2)) - 3 = RSo,a = (R + 3)/(1 + sqrt(3/2) + sqrt(2))Now, let's compute the denominator:1 + sqrt(3/2) + sqrt(2)Compute sqrt(3/2) ≈ 1.2247sqrt(2) ≈ 1.4142So,1 + 1.2247 + 1.4142 ≈ 3.6389But let's keep it exact for now.So,a = (R + 3)/(1 + sqrt(3/2) + sqrt(2))Therefore,x = a - 1 = [(R + 3)/(1 + sqrt(3/2) + sqrt(2))] - 1Similarly,y = sqrt(3/2) a - 1 = sqrt(3/2)*[(R + 3)/(1 + sqrt(3/2) + sqrt(2))] - 1z = sqrt(2) a - 1 = sqrt(2)*[(R + 3)/(1 + sqrt(3/2) + sqrt(2))] - 1Alternatively, we can factor out the denominator:Let me denote D = 1 + sqrt(3/2) + sqrt(2)So,x = (R + 3)/D - 1 = (R + 3 - D)/DSimilarly,y = sqrt(3/2)*(R + 3)/D - 1 = [sqrt(3/2)(R + 3) - D]/Dz = sqrt(2)*(R + 3)/D - 1 = [sqrt(2)(R + 3) - D]/DBut perhaps it's better to leave it as:x = (R + 3)/D - 1y = sqrt(3/2)*(R + 3)/D - 1z = sqrt(2)*(R + 3)/D - 1Where D = 1 + sqrt(3/2) + sqrt(2)Alternatively, we can rationalize or simplify further, but it might complicate things.Alternatively, we can express sqrt(3/2) as sqrt(6)/2, and sqrt(2) as sqrt(2).So,D = 1 + sqrt(6)/2 + sqrt(2)So,D = 1 + (sqrt(6) + 2 sqrt(2))/2But perhaps not necessary.Alternatively, we can write all terms over a common denominator.But maybe it's fine as is.So, the optimal allocation is:x = (R + 3)/D - 1y = sqrt(3/2)*(R + 3)/D - 1z = sqrt(2)*(R + 3)/D - 1Where D = 1 + sqrt(3/2) + sqrt(2)Alternatively, we can write:x = (R + 3 - D)/DSimilarly for y and z.But perhaps it's better to leave it in terms of a.Wait, let me check if this makes sense.We have x, y, z expressed in terms of a, which is a function of R.But let me verify the constraint.Compute x + y + z:x + y + z = (a -1) + (sqrt(3/2)a -1) + (sqrt(2)a -1) = a(1 + sqrt(3/2) + sqrt(2)) - 3Which equals R, as per the constraint.So, that's correct.Therefore, the optimal allocation is as above.Alternatively, we can express the fractions in terms of R.But perhaps it's better to leave it in terms of D.So, summarizing:For part 1, the optimal allocation is:x = (24R +1)/52y = (12R +7)/52z = (4R -2)/13For part 2, the optimal allocation is:x = (R + 3)/D - 1y = sqrt(3/2)*(R + 3)/D - 1z = sqrt(2)*(R + 3)/D - 1Where D = 1 + sqrt(3/2) + sqrt(2)Alternatively, we can write D as 1 + sqrt(6)/2 + sqrt(2), but it's the same thing.So, that's the solution.</think>