🧠:<think>Okay, so I have this problem about optimizing a cryptocurrency portfolio. There are two parts. Let me try to tackle them one by one.Starting with part 1: I need to find the time ( t ) within the first year (so between 0 and 12 months) where the combined monthly return from both Crypto A and Crypto B is maximized. The returns are given by functions ( R_A(t) = 5t^2 - 3t + 2 ) and ( R_B(t) = 4t^3 - 10t + 6 ). So, the combined return is ( R_A(t) + R_B(t) ).First, I should write out the combined return function:( R(t) = R_A(t) + R_B(t) = (5t^2 - 3t + 2) + (4t^3 - 10t + 6) )Let me simplify that:Combine like terms:- The ( t^3 ) term: 4t^3- The ( t^2 ) term: 5t^2- The ( t ) terms: -3t -10t = -13t- The constants: 2 + 6 = 8So, ( R(t) = 4t^3 + 5t^2 - 13t + 8 )Now, to find the maximum of this function on the interval [0, 12], I need to find its critical points and evaluate the function at those points as well as at the endpoints.Critical points occur where the derivative is zero or undefined. Since this is a polynomial, the derivative will be defined everywhere, so just find where the derivative is zero.Let me compute the derivative ( R'(t) ):( R'(t) = d/dt [4t^3 + 5t^2 - 13t + 8] = 12t^2 + 10t - 13 )So, set ( R'(t) = 0 ):( 12t^2 + 10t - 13 = 0 )This is a quadratic equation. Let me use the quadratic formula to solve for ( t ):( t = [-b pm sqrt{b^2 - 4ac}]/(2a) )Here, ( a = 12 ), ( b = 10 ), ( c = -13 )Compute discriminant:( D = b^2 - 4ac = 10^2 - 4*12*(-13) = 100 + 624 = 724 )So,( t = [-10 pm sqrt{724}]/24 )Compute sqrt(724). Let me see, 26^2 is 676, 27^2 is 729, so sqrt(724) is approximately 26.9.So,First solution:( t = [-10 + 26.9]/24 ≈ 16.9 /24 ≈ 0.704 ) monthsSecond solution:( t = [-10 - 26.9]/24 ≈ (-36.9)/24 ≈ -1.5375 ) monthsBut since time ( t ) can't be negative, we discard the negative solution.So, the critical point is at approximately 0.704 months.Now, we need to check the value of ( R(t) ) at this critical point and at the endpoints t=0 and t=12.Let me compute R(t) at t=0, t≈0.704, and t=12.First, t=0:( R(0) = 4*0 + 5*0 -13*0 +8 = 8 )Next, t≈0.704:Let me compute each term:4t^3: 4*(0.704)^3 ≈ 4*(0.349) ≈ 1.3965t^2: 5*(0.704)^2 ≈ 5*(0.495) ≈ 2.475-13t: -13*(0.704) ≈ -9.152+8: 8Add them up:1.396 + 2.475 = 3.8713.871 - 9.152 = -5.281-5.281 + 8 = 2.719So, R(0.704) ≈ 2.719Wait, that seems lower than R(0). Hmm, that can't be right. Maybe I made a calculation error.Wait, let me recalculate R(0.704):Compute each term step by step:t = 0.704t^3 = (0.704)^3 ≈ 0.704 * 0.704 = 0.495; 0.495 * 0.704 ≈ 0.348So, 4t^3 ≈ 4*0.348 ≈ 1.392t^2 = (0.704)^2 ≈ 0.4955t^2 ≈ 5*0.495 ≈ 2.475-13t ≈ -13*0.704 ≈ -9.152+8So, adding up:1.392 + 2.475 = 3.8673.867 - 9.152 = -5.285-5.285 + 8 = 2.715So, R(0.704) ≈ 2.715Wait, that's still lower than R(0)=8. That seems odd because if the derivative is zero at t≈0.7, and the function is increasing before that point, but if R(t) is lower at t≈0.7 than at t=0, that would mean it's a minimum, not a maximum.Wait, perhaps I made a mistake in computing the derivative or the function.Wait, let me double-check the derivative:R(t) = 4t^3 + 5t^2 -13t +8Derivative is 12t^2 + 10t -13. That's correct.So, setting derivative to zero, we found t≈0.704.But R(t) at that point is lower than at t=0. So, that suggests that the function is decreasing at t=0.704, but since it's a critical point, maybe it's a local minimum.Wait, let's check the second derivative to determine concavity.Second derivative R''(t) = 24t + 10At t≈0.704, R''(t) ≈24*(0.704) +10 ≈16.896 +10≈26.896>0So, since the second derivative is positive, it's a local minimum.Wait, so that means the function has a local minimum at t≈0.704. So, the maximum must be at one of the endpoints.So, let's compute R(t) at t=12.Compute R(12):4*(12)^3 +5*(12)^2 -13*(12) +8Compute each term:12^3=1728, so 4*1728=691212^2=144, so 5*144=720-13*12= -156+8So, total R(12)=6912 +720 -156 +8Compute:6912 +720=76327632 -156=74767476 +8=7484So, R(12)=7484Compare to R(0)=8. So, clearly, the function is increasing from t=0 to t=12, with a dip at t≈0.704, but the maximum is at t=12.Wait, but that seems counterintuitive because the function is a cubic, which can have different behaviors. Let me confirm.Wait, the function is R(t)=4t^3 +5t^2 -13t +8As t approaches infinity, 4t^3 dominates, so it goes to infinity. So, as t increases, the function tends to infinity. So, on the interval [0,12], the maximum should be at t=12.But wait, the derivative at t=12 is R'(12)=12*(12)^2 +10*(12) -13=12*144 +120 -13=1728 +120=1848 -13=1835>0So, the function is increasing at t=12, meaning that beyond t=12, it would keep increasing. But since we're limited to t=12, the maximum is at t=12.But wait, let me check the behavior between t=0 and t=12.At t=0, R(t)=8At t=0.704, R(t)=~2.715At t=12, R(t)=7484So, the function decreases from t=0 to t≈0.704, reaching a local minimum, then increases beyond that.So, the maximum on [0,12] is at t=12.Therefore, the maximum combined return occurs at t=12 months, with R(12)=7484.Wait, but 7484 seems like a very high return. Let me check the calculations again.Compute R(12):4*(12)^3 =4*1728=69125*(12)^2=5*144=720-13*(12)= -156+8So, 6912 +720=76327632 -156=74767476 +8=7484Yes, that's correct.So, the maximum combined return is 7484 at t=12.Wait, but that seems extremely high. Is that realistic? Maybe in the context of cryptocurrencies, which can have high volatility, but still, 7484% return in a month? That seems unrealistic, but perhaps the functions are just mathematical models.So, perhaps the answer is t=12, with R=7484.Wait, but let me check if there's a higher value somewhere else in the interval.Wait, the function is a cubic, so it can have a local maximum and minimum. But in this case, the derivative only had one real positive root at t≈0.704, which was a local minimum.So, the function is decreasing from t=0 to t≈0.704, then increasing beyond that. So, on [0,12], the maximum is at t=12.Therefore, the answer to part 1 is t=12 months, with a combined return of 7484.Wait, but let me check if I made a mistake in the combined function.Original functions:R_A(t)=5t^2 -3t +2R_B(t)=4t^3 -10t +6So, R(t)=4t^3 +5t^2 -13t +8Yes, that's correct.So, moving on to part 2.The accountant wants to minimize risk by adjusting the proportion x in Crypto A and (1-x) in Crypto B. The risk function is given by:σ(x)=0.2x² +0.1(1−x)² +0.3x(1−x)We need to find x that minimizes σ(x).First, let me expand σ(x):σ(x)=0.2x² +0.1(1 - 2x +x²) +0.3x(1 -x)Compute each term:0.2x²0.1*(1 -2x +x²)=0.1 -0.2x +0.1x²0.3x*(1 -x)=0.3x -0.3x²Now, combine all terms:0.2x² +0.1 -0.2x +0.1x² +0.3x -0.3x²Combine like terms:x² terms: 0.2x² +0.1x² -0.3x² = (0.2 +0.1 -0.3)x²=0x²=0x terms: -0.2x +0.3x=0.1xconstants: 0.1So, σ(x)=0x² +0.1x +0.1=0.1x +0.1Wait, that can't be right. Let me check the expansion again.Wait, 0.2x² +0.1*(1 -2x +x²) +0.3x*(1 -x)=0.2x² +0.1 -0.2x +0.1x² +0.3x -0.3x²Now, combine x² terms:0.2x² +0.1x² -0.3x² = (0.2 +0.1 -0.3)x²=0x²x terms:-0.2x +0.3x=0.1xconstants: 0.1So, σ(x)=0.1x +0.1Wait, that's a linear function. So, to minimize σ(x)=0.1x +0.1, which is a straight line with a positive slope, the minimum occurs at the smallest possible x.But x is the proportion of investment in Crypto A, so x is between 0 and 1.Therefore, the minimum occurs at x=0.But that seems odd because the risk function was given as a quadratic, but after expansion, it's linear.Wait, perhaps I made a mistake in expanding.Let me re-express σ(x):σ(x)=0.2x² +0.1(1−x)² +0.3x(1−x)Let me compute each term step by step.First term: 0.2x²Second term: 0.1*(1 -x)^2=0.1*(1 -2x +x²)=0.1 -0.2x +0.1x²Third term: 0.3x*(1 -x)=0.3x -0.3x²Now, add them all together:0.2x² + (0.1 -0.2x +0.1x²) + (0.3x -0.3x²)Combine like terms:x²: 0.2x² +0.1x² -0.3x²= (0.2+0.1-0.3)x²=0x²x terms: -0.2x +0.3x=0.1xconstants: 0.1So, indeed, σ(x)=0.1x +0.1So, it's a linear function with slope 0.1, which is positive. Therefore, the minimum occurs at x=0, giving σ(x)=0.1.Wait, but that seems strange because usually, portfolio risk is a convex function, and the minimum would be somewhere inside the interval. But in this case, the risk function simplifies to a linear function, so the minimum is at x=0.Therefore, the minimal risk occurs when x=0, i.e., all investment in Crypto B.But let me double-check the expansion.Wait, 0.2x² +0.1(1−x)² +0.3x(1−x)=0.2x² +0.1(1 -2x +x²) +0.3x -0.3x²=0.2x² +0.1 -0.2x +0.1x² +0.3x -0.3x²Now, combining x² terms:0.2x² +0.1x² -0.3x²=0x²x terms:-0.2x +0.3x=0.1xconstants: 0.1Yes, correct. So, σ(x)=0.1x +0.1Therefore, the minimal risk is at x=0, giving σ=0.1.Wait, but that seems counterintuitive because usually, diversification reduces risk, but in this case, the risk function is linear, so the minimal risk is achieved by putting all in the asset with lower risk. Wait, but in this case, the risk function is linear, so the minimal risk is at x=0.Alternatively, perhaps the problem is designed such that the risk function is linear, so the answer is x=0.Alternatively, maybe I made a mistake in the problem statement.Wait, let me check the problem statement again.The risk function is:σ(x)=0.2x² +0.1(1−x)² +0.3x(1−x)Yes, that's correct.So, expanding it, we get σ(x)=0.1x +0.1, which is linear.Therefore, the minimal value is at x=0.So, the answer is x=0.But that seems odd, but mathematically, that's correct.Alternatively, perhaps the problem intended for the risk function to be quadratic, but due to coefficients, it simplified to linear.So, perhaps the answer is x=0.Alternatively, maybe I made a mistake in the expansion.Wait, let me compute σ(x) at x=0 and x=1.At x=0:σ(0)=0.2*0 +0.1*(1)^2 +0.3*0*(1)=0 +0.1 +0=0.1At x=1:σ(1)=0.2*1 +0.1*0 +0.3*1*0=0.2 +0 +0=0.2So, σ(x) increases from 0.1 at x=0 to 0.2 at x=1, with a linear function in between.Therefore, the minimal risk is at x=0.So, the answer is x=0.Wait, but that seems to suggest that putting all in Crypto B gives the minimal risk, which is 0.1, while any investment in Crypto A increases the risk.So, the minimal risk is achieved when x=0.Therefore, the answer to part 2 is x=0.Wait, but let me think again. Maybe the problem intended for the risk function to be convex, but due to the coefficients, it's linear. So, perhaps the answer is x=0.Alternatively, perhaps I made a mistake in the expansion.Wait, let me recompute σ(x):σ(x)=0.2x² +0.1(1−x)² +0.3x(1−x)Compute each term:0.2x²0.1*(1 -2x +x²)=0.1 -0.2x +0.1x²0.3x*(1 -x)=0.3x -0.3x²Now, sum all terms:0.2x² +0.1 -0.2x +0.1x² +0.3x -0.3x²Now, combine like terms:x²: 0.2 +0.1 -0.3=0x: -0.2 +0.3=0.1constants:0.1So, σ(x)=0.1x +0.1Yes, that's correct.Therefore, the minimal risk is at x=0, with σ=0.1.So, the answers are:1. t=12 months, R=74842. x=0But let me just confirm part 1 again because 7484 seems extremely high.Wait, R(t)=4t^3 +5t^2 -13t +8At t=12, R(12)=4*(12)^3 +5*(12)^2 -13*12 +8=4*1728 +5*144 -156 +8=6912 +720 -156 +8=6912+720=7632; 7632-156=7476; 7476+8=7484.Yes, that's correct.So, the maximum combined return is 7484 at t=12.Therefore, the answers are:1. t=12, R=74842. x=0</think>

🧠:<think>Alright, so I have this problem about a film student inspired by the TV show *House*. They want to analyze the structure of the episodes to plan their film. There are two sub-problems here. Let me tackle them one by one.Sub-problem 1:The student notices that each episode follows an exponential decay model for the frequency of new clues. The number of new clues decreases by a factor of 1/3 every 15 minutes. The initial number of clues in the first 15 minutes is 9. I need to express the number of new clues ( C(t) ) as a function of time ( t ) in minutes and then calculate the total number of new clues in a 45-minute episode.Okay, exponential decay models are of the form ( C(t) = C_0 cdot e^{-kt} ), where ( C_0 ) is the initial amount, ( k ) is the decay constant, and ( t ) is time. But here, it's given that the number decreases by a factor of 1/3 every 15 minutes. So, maybe it's better to model it using a base that's 1/3 and time in 15-minute intervals.Let me think. If every 15 minutes, the number of clues is multiplied by 1/3, then after ( n ) intervals of 15 minutes, the number of clues would be ( C(n) = 9 cdot (1/3)^n ).But since ( t ) is in minutes, I need to express ( n ) in terms of ( t ). Since each interval is 15 minutes, ( n = t / 15 ). Therefore, the function becomes:( C(t) = 9 cdot (1/3)^{t/15} ).Alternatively, this can be written using the exponential function with base ( e ). Let me recall that ( a^b = e^{b ln a} ). So, ( (1/3)^{t/15} = e^{(t/15) ln(1/3)} = e^{-(t/15) ln 3} ). Therefore, another form is:( C(t) = 9 cdot e^{-(ln 3 / 15) t} ).But since the problem mentions an exponential decay model, either form should be acceptable. However, since the decay factor is given per 15 minutes, the first form might be more straightforward.Now, to find the total number of new clues in a 45-minute episode, I need to sum the clues over each 15-minute interval. Since it's a decay, each subsequent 15-minute interval contributes fewer clues.So, let's break it down:- First 15 minutes: 9 clues.- Second 15 minutes (15-30): 9 * (1/3) = 3 clues.- Third 15 minutes (30-45): 3 * (1/3) = 1 clue.Therefore, total clues = 9 + 3 + 1 = 13.Wait, but is this the correct approach? Because exponential decay can also be continuous, but in this case, the problem states that the number decreases by a factor every 15 minutes, which suggests it's discrete, i.e., happening at each interval. So, yes, summing the clues at each interval makes sense.Alternatively, if it were continuous, we might integrate the function over 45 minutes. But since the decay is given per 15-minute intervals, I think the discrete approach is appropriate here.So, total clues in 45 minutes: 9 + 3 + 1 = 13.Sub-problem 2:The student wants to create a scene with multiple hypotheses, 5 potential diagnoses, each with a probability of being correct: 0.10, 0.15, 0.20, 0.25, and 0.30. The probabilities are independent. They need the expected number of correct diagnoses in a 5-episode sequence and the variance.Wait, hold on. Is it 5 potential diagnoses per episode, and they're looking at 5 episodes? Or is it 5 diagnoses across 5 episodes? The wording says "in a 5-episode sequence," so I think it's 5 episodes, each with these 5 diagnoses.But the problem says "the expected number of correct diagnoses in a 5-episode sequence." So, per episode, they have 5 diagnoses, each with their own probability of being correct. So, over 5 episodes, how many correct diagnoses are expected?Wait, but each episode is independent? Or is it the same set of diagnoses across episodes? Hmm.Wait, the problem says "the expected number of correct diagnoses in a 5-episode sequence." So, perhaps each episode has 5 diagnoses, each with their own probability, and we need the total expected number across all 5 episodes.But let me parse the problem again:"Assuming the probabilities are independent, what is the expected number of correct diagnoses in a 5-episode sequence? Additionally, calculate the variance of the number of correct diagnoses across the 5 episodes."Hmm, so maybe each episode has 5 diagnoses, each with probabilities 0.10, 0.15, 0.20, 0.25, 0.30. So, per episode, the expected number of correct diagnoses is the sum of the probabilities.Therefore, for one episode, expected correct diagnoses = 0.10 + 0.15 + 0.20 + 0.25 + 0.30 = 1.0.Then, over 5 episodes, since each episode is independent, the expected total is 5 * 1.0 = 5.0.Similarly, the variance for one episode: since each diagnosis is a Bernoulli trial with probability p_i, the variance for one episode is the sum of p_i*(1 - p_i).So, for one episode:Variance = 0.10*(1 - 0.10) + 0.15*(1 - 0.15) + 0.20*(1 - 0.20) + 0.25*(1 - 0.25) + 0.30*(1 - 0.30)Calculating each term:0.10*0.90 = 0.090.15*0.85 = 0.12750.20*0.80 = 0.160.25*0.75 = 0.18750.30*0.70 = 0.21Adding them up: 0.09 + 0.1275 = 0.2175; 0.2175 + 0.16 = 0.3775; 0.3775 + 0.1875 = 0.565; 0.565 + 0.21 = 0.775.So, variance per episode is 0.775.Therefore, over 5 episodes, since the episodes are independent, the total variance is 5 * 0.775 = 3.875.Hence, expected number is 5, variance is 3.875.Wait, but let me double-check if I interpreted the problem correctly. The problem says "the expected number of correct diagnoses in a 5-episode sequence." So, does that mean 5 episodes, each with 5 diagnoses, and we're summing all correct ones? Or is it 5 diagnoses across 5 episodes?Wait, the problem says: "5 potential diagnoses" and "each diagnosis has a probability of being correct based on prior episodes' data: 0.10, 0.15, 0.20, 0.25, and 0.30, respectively." So, each diagnosis has its own probability, and they are independent.So, if it's a 5-episode sequence, perhaps each episode has 5 diagnoses, each with their respective probabilities. So, per episode, the expected number is 1.0, as calculated. Then, over 5 episodes, it's 5.0.Alternatively, if it's 5 diagnoses across 5 episodes, meaning each episode has one diagnosis, but that doesn't make much sense because the probabilities are given per diagnosis, not per episode.Wait, the problem says: "5 potential diagnoses" and each has a probability. So, perhaps in each episode, they consider 5 diagnoses, each with their own probability. So, over 5 episodes, each episode has 5 diagnoses, each with their own probability, and we need the total expected number of correct diagnoses across all 5 episodes.Therefore, per episode, expected correct is 1.0, so 5 episodes would be 5.0.Similarly, variance per episode is 0.775, so total variance is 5 * 0.775 = 3.875.Alternatively, if the 5 probabilities are for 5 episodes, each with one diagnosis, then the expected number would be the sum of the probabilities: 0.10 + 0.15 + 0.20 + 0.25 + 0.30 = 1.0, and variance would be the sum of p_i*(1 - p_i) = 0.775.But the problem says "5 potential diagnoses" and "in a 5-episode sequence." So, I think it's 5 diagnoses per episode, over 5 episodes.Wait, the problem says: "the expected number of correct diagnoses in a 5-episode sequence." So, perhaps each episode has 5 diagnoses, each with their own probability, and we need the total expected number across all 5 episodes.So, per episode, expected correct is 1.0, so 5 episodes: 5.0.Variance per episode is 0.775, so total variance is 5 * 0.775 = 3.875.Alternatively, if it's 5 diagnoses across 5 episodes, meaning each episode has one diagnosis, but then the probabilities are given per diagnosis, not per episode.Wait, the problem says: "5 potential diagnoses" and "each diagnosis has a probability of being correct based on prior episodes' data: 0.10, 0.15, 0.20, 0.25, and 0.30, respectively."So, each diagnosis (there are 5 of them) has its own probability. So, if they are considering these 5 diagnoses across 5 episodes, perhaps each episode has one diagnosis, each with their respective probability.But the problem says "in a 5-episode sequence." So, maybe each episode has all 5 diagnoses, each with their own probability, and we need the total expected number of correct diagnoses across all 5 episodes.Wait, that would mean per episode, they have 5 diagnoses, each with their own probability, so per episode, expected correct is 1.0, and over 5 episodes, it's 5.0.Alternatively, if each episode has one diagnosis, but the 5 probabilities are for 5 different episodes, then the expected number would be the sum of the probabilities, which is 1.0, and variance is 0.775.But the problem says "5 potential diagnoses," which suggests that in each episode, there are 5 diagnoses, each with their own probability. So, per episode, expected correct is 1.0, over 5 episodes, it's 5.0.I think that's the correct interpretation.So, to recap:- Each episode has 5 diagnoses, each with probabilities 0.10, 0.15, 0.20, 0.25, 0.30.- Therefore, per episode, expected correct = 0.10 + 0.15 + 0.20 + 0.25 + 0.30 = 1.0.- Over 5 episodes, expected total correct = 5 * 1.0 = 5.0.- Variance per episode: sum of p_i*(1 - p_i) = 0.775.- Total variance over 5 episodes: 5 * 0.775 = 3.875.Therefore, the expected number is 5, variance is 3.875.I think that's solid.Final AnswerSub-problem 1: The total number of new clues in a 45-minute episode is boxed{13}.Sub-problem 2: The expected number of correct diagnoses is boxed{5} and the variance is boxed{3.875}.</think>

🧠:<think>Alright, so Alex is looking at setting up these student-run kiosks on campus. There are two main parts to this problem: first, figuring out how many units each kiosk needs to sell daily to maximize profit, and then determining how many kiosks can be set up without exceeding a certain student engagement level. Let me tackle each part step by step.Starting with the first part: maximizing profit for a single kiosk. I remember that profit is calculated as revenue minus cost. So, if we have the revenue function ( R(x) = 500x - 20x^2 ) and the cost function ( C(x) = 200 + 100x ), then the profit function ( P(x) ) should be ( R(x) - C(x) ).Let me write that out:( P(x) = R(x) - C(x) = (500x - 20x^2) - (200 + 100x) )Simplifying that:( P(x) = 500x - 20x^2 - 200 - 100x )Combine like terms:( P(x) = (500x - 100x) - 20x^2 - 200 )( P(x) = 400x - 20x^2 - 200 )Hmm, so that's a quadratic function in terms of x. Since the coefficient of ( x^2 ) is negative (-20), the parabola opens downward, which means the vertex is the maximum point. To find the maximum profit, we need to find the vertex of this parabola.The general form of a quadratic is ( ax^2 + bx + c ), so in this case, ( a = -20 ) and ( b = 400 ). The x-coordinate of the vertex is given by ( -b/(2a) ).Calculating that:( x = -400 / (2 * -20) )( x = -400 / (-40) )( x = 10 )So, selling 10 units per day will maximize the profit. Now, let's find the maximum profit by plugging x = 10 back into the profit function.( P(10) = 400*10 - 20*(10)^2 - 200 )Calculating each term:400*10 = 400020*(10)^2 = 20*100 = 2000So,( P(10) = 4000 - 2000 - 200 = 4000 - 2200 = 1800 )Wait, that seems high. Let me double-check my calculations.Wait, 400*10 is indeed 4000. 20*(10)^2 is 2000. So 4000 - 2000 is 2000, minus 200 is 1800. Hmm, okay, that seems correct. So the maximum profit is 1800 per kiosk per day when selling 10 units.Wait, but let me think again. The revenue function is 500x - 20x². At x=10, R(10)=500*10 - 20*100=5000 - 2000=3000.The cost function is 200 + 100x. At x=10, C(10)=200 + 1000=1200.So profit is 3000 - 1200=1800. Yeah, that's correct. So, 10 units, 1800 profit. Okay, that seems solid.Moving on to the second part: determining the maximum number of kiosks ( n ) such that the engagement level ( E(n) = 100log(n+1) + 2n^2 ) stays below 250.So, we need to solve for ( n ) in the inequality:( 100log(n+1) + 2n^2 < 250 )Hmm, this is a bit trickier because it's a transcendental equation—it has both a logarithmic term and a quadratic term. These types of equations usually can't be solved algebraically, so we might need to use numerical methods or trial and error to find the maximum integer ( n ) that satisfies the inequality.Let me try plugging in some integer values for ( n ) and see where ( E(n) ) crosses 250.Starting with n=0:( E(0) = 100log(1) + 0 = 0 + 0 = 0 ) which is way below 250.n=1:( E(1) = 100log(2) + 2(1)^2 ≈ 100*0.6931 + 2 ≈ 69.31 + 2 = 71.31 )Still below 250.n=2:( E(2) = 100log(3) + 2(4) ≈ 100*1.0986 + 8 ≈ 109.86 + 8 = 117.86 )n=3:( E(3) = 100log(4) + 2(9) ≈ 100*1.3863 + 18 ≈ 138.63 + 18 = 156.63 )n=4:( E(4) = 100log(5) + 2(16) ≈ 100*1.6094 + 32 ≈ 160.94 + 32 = 192.94 )n=5:( E(5) = 100log(6) + 2(25) ≈ 100*1.7918 + 50 ≈ 179.18 + 50 = 229.18 )Still below 250.n=6:( E(6) = 100log(7) + 2(36) ≈ 100*1.9459 + 72 ≈ 194.59 + 72 = 266.59 )Oh, that's above 250. So, n=6 gives E(n)=266.59, which is over the threshold.So, n=5 gives E(n)=229.18, which is under 250. Let me check n=5.5 just to see how close we can get.But since n must be an integer (you can't have half a kiosk), n=5 is the maximum. But wait, let me check n=5. Is there a decimal value between 5 and 6 where E(n)=250?Let me set up the equation:( 100log(n+1) + 2n^2 = 250 )Let me denote ( f(n) = 100log(n+1) + 2n^2 ). We know f(5)=229.18 and f(6)=266.59. So, somewhere between 5 and 6, f(n) crosses 250.To find the exact point, we can use linear approximation or more precise methods, but since n must be an integer, the maximum n is 5 because 6 exceeds the threshold. Therefore, the maximum number of kiosks is 5.But just to be thorough, let me try n=5. Let's see:At n=5, E(n)=229.18.At n=6, it's 266.59.The difference between n=5 and n=6 in E(n) is about 37.41.We need to find the n where E(n)=250. So, 250 - 229.18 = 20.82.So, 20.82 / 37.41 ≈ 0.556. So, approximately 0.556 of the way from n=5 to n=6.So, n≈5 + 0.556≈5.556.But since n must be an integer, 5.556 is approximately 5.56, which is still less than 6, but since we can't have a fraction of a kiosk, the maximum integer n is 5.Therefore, the maximum number of kiosks without exceeding the engagement threshold is 5.Wait, but let me check n=5.5 just to see:E(5.5)=100*log(6.5) + 2*(5.5)^2.Calculate log(6.5). Let me recall that log(6)=1.7918, log(7)=1.9459. So, log(6.5) is approximately (1.7918 + 1.9459)/2≈1.8689.So, 100*1.8689≈186.89.2*(5.5)^2=2*30.25=60.5.So, E(5.5)=186.89 + 60.5≈247.39, which is still below 250.So, at n=5.5, E(n)=247.39.Then, n=5.6:log(6.6). Let me estimate log(6.6). Since log(6)=1.7918, log(7)=1.9459. The difference is 0.1541 over 1 unit. So, 0.6 beyond 6 is 0.6*0.1541≈0.0925. So, log(6.6)≈1.7918 + 0.0925≈1.8843.100*log(6.6)=188.43.2*(5.6)^2=2*31.36=62.72.So, E(5.6)=188.43 + 62.72≈251.15.That's just above 250. So, somewhere between 5.5 and 5.6, E(n) crosses 250.But since n must be an integer, the maximum n is 5.Therefore, the maximum number of kiosks is 5.Wait, but let me confirm with n=5.55:log(6.55). Hmm, log(6.5)=1.8689, log(6.6)=1.8843. So, 6.55 is halfway between 6.5 and 6.6, so log(6.55)≈(1.8689 + 1.8843)/2≈1.8766.100*log(6.55)=187.66.2*(5.55)^2=2*(30.8025)=61.605.So, E(5.55)=187.66 + 61.605≈249.265, which is still below 250.n=5.56:log(6.56). Let's see, from 6.55 to 6.6, log increases by about 0.0154 per 0.05 increase in n. Wait, actually, the derivative of log(n+1) is 1/(n+1). So, at n=5.55, derivative is 1/(6.55)≈0.1526 per unit increase in n. So, for a small delta, the change in log(n+1) is approximately delta * 0.1526.So, if we go from n=5.55 to n=5.56, delta=0.01, so log increases by approximately 0.01*0.1526≈0.001526.So, log(6.56)≈log(6.55) + 0.001526≈1.8766 + 0.0015≈1.8781.100*log(6.56)=187.81.2*(5.56)^2=2*(30.9136)=61.8272.So, E(5.56)=187.81 + 61.8272≈249.6372.Still below 250.n=5.57:log(6.57)=log(6.56) + 0.01*0.1526≈1.8781 + 0.0015≈1.8796.100*log(6.57)=187.96.2*(5.57)^2=2*(31.0249)=62.0498.E(5.57)=187.96 + 62.0498≈250.0098.Oh, that's just over 250. So, at n≈5.57, E(n)=250.0098.Therefore, the exact value is around 5.57, but since n must be an integer, the maximum n is 5.So, summarizing:1. For each kiosk, selling 10 units daily maximizes profit, yielding 1800.2. The maximum number of kiosks without exceeding the engagement level of 250 is 5.I think that's it. Let me just recap to make sure I didn't miss anything.For part 1, profit is revenue minus cost. We set up the profit function, found it was a quadratic, calculated the vertex to find the maximum. That gave us x=10, profit=1800.For part 2, we had an engagement function that's a combination of log and quadratic. We tested integer values and found that at n=5, E(n)=229.18, and at n=6, it's 266.59. Since 250 is between these, we approximated where E(n)=250 and found it's around 5.57, but since n must be integer, 5 is the maximum.Yes, that seems correct.</think>

🧠:<think>Okay, so I have two problems to solve here. Let me start with the first one about the Sierpinski triangle.1. The painter is creating a mural on a wall that's 24 meters long and 18 meters high. They're using a grid system with 1-meter squares. They want to incorporate a fractal pattern inspired by the Sierpinski triangle. The question is asking how many 1-meter triangular sections are removed after 5 iterations of the Sierpinski process.Alright, so first, I need to recall what a Sierpinski triangle is. It's a fractal created by recursively removing triangles. Starting with a large equilateral triangle, you divide it into four smaller equilateral triangles and remove the central one. Then you repeat this process for each of the remaining three triangles, and so on.But wait, the wall is a rectangle, not an equilateral triangle. So the first thing is to figure out the largest equilateral triangle that can fit into a 24x18 meter rectangle.Hmm, an equilateral triangle has all sides equal and all angles 60 degrees. The height of an equilateral triangle is given by ( h = frac{sqrt{3}}{2} times text{side length} ).So, if we're fitting an equilateral triangle into a rectangle, the height of the triangle can't exceed the height of the rectangle, which is 18 meters, and the base can't exceed the length of the rectangle, which is 24 meters.So, let's calculate the maximum possible side length of the equilateral triangle.If we consider the height constraint: ( h = frac{sqrt{3}}{2} s leq 18 ). Solving for s: ( s leq frac{18 times 2}{sqrt{3}} = frac{36}{sqrt{3}} = 12sqrt{3} approx 20.78 ) meters.But the base of the triangle can't exceed 24 meters. So, 12√3 is approximately 20.78, which is less than 24, so the limiting factor is the height. Therefore, the largest equilateral triangle that can fit into the rectangle has a side length of 12√3 meters.Wait, but the grid is 1-meter squares. So each triangular section is 1-meter on each side? Or is it a 1-meter square divided into triangles?Wait, the problem says "1-meter triangular sections." So, each section is a small equilateral triangle with sides of 1 meter. So, the entire wall is divided into 1x1 meter squares, but the painter is using triangular sections. So, each square can be divided into two triangles, perhaps?But maybe not. Let me think. If the grid is 1x1 meter squares, and the fractal is made up of 1-meter triangles, then each iteration will involve removing central triangles from larger triangles.But the initial triangle is 12√3 meters on each side. Since each small triangle is 1 meter, the number of small triangles along one side is 12√3. But 12√3 is approximately 20.78, which isn't an integer. Hmm, that might be a problem because we can't have a fraction of a triangle.Wait, maybe I need to adjust the initial triangle to fit within the grid such that the side length is an integer number of 1-meter triangles. So, perhaps the largest equilateral triangle that can fit into the 24x18 rectangle with integer side length.So, the side length s must satisfy ( frac{sqrt{3}}{2} s leq 18 ) and ( s leq 24 ). So, solving for s:From the height constraint: ( s leq frac{18 times 2}{sqrt{3}} approx 20.78 ). So, the maximum integer s is 20.But 20 is less than 24, so that's okay. So, the largest equilateral triangle that can fit into the rectangle has a side length of 20 meters.But wait, 20 meters is the side length, so the height would be ( frac{sqrt{3}}{2} times 20 approx 17.32 ) meters, which is less than 18, so that works.So, the initial equilateral triangle has a side length of 20 meters, which is 20 small triangles on each side.In the Sierpinski triangle, each iteration removes the central triangle. The number of triangles removed at each iteration follows a pattern.At iteration 1: 1 triangle removed.At iteration 2: 3 triangles removed.At iteration 3: 9 triangles removed.Wait, no, actually, each iteration removes 3^(n-1) triangles at the nth iteration.Wait, let me think again.The Sierpinski triangle starts with 1 triangle. After the first iteration, you remove 1 triangle, leaving 3. After the second iteration, you remove 3 triangles, leaving 9. After the third iteration, you remove 9 triangles, leaving 27, and so on.So, the number of triangles removed at each iteration is 3^(n-1), where n is the iteration number.Therefore, after 5 iterations, the total number of triangles removed is the sum from n=1 to n=5 of 3^(n-1).So, that's a geometric series: 1 + 3 + 9 + 27 + 81.Calculating that: 1 + 3 = 4, 4 + 9 = 13, 13 + 27 = 40, 40 + 81 = 121.So, 121 triangles removed after 5 iterations.But wait, each triangle is 1 meter on each side, so each small triangle is 1x1x1 meters.But the initial triangle is 20 meters on each side, so the number of small triangles in the initial triangle is (20)^2 / (sqrt(3)/2 * 1^2) ?Wait, no, the area of an equilateral triangle is ( frac{sqrt{3}}{4} s^2 ). So, the number of small triangles would be the area of the large triangle divided by the area of a small triangle.Area of large triangle: ( frac{sqrt{3}}{4} times 20^2 = frac{sqrt{3}}{4} times 400 = 100sqrt{3} ).Area of small triangle: ( frac{sqrt{3}}{4} times 1^2 = frac{sqrt{3}}{4} ).So, number of small triangles: ( frac{100sqrt{3}}{sqrt{3}/4} = 100 times 4 = 400 ).So, the initial triangle is made up of 400 small triangles.But in the Sierpinski process, each iteration removes 3^(n-1) triangles. So, after 5 iterations, 121 triangles are removed. But wait, 121 is less than 400, so that seems possible.But let me confirm. The number of triangles removed after n iterations is (3^n - 1)/2. Wait, is that correct?Wait, no, actually, the total number of removed triangles after n iterations is (3^n - 1)/2. Let me check:At n=1: (3 -1)/2 = 1, correct.n=2: (9 -1)/2 = 4, but earlier I thought it was 1 + 3 = 4, correct.n=3: (27 -1)/2 = 13, which is 1 + 3 + 9 =13, correct.n=4: (81 -1)/2 =40, which is 1 + 3 + 9 + 27=40, correct.n=5: (243 -1)/2=121, which is 1 + 3 + 9 + 27 +81=121, correct.So, yes, the formula is (3^n -1)/2.Therefore, after 5 iterations, 121 triangles are removed.But wait, the initial triangle is 400 small triangles. So, 121 removed, leaving 400 - 121 = 279 triangles.But the question is asking how many 1-meter triangular sections are removed after 5 iterations.So, the answer is 121.But wait, hold on. The initial triangle is 20 meters on each side, which is 20 small triangles per side. So, each iteration divides the triangles into smaller ones.But in the Sierpinski triangle, each iteration replaces each existing triangle with three smaller ones, removing the central one.So, the number of triangles removed at each iteration is 3^(n-1), as we thought.So, after 5 iterations, total removed is 121.But wait, is this correct? Because the initial number is 400, and 121 is the number of removed triangles, but each removed triangle is 1x1x1.Wait, but in the Sierpinski process, each iteration removes triangles that are 1/4 the size of the previous iteration's triangles.Wait, no, actually, in the Sierpinski triangle, each iteration removes the central triangle from each existing triangle, so the number of removed triangles increases by a factor of 3 each time.So, the first iteration removes 1 triangle, the second removes 3, the third removes 9, etc.So, the total removed after 5 iterations is 1 + 3 + 9 + 27 + 81 = 121.Therefore, the answer is 121.But wait, let me make sure that the initial triangle is indeed 20 small triangles on each side.Given the wall is 24 meters long and 18 meters high, the maximum equilateral triangle that can fit has a side length s such that the height ( h = frac{sqrt{3}}{2} s leq 18 ).So, s = (18 * 2)/sqrt(3) = 36 / 1.732 ≈ 20.78 meters.But since we can't have a fraction of a triangle, we take s=20 meters.So, the initial triangle is 20 small triangles on each side, which is 20 meters.So, the initial number of small triangles is 400, as calculated earlier.Therefore, after 5 iterations, 121 triangles are removed.So, the answer is 121.Now, moving on to the second problem.2. The painter listens to indie pop music with intensity modeled by ( I(t) = I_0 sin(omega t + phi) ). Given ( I_0 = 10 ) units, period ( T = 120 ) seconds, and phase shift ( phi = frac{pi}{4} ). Find the average intensity over one period.Okay, so average intensity over one period. For a sinusoidal function, the average value over one period is zero because the positive and negative areas cancel out. But wait, intensity is a measure of power, which is proportional to the square of the amplitude. But in this case, the function given is just the intensity as a sine function, not the square.Wait, but the question says "intensity" is modeled by ( I(t) = I_0 sin(omega t + phi) ). So, it's a sinusoidal function, but intensity is typically a non-negative quantity. However, the function given can take negative values because sine oscillates between -1 and 1. So, that might be an issue.But perhaps in this context, they're just using the sine function to model the intensity, and they might be considering the absolute value or something else. But the problem doesn't specify, so I have to assume that the intensity is given by this function, even though it can be negative.But average intensity over one period. If we take the average of ( sin(omega t + phi) ) over one period, it's zero. But that doesn't make physical sense for intensity, which should be non-negative. So, maybe the question is expecting the average of the absolute value, or perhaps the root mean square (RMS) value.But the problem says "average intensity", so I think it's just the average value of the function over one period, regardless of the physical interpretation.So, the average value of a function over an interval [a, b] is ( frac{1}{b - a} int_{a}^{b} f(t) dt ).Since the function is periodic with period T, the average over one period is ( frac{1}{T} int_{0}^{T} I(t) dt ).So, let's compute that.Given ( I(t) = 10 sin(omega t + frac{pi}{4}) ).First, we need to find ( omega ). The angular frequency ( omega = frac{2pi}{T} ). Given T=120 seconds, so ( omega = frac{2pi}{120} = frac{pi}{60} ) radians per second.So, ( I(t) = 10 sinleft( frac{pi}{60} t + frac{pi}{4} right) ).Now, the average intensity ( bar{I} ) is:( bar{I} = frac{1}{120} int_{0}^{120} 10 sinleft( frac{pi}{60} t + frac{pi}{4} right) dt ).Let me compute this integral.Let me make a substitution to simplify the integral. Let ( u = frac{pi}{60} t + frac{pi}{4} ).Then, ( du = frac{pi}{60} dt ), so ( dt = frac{60}{pi} du ).When t=0, u= ( frac{pi}{4} ).When t=120, u= ( frac{pi}{60} times 120 + frac{pi}{4} = 2pi + frac{pi}{4} = frac{9pi}{4} ).So, the integral becomes:( bar{I} = frac{10}{120} times frac{60}{pi} int_{pi/4}^{9pi/4} sin(u) du ).Simplify constants:( frac{10}{120} times frac{60}{pi} = frac{10 times 60}{120 pi} = frac{600}{120 pi} = frac{5}{pi} ).So, ( bar{I} = frac{5}{pi} int_{pi/4}^{9pi/4} sin(u) du ).Compute the integral:( int sin(u) du = -cos(u) + C ).So,( int_{pi/4}^{9pi/4} sin(u) du = -cos(9pi/4) + cos(pi/4) ).Compute ( cos(9pi/4) ) and ( cos(pi/4) ).Note that ( 9pi/4 = 2pi + pi/4 ), so ( cos(9pi/4) = cos(pi/4) = frac{sqrt{2}}{2} ).Similarly, ( cos(pi/4) = frac{sqrt{2}}{2} ).So,( -cos(9pi/4) + cos(pi/4) = -frac{sqrt{2}}{2} + frac{sqrt{2}}{2} = 0 ).Therefore, the integral is zero, so ( bar{I} = frac{5}{pi} times 0 = 0 ).But that's the average value of the sine function over one period, which is indeed zero. However, as I thought earlier, intensity shouldn't be negative, so maybe the question expects the average of the absolute value or the RMS value.But the problem specifically says "average intensity", so unless specified otherwise, I think it's just the average value, which is zero.But that seems odd because intensity is typically a non-negative quantity. Maybe the function should be the absolute value of the sine function, but the problem didn't specify that.Alternatively, perhaps the average is taken over the absolute value. Let me check.If we consider the average of |I(t)| over one period, then it's different.The average of |sin(x)| over one period is ( frac{2}{pi} ).So, in that case, the average intensity would be ( 10 times frac{2}{pi} approx 6.366 ) units.But since the problem didn't specify absolute value, I think the answer is zero.But maybe I should consider that intensity is the square of the amplitude, so perhaps the average intensity is the RMS value.The RMS value of a sine wave is ( frac{I_0}{sqrt{2}} ).So, in that case, the average intensity would be ( frac{10}{sqrt{2}} approx 7.071 ) units.But again, the problem didn't specify that. It just said "average intensity".Hmm, this is a bit confusing. Let me read the problem again."Given that the maximum intensity ( I_0 ) is 10 units, the period of the sinusoid is 120 seconds, and the phase shift ( phi ) is ( frac{pi}{4} ), find the average intensity of the music over one period."So, it's just the average of ( I(t) ) over one period. Since ( I(t) ) is a sine function, its average over one period is zero.But in reality, intensity can't be negative, so maybe the function should be the square of the sine function, but the problem says it's modeled by ( I(t) = I_0 sin(omega t + phi) ).Alternatively, perhaps the average is intended to be the RMS value, which is a common measure for such oscillating quantities.But unless specified, I think the average value is zero.But let me think again. If the intensity is modeled by a sine function, which does take negative values, but intensity is a physical quantity that can't be negative. So, perhaps the model is incorrect, or perhaps they just want the average of the absolute value.But the problem doesn't specify, so I think the safest answer is zero.But wait, let me check the integral again.I did the integral correctly, right?Yes, substitution was correct, and the integral of sin over a full period is zero.So, unless the question is expecting something else, the average intensity is zero.But that seems counterintuitive because intensity is a positive quantity. So, maybe the question is expecting the average of the absolute value, which would be non-zero.Let me compute that as well.Average of |I(t)| over one period is:( frac{1}{T} int_{0}^{T} |I(t)| dt ).So, for ( I(t) = 10 sin(omega t + phi) ), the average of |sin| over one period is ( frac{2}{pi} times I_0 ).So, ( frac{2}{pi} times 10 approx 6.366 ) units.Alternatively, if we compute it directly:( frac{1}{120} int_{0}^{120} |10 sin(frac{pi}{60} t + frac{pi}{4})| dt ).This integral is equal to ( frac{10}{120} times 2 times frac{pi}{2} times frac{2}{pi} times 120 )?Wait, no, let me use substitution again.Let ( u = frac{pi}{60} t + frac{pi}{4} ), then ( du = frac{pi}{60} dt ), so ( dt = frac{60}{pi} du ).When t=0, u= ( frac{pi}{4} ).When t=120, u= ( frac{pi}{60} times 120 + frac{pi}{4} = 2pi + frac{pi}{4} = frac{9pi}{4} ).So, the integral becomes:( frac{10}{120} times frac{60}{pi} int_{pi/4}^{9pi/4} |sin(u)| du ).Simplify constants:( frac{10}{120} times frac{60}{pi} = frac{5}{pi} ).So, ( frac{5}{pi} int_{pi/4}^{9pi/4} |sin(u)| du ).Now, the integral of |sin(u)| over [π/4, 9π/4] is equal to 4, because over each π interval, the integral of |sin(u)| is 2, and from π/4 to 9π/4 is 2π, so 2 intervals of π, each contributing 2, so total 4.Wait, let me compute it step by step.The function |sin(u)| has a period of π, and over each period, the integral is 2.So, from π/4 to 9π/4 is 2π, which is two full periods of |sin(u)|.Therefore, the integral is 2 * 2 = 4.So, the integral is 4.Therefore, the average is ( frac{5}{pi} times 4 = frac{20}{pi} approx 6.366 ).So, approximately 6.366 units.But the problem didn't specify whether to take the absolute value or not. So, I'm confused.But in physics, when we talk about average intensity of a wave, it's usually the average of the square of the amplitude, which is the RMS value. But in this case, the function is given as intensity proportional to sine, which is unusual because intensity is proportional to the square of the amplitude.But the problem says "intensity is modeled by a sinusoidal function", so perhaps they just want the average of the sine function, which is zero.But that seems odd because intensity can't be negative.Alternatively, maybe the function is actually the amplitude, and intensity is proportional to the square of the amplitude, but the problem says intensity is modeled by the sine function.This is a bit ambiguous.But given that the problem says "intensity is modeled by a sinusoidal function", and doesn't specify taking absolute value or squaring, I think the answer is zero.But to be thorough, let me consider both cases.Case 1: Average of I(t) over one period: 0.Case 2: Average of |I(t)| over one period: ( frac{20}{pi} approx 6.366 ).Case 3: RMS value: ( frac{10}{sqrt{2}} approx 7.071 ).But the problem says "average intensity", so unless specified, it's likely case 1, which is zero.But in reality, intensity is a non-negative quantity, so maybe the question expects case 2 or 3.But since the function is given as a sine function, and the average of a sine function over one period is zero, I think the answer is zero.But I'm not entirely sure. Maybe the problem expects the average of the absolute value.But given that the problem didn't specify, I think the safest answer is zero.Wait, but let me think again. If the intensity is modeled by a sine function, which can be negative, but intensity can't be negative, perhaps the function is actually the amplitude, and intensity is the square of that.But the problem says "intensity is modeled by a sinusoidal function", so perhaps it's just a sine function, and the average is zero.Alternatively, maybe the phase shift affects the average, but no, the average over a full period is still zero regardless of phase shift.So, I think the answer is zero.But I'm still a bit uncertain because intensity is a non-negative quantity, but the function given is a sine function which can be negative.But unless the problem specifies taking absolute value or squaring, I think the average is zero.So, to sum up:Problem 1: 121 triangles removed.Problem 2: Average intensity is 0 units.But wait, let me double-check problem 1.The initial triangle is 20 small triangles on each side, so 20x20 grid? Wait, no, it's an equilateral triangle, so the number of small triangles is different.Wait, earlier I calculated the number of small triangles as 400, but that's based on area.But in the Sierpinski triangle, each iteration removes 3^(n-1) triangles.After 5 iterations, total removed is 121.But wait, 121 is less than 400, so that seems okay.But let me think about the scaling.Each iteration divides the triangle into smaller triangles. So, the number of small triangles after n iterations is 3^n.But the number of triangles removed is (3^n -1)/2.Wait, no, the number of removed triangles after n iterations is (3^n -1)/2.So, for n=5, it's (243 -1)/2=121.So, yes, that's correct.Therefore, the answer is 121.So, final answers:1. 1212. 0But wait, for problem 2, if the average is zero, but intensity can't be negative, maybe the question expects the average of the absolute value, which is ( frac{20}{pi} ).But since the problem didn't specify, I think it's safer to go with zero.Alternatively, maybe the question is expecting the RMS value, which is ( frac{10}{sqrt{2}} approx 7.071 ).But again, the problem didn't specify.Hmm, this is tricky.Wait, let me check the definition of average intensity.In physics, average intensity over a period is indeed the RMS value, which is the square root of the average of the square of the intensity.But in this problem, the intensity is given as a sine function, so the average intensity would be the RMS value.So, perhaps the problem expects the RMS value.So, let's compute that.The RMS value of ( I(t) = I_0 sin(omega t + phi) ) is ( frac{I_0}{sqrt{2}} ).So, ( frac{10}{sqrt{2}} = 5sqrt{2} approx 7.071 ).But the problem says "average intensity", not RMS intensity.But in physics, average intensity is often considered as the RMS value.So, maybe the answer is ( 5sqrt{2} ).But the problem didn't specify, so I'm not sure.Alternatively, maybe the average of the absolute value, which is ( frac{2}{pi} I_0 approx 6.366 ).But again, the problem didn't specify.Given the ambiguity, I think the safest answer is zero, as the average of the given function over one period.But I'm not entirely confident.Alternatively, if I consider that intensity is always positive, perhaps the function should be the absolute value of the sine function, in which case the average is ( frac{2}{pi} I_0 ).But the problem didn't specify that.Given that, I think the answer is zero.But I'm still uncertain.Wait, let me check the integral again.If I(t) = 10 sin(ωt + φ), then the average over one period is zero.But if I(t) is intensity, which is a non-negative quantity, then perhaps the model is incorrect, and they should have used the absolute value or the square.But since the problem says it's modeled by a sinusoidal function, I think we have to take it as is.Therefore, the average intensity is zero.So, final answers:1. 1212. 0But I'm still a bit unsure about problem 2. Maybe I should go with the RMS value.Wait, let me think about the definition.Average intensity is typically the time average of the intensity over a period, which for a sinusoidal wave is the RMS value.But in this case, the intensity is given as a sine function, which is unusual because intensity is proportional to the square of the electric field or something like that.So, perhaps the function should be ( I(t) = I_0 sin^2(omega t + phi) ), but the problem says it's a sinusoidal function, not a squared one.Therefore, perhaps the average is zero.But again, intensity can't be negative, so maybe the function is actually the absolute value.But the problem didn't specify.Given that, I think the answer is zero.But to be thorough, I'll note both possibilities.But since the problem didn't specify, I think the answer is zero.So, final answers:1. boxed{121}2. boxed{0}</think>