🧠:<think>Okay, so I have this problem about a restaurant that's trying to recover from a drop in customers. They implemented a new marketing strategy and changed their menu, and now I need to figure out some math stuff about their customer numbers and revenue. Let me break this down step by step.First, part 1 is about finding the quadratic function that models the number of customers over time. The function is given as ( C(t) = at^2 + bt + c ). I know that ( c ) is the initial number of customers after the setback, which is 50. So, that's one value already: ( c = 50 ).Now, they gave me two more points: after 10 days, the number of customers was 150, and after 20 days, it was 300. So, I can plug these into the equation to form a system of equations.Let me write that out.For day 10:( C(10) = a(10)^2 + b(10) + 50 = 150 )Which simplifies to:( 100a + 10b + 50 = 150 )Subtract 50 from both sides:( 100a + 10b = 100 )I can divide this equation by 10 to make it simpler:( 10a + b = 10 ) -- Let's call this Equation 1.For day 20:( C(20) = a(20)^2 + b(20) + 50 = 300 )Which simplifies to:( 400a + 20b + 50 = 300 )Subtract 50 from both sides:( 400a + 20b = 250 )I can divide this equation by 10 as well:( 40a + 2b = 25 ) -- Let's call this Equation 2.Now, I have two equations:1. ( 10a + b = 10 )2. ( 40a + 2b = 25 )I can solve this system of equations using substitution or elimination. Let me try elimination.First, let me multiply Equation 1 by 2 to make the coefficients of ( b ) the same:1. ( 20a + 2b = 20 )2. ( 40a + 2b = 25 )Now, subtract Equation 1 from Equation 2:( (40a + 2b) - (20a + 2b) = 25 - 20 )Simplify:( 20a = 5 )So, ( a = 5 / 20 = 1/4 = 0.25 )Now, plug ( a = 0.25 ) back into Equation 1:( 10(0.25) + b = 10 )Which is:( 2.5 + b = 10 )Subtract 2.5:( b = 7.5 )So, the coefficients are:( a = 0.25 )( b = 7.5 )( c = 50 )Let me double-check these values with the given points.For day 10:( C(10) = 0.25(100) + 7.5(10) + 50 = 25 + 75 + 50 = 150 ). That's correct.For day 20:( C(20) = 0.25(400) + 7.5(20) + 50 = 100 + 150 + 50 = 300 ). That's also correct.Great, so part 1 is done. Now, moving on to part 2.Part 2 is about the revenue function, which is linear and depends on the number of customers. The function is ( R(C) = mC + n ). They gave me two points: on day 15, the revenue was 4500, and on day 25, it was 8000.But wait, the revenue is a function of the number of customers, not time. So, I need to find ( R(C) ), but I also need to relate it to the time. Since ( C(t) ) is quadratic, I can find ( C(15) ) and ( C(25) ) using the function we found in part 1.So, first, let me find ( C(15) ) and ( C(25) ).Using ( C(t) = 0.25t^2 + 7.5t + 50 ).For day 15:( C(15) = 0.25(225) + 7.5(15) + 50 )Calculate each term:0.25 * 225 = 56.257.5 * 15 = 112.5Adding up: 56.25 + 112.5 + 50 = 218.75So, on day 15, there were 218.75 customers. Hmm, but you can't have a fraction of a customer, but since we're dealing with models, it's okay.Similarly, for day 25:( C(25) = 0.25(625) + 7.5(25) + 50 )Calculate each term:0.25 * 625 = 156.257.5 * 25 = 187.5Adding up: 156.25 + 187.5 + 50 = 393.75So, on day 25, there were 393.75 customers.Now, the revenue on day 15 was 4500, which corresponds to 218.75 customers. So, ( R(218.75) = 4500 ).Similarly, on day 25, ( R(393.75) = 8000 ).So, we can set up two equations:1. ( 4500 = m(218.75) + n )2. ( 8000 = m(393.75) + n )Now, let's write these equations:Equation 1: ( 218.75m + n = 4500 )Equation 2: ( 393.75m + n = 8000 )To solve for ( m ) and ( n ), I can subtract Equation 1 from Equation 2 to eliminate ( n ):( (393.75m + n) - (218.75m + n) = 8000 - 4500 )Simplify:( 175m = 3500 )So, ( m = 3500 / 175 = 20 )Now, plug ( m = 20 ) back into Equation 1:( 218.75(20) + n = 4500 )Calculate 218.75 * 20:218.75 * 20 = 4375So:4375 + n = 4500Subtract 4375:n = 4500 - 4375 = 125Therefore, ( m = 20 ) and ( n = 125 ).So, the revenue function is ( R(C) = 20C + 125 ).Now, the question also asks to predict the revenue on the 30th day. So, first, I need to find ( C(30) ) using the quadratic function, then plug that into the revenue function.Let's compute ( C(30) ):( C(30) = 0.25(30)^2 + 7.5(30) + 50 )Calculate each term:0.25 * 900 = 2257.5 * 30 = 225Adding up: 225 + 225 + 50 = 500So, on day 30, there are 500 customers.Now, plug into ( R(C) ):( R(500) = 20(500) + 125 = 10000 + 125 = 10125 )So, the predicted revenue on day 30 is 10,125.Let me just double-check my calculations to make sure I didn't make any errors.For part 1, the quadratic function:At t=10: 0.25*100 + 7.5*10 +50=25+75+50=150. Correct.At t=20: 0.25*400 +7.5*20 +50=100+150+50=300. Correct.For part 2, revenue function:At t=15, C=218.75, R=4500:20*218.75=4375, 4375+125=4500. Correct.At t=25, C=393.75, R=8000:20*393.75=7875, 7875+125=8000. Correct.Revenue on day 30: C=500, R=20*500+125=10000+125=10125. Correct.Everything seems to check out.Final Answer1. The values are ( a = boxed{0.25} ), ( b = boxed{7.5} ), and ( c = boxed{50} ).2. The values are ( m = boxed{20} ) and ( n = boxed{125} ), and the predicted revenue on the 30th day is ( boxed{10125} ) dollars.</think>

🧠:<think>Okay, so I have this problem about a sports bar near Madison Square Garden. The restaurateur is trying to figure out the total revenue on game days and non-game days. Let me try to break this down step by step.First, the bar has 150 seats and is open 7 days a week. There are 3 game days each week, so that leaves 4 non-game days. Starting with part 1: On game days, the average revenue per customer is given by the function ( R_g(x) = 50 + 10sinleft(frac{pi x}{6}right) ) dollars, where ( x ) is the number of hours the game is being broadcasted. The game lasts for 3 hours, and the bar operates at full capacity. I need to calculate the total revenue generated on that game day.Hmm, okay. So, the bar is at full capacity, which is 150 seats. But wait, does that mean 150 customers? Or is it 150 seats, so maybe more customers come in over the day? Hmm, the problem says "operates at full capacity," so I think that means all seats are occupied, so 150 customers. But wait, the bar is open for the duration of the game, which is 3 hours. So, does that mean 150 customers per hour, or 150 customers total for the 3 hours?Wait, the problem says "the bar operates at full capacity," so maybe it's 150 customers in total for the game day. Or perhaps it's 150 customers per hour? Hmm, I need to clarify this.Looking back at the problem: "the bar operates at full capacity." Since it's a sports bar, it's likely that during the game, the bar is packed, so maybe all 150 seats are occupied for the duration of the game. So, perhaps 150 customers, each staying for the entire 3 hours? Or maybe 150 customers per hour? Hmm, the problem isn't entirely clear.Wait, the function ( R_g(x) ) is given as the average revenue per customer, where ( x ) is the number of hours the game is being broadcasted. So, if the game lasts 3 hours, ( x = 3 ). So, the average revenue per customer is ( R_g(3) ).So, first, let's compute ( R_g(3) ):( R_g(3) = 50 + 10sinleft(frac{pi times 3}{6}right) )Simplify the sine argument:( frac{pi times 3}{6} = frac{pi}{2} )So, ( sinleft(frac{pi}{2}right) = 1 )Therefore, ( R_g(3) = 50 + 10 times 1 = 60 ) dollars per customer.Okay, so each customer generates an average revenue of 60 on that game day.Now, how many customers are there? The bar has 150 seats and operates at full capacity. So, does that mean 150 customers? Or is it 150 customers per hour?Wait, the game is 3 hours long. If the bar is operating at full capacity, it's likely that all 150 seats are occupied during the game. So, perhaps 150 customers for the entire duration of the game. But then, is the revenue per customer based on the time they spend there?Wait, the average revenue per customer is given as a function of the number of hours the game is broadcasted. So, if the game is 3 hours, each customer contributes 60 in revenue. So, if the bar is at full capacity, meaning 150 customers, each contributing 60, then total revenue is 150 * 60.But wait, is it 150 customers in total, or 150 customers per hour? Because if it's 150 per hour, over 3 hours, that would be 450 customers. But the problem says "operates at full capacity," which probably refers to the number of customers at any given time, not the total over the day.Wait, the bar has 150 seats, so it can seat 150 people at a time. So, if it's operating at full capacity during the game, which is 3 hours, then perhaps 150 customers are there for the entire 3 hours. So, the total number of customers is 150, each contributing 60, so total revenue is 150 * 60 = 9,000.But wait, another thought: if the bar is open for 3 hours, and operates at full capacity, maybe it's 150 customers per hour. So, over 3 hours, it's 150 * 3 = 450 customers. But then, the revenue per customer is 60, so total revenue would be 450 * 60 = 27,000.Hmm, now I'm confused. Which interpretation is correct?Looking back at the problem statement: "the bar operates at full capacity." It doesn't specify whether it's per hour or total. But in the context of a sports bar during a game, it's more likely that they are serving 150 customers during the game, not 150 per hour. Because if it's 150 per hour, over 3 hours, that would be a lot, and the bar might not have the staff or space to handle that.Alternatively, maybe the bar is open for the entire day, but the game is only 3 hours. So, perhaps during the game, it's at full capacity, but outside the game, it's not. But the problem says "on a particular game day, the game lasts for 3 hours and the bar operates at full capacity." So, maybe the bar is open for the entire day, but during the 3-hour game, it's at full capacity. But the problem doesn't specify the total hours the bar is open on game days.Wait, actually, the problem says "the bar is open 7 days a week," but doesn't specify the hours. However, in part 2, it says the bar is open for 12 hours each day on non-game days. So, maybe on game days, it's open for more hours? Or the same?Wait, the problem doesn't specify the operating hours on game days. It only says that on game days, the number of customers is higher, and the function ( R_g(x) ) depends on the number of hours the game is broadcasted, which is 3 hours.So, perhaps the bar is open for the duration of the game, which is 3 hours, and during those 3 hours, it operates at full capacity, meaning 150 customers. So, 150 customers for 3 hours, each contributing 60, so total revenue is 150 * 60 = 9,000.Alternatively, if the bar is open for more hours on game days, but the problem doesn't specify, so maybe it's only open for the 3 hours of the game. So, I think the first interpretation is correct: 150 customers for 3 hours, each contributing 60, so total revenue is 9,000.Wait, but another angle: the average revenue per customer is 60, and if the bar is at full capacity, which is 150 seats, but how many customers does that translate to? If each seat is occupied, then 150 customers. So, total revenue is 150 * 60 = 9,000.Yes, that seems reasonable. So, I think the total revenue on that game day is 9,000.Moving on to part 2: On non-game days, the average revenue per customer is modeled by ( R_n(x) = 30 + 5cosleft(frac{pi x}{12}right) ) dollars, where ( x ) is the number of hours the bar is open. The bar is open for 12 hours each day and operates at 80% capacity on non-game days. I need to calculate the total weekly revenue generated on non-game days.First, let's figure out the average revenue per customer on non-game days. Since the bar is open for 12 hours, ( x = 12 ). So, plug that into the function:( R_n(12) = 30 + 5cosleft(frac{pi times 12}{12}right) )Simplify the cosine argument:( frac{pi times 12}{12} = pi )So, ( cos(pi) = -1 )Therefore, ( R_n(12) = 30 + 5 times (-1) = 30 - 5 = 25 ) dollars per customer.Okay, so each customer on non-game days generates an average revenue of 25.Now, the bar operates at 80% capacity on non-game days. The total capacity is 150 seats, so 80% of that is 0.8 * 150 = 120 customers.But wait, is that 120 customers per hour or total for the day? The problem says "operates at 80% capacity," which is a bit ambiguous. But in the context of a bar open for 12 hours, it's more likely that 80% capacity refers to the number of customers per hour. So, 120 customers per hour.But wait, the problem doesn't specify whether the capacity is per hour or total. Hmm.Wait, in part 1, the bar operates at full capacity during the game, which is 3 hours, and we interpreted that as 150 customers in total. So, maybe on non-game days, operating at 80% capacity would mean 120 customers in total for the day.But that seems low because the bar is open for 12 hours. If it's 120 customers total, that's 10 customers per hour, which seems too low for a popular bar. Alternatively, if it's 120 customers per hour, that's 120 * 12 = 1,440 customers in a day, which seems high.Wait, perhaps the capacity is the number of customers that can be seated at a time, so 150 seats. So, 80% capacity would be 120 customers at a time. So, if the bar is open for 12 hours, and each hour, it can serve 120 customers, but actually, in reality, the number of customers per hour might vary.But the problem says "operates at 80% capacity," so maybe it's 120 customers per hour on average. So, over 12 hours, that would be 120 * 12 = 1,440 customers.But wait, that might not be correct because the bar can't serve 120 customers every hour if it's only seating 120 at a time. Unless customers are coming in and out, like a revolving door. So, the number of customers per hour would be higher than the seating capacity.Wait, this is getting complicated. Maybe I need to think differently.The problem says "operates at 80% capacity on non-game days." So, if the bar has 150 seats, 80% is 120. So, does that mean 120 customers in total for the day? Or 120 customers per hour?Wait, in part 1, the bar operates at full capacity during the game, which is 3 hours, and we assumed that meant 150 customers in total. So, by analogy, on non-game days, operating at 80% capacity for 12 hours would mean 120 customers in total for the day.But that seems inconsistent because on game days, the bar is open for 3 hours and serves 150 customers, while on non-game days, it's open for 12 hours and serves only 120 customers? That doesn't make sense because non-game days are longer, so they should have more customers.Alternatively, maybe the capacity refers to the number of customers per hour. So, on game days, during the 3-hour game, they serve 150 customers per hour, so total 450 customers. On non-game days, they serve 120 customers per hour, over 12 hours, so 1,440 customers.But that seems high. Let me think again.Wait, the problem says "the bar operates at full capacity," which is 150 seats. So, if it's full capacity, that's 150 customers at a time. So, during the game, which is 3 hours, the bar is full, so 150 customers for the entire 3 hours. So, total customers would be 150, each contributing 60, so 9,000.On non-game days, the bar operates at 80% capacity, so 120 customers at a time. But the bar is open for 12 hours. So, how many customers does that translate to?If the bar can seat 120 customers at a time, and it's open for 12 hours, the total number of customers would depend on how often customers come and go. If customers stay for, say, 1 hour on average, then the number of customers per hour would be 120, so over 12 hours, it's 120 * 12 = 1,440 customers.But the problem doesn't specify the average time a customer spends in the bar. Hmm.Wait, maybe the problem is simpler. It says "operates at 80% capacity," which is 120 customers. But it's open for 12 hours. So, perhaps the number of customers is 120 per hour, so total customers is 120 * 12 = 1,440.But then, the average revenue per customer is 25, so total revenue would be 1,440 * 25 = 36,000 per day.But wait, that seems high for a non-game day. Let me check.Alternatively, maybe "operates at 80% capacity" means that the bar is serving 80% of its maximum number of customers per day. So, if on a game day, they serve 150 customers in 3 hours, then on a non-game day, they serve 120 customers in 12 hours. So, total revenue would be 120 * 25 = 3,000 per day.But that seems low because the bar is open longer. Hmm.Wait, perhaps I need to consider the average revenue per customer as a function of the number of hours the bar is open. So, ( R_n(x) ) is the average revenue per customer when the bar is open for ( x ) hours. So, if the bar is open for 12 hours, the average revenue per customer is 25, as we calculated.Now, the number of customers on non-game days is 80% of capacity. Capacity is 150 seats, so 120 customers. But is that per hour or total?Wait, if the bar is open for 12 hours, and operates at 80% capacity, that would mean 120 customers per hour. So, over 12 hours, that's 120 * 12 = 1,440 customers.But then, each customer contributes 25, so total revenue is 1,440 * 25 = 36,000 per day.But wait, that seems high because on game days, they only make 9,000, but on non-game days, they make 36,000? That doesn't make sense because game days should have higher revenue.Wait, no, actually, the average revenue per customer on game days is higher (60) compared to non-game days (25). So, even though non-game days have more customers, the total revenue might be higher or lower depending on the numbers.Wait, let's do the math:Game day revenue: 9,000Non-game day revenue: 36,000So, weekly revenue would be 3 * 9,000 + 4 * 36,000 = 27,000 + 144,000 = 171,000.But that seems plausible, but I'm not sure if the interpretation is correct.Alternatively, if on non-game days, the bar serves 120 customers in total for the day, then total revenue is 120 * 25 = 3,000 per day. Then, weekly non-game revenue would be 4 * 3,000 = 12,000.But that seems low compared to game days.Wait, maybe the capacity refers to the number of customers that can be seated at a time, not the total number served in a day. So, if the bar can seat 150 at a time, and operates at 80% capacity, that's 120 customers at a time. If the bar is open for 12 hours, and assuming customers stay for an average of, say, 2 hours, then the number of customers per hour would be 120 / 2 = 60. So, over 12 hours, that's 60 * 12 = 720 customers.But this is getting too speculative. The problem doesn't specify the average time a customer spends in the bar, so maybe we need to assume that the capacity refers to the total number of customers per day.Wait, but in part 1, the bar operates at full capacity during the game, which is 3 hours, and we assumed that meant 150 customers in total. So, by analogy, on non-game days, operating at 80% capacity for 12 hours would mean 120 customers in total for the day.So, total revenue per non-game day would be 120 * 25 = 3,000.Then, weekly non-game revenue would be 4 * 3,000 = 12,000.But that seems low because the bar is open longer on non-game days, so they should have more customers. But maybe the function ( R_n(x) ) already accounts for the number of hours open, so the average revenue per customer is 25 regardless of the time spent.Wait, the function ( R_n(x) ) is given as a function of ( x ), the number of hours the bar is open. So, when ( x = 12 ), ( R_n(12) = 25 ). So, that's the average revenue per customer when the bar is open for 12 hours.So, if the bar is open for 12 hours, and operates at 80% capacity, which is 120 customers, but is that 120 customers in total or per hour?Wait, the problem says "operates at 80% capacity on non-game days." Since capacity is 150 seats, 80% is 120. So, that would mean 120 customers at a time. But the bar is open for 12 hours. So, the total number of customers would depend on how often customers come and go.But without information on the average time a customer spends, we can't calculate the total number of customers. So, perhaps the problem is assuming that the number of customers is 80% of capacity, which is 120, and that's the total for the day.Alternatively, maybe the bar serves 120 customers per hour, so over 12 hours, that's 1,440 customers.But the problem doesn't specify, so I'm stuck.Wait, let's look at part 1 again. On game days, the bar operates at full capacity, which is 150 seats, and the game is 3 hours. We assumed that meant 150 customers in total, not per hour. So, by analogy, on non-game days, operating at 80% capacity (120 customers) would mean 120 customers in total for the day, not per hour.Therefore, total revenue per non-game day is 120 * 25 = 3,000.Since there are 4 non-game days, total weekly revenue from non-game days is 4 * 3,000 = 12,000.But wait, that seems inconsistent because the bar is open longer on non-game days, so shouldn't they have more customers?Alternatively, maybe the capacity refers to the number of customers per hour. So, on game days, full capacity is 150 per hour, over 3 hours, so 450 customers. On non-game days, 80% capacity is 120 per hour, over 12 hours, so 1,440 customers.But then, total revenue would be 1,440 * 25 = 36,000 per day, which seems high.Wait, but the problem says "the bar operates at full capacity," which is 150 seats. So, if it's full, that's 150 customers at a time. So, during the game, which is 3 hours, the bar is full, so 150 customers for the entire 3 hours. So, total customers is 150, each contributing 60, so 9,000.On non-game days, the bar operates at 80% capacity, so 120 customers at a time. But the bar is open for 12 hours. So, how many customers does that translate to?If the bar is open for 12 hours and can seat 120 customers at a time, the total number of customers would be 120 * 12 = 1,440, assuming customers come in and out every hour. But that's a big assumption.Alternatively, if customers stay for the entire 12 hours, which is unlikely, then total customers would be 120.But that doesn't make sense because the bar is open for 12 hours, so customers come and go.Wait, maybe the problem is considering the bar's capacity as the number of customers it can serve in a day, not per hour. So, on game days, full capacity is 150 customers in 3 hours, so 50 per hour. On non-game days, 80% capacity is 120 customers in 12 hours, so 10 per hour. But that seems low.Wait, this is getting too confusing. Maybe I need to make an assumption based on part 1.In part 1, the bar operates at full capacity during the game, which is 3 hours, and we assumed that meant 150 customers in total. So, by analogy, on non-game days, operating at 80% capacity for 12 hours would mean 120 customers in total for the day.Therefore, total revenue per non-game day is 120 * 25 = 3,000.Weekly non-game revenue is 4 * 3,000 = 12,000.But wait, that seems low because the bar is open longer on non-game days. Maybe the problem is considering the average revenue per customer as a function of the number of hours the bar is open, so when the bar is open for 12 hours, the average revenue per customer is 25, and the number of customers is 80% of capacity, which is 120. So, total revenue is 120 * 25 = 3,000 per day.But then, over 4 days, that's 12,000.Alternatively, maybe the number of customers is 80% of the total capacity per hour. So, 120 customers per hour, over 12 hours, so 1,440 customers, each contributing 25, so total revenue is 1,440 * 25 = 36,000 per day.But that seems high.Wait, maybe the problem is using "capacity" as the number of customers that can be served in a day, not per hour. So, on game days, full capacity is 150 customers in 3 hours, so 50 per hour. On non-game days, 80% capacity is 120 customers in 12 hours, so 10 per hour. That seems inconsistent.Alternatively, maybe the capacity is the number of customers that can be seated at a time, so 150 seats. So, on game days, full capacity is 150 customers at a time, over 3 hours, so total customers is 150. On non-game days, 80% capacity is 120 customers at a time, over 12 hours, so total customers is 120.But that would mean the bar only serves 120 customers in 12 hours, which is 10 per hour, which seems low.Wait, maybe the problem is considering that on non-game days, the bar is open for 12 hours, and operates at 80% capacity, which is 120 customers in total for the day.So, total revenue per non-game day is 120 * 25 = 3,000.Therefore, weekly non-game revenue is 4 * 3,000 = 12,000.But I'm still not sure. Maybe I need to think differently.Wait, perhaps the problem is considering that on non-game days, the bar is open for 12 hours, and the average revenue per customer is 25, and the number of customers is 80% of the total capacity per day.But what's the total capacity per day? On game days, it's 150 customers in 3 hours. So, per hour, that's 50 customers. So, over 12 hours, the capacity would be 50 * 12 = 600 customers. So, 80% of that is 480 customers.Therefore, total revenue per non-game day is 480 * 25 = 12,000.But that seems high.Wait, but the problem says "the bar operates at 80% capacity on non-game days." So, if the bar's capacity is 150 seats, 80% is 120. So, that's 120 customers at a time. But the bar is open for 12 hours. So, the total number of customers would be 120 * 12 = 1,440.But that's assuming customers come in and out every hour, which might not be realistic, but perhaps that's the assumption we need to make.So, total revenue per non-game day is 1,440 * 25 = 36,000.But that seems high because on game days, they only make 9,000, but on non-game days, they make 36,000. That would mean non-game days are more profitable, which might not make sense because game days are supposed to be busier.Wait, but the average revenue per customer on game days is higher (60) compared to non-game days (25). So, even though non-game days have more customers, the total revenue might be higher or lower depending on the numbers.Wait, let's calculate:Game day revenue: 3 days * 9,000 = 27,000Non-game day revenue: 4 days * 36,000 = 144,000Total weekly revenue: 27,000 + 144,000 = 171,000But that seems plausible, but I'm not sure if the interpretation is correct.Alternatively, if on non-game days, the bar serves 120 customers in total, then total revenue is 4 * 3,000 = 12,000, which is much lower.Given the ambiguity, I think the problem expects us to assume that the number of customers is 80% of capacity per day, so 120 customers, each contributing 25, so 3,000 per day, and weekly revenue is 12,000.But I'm not entirely confident. Maybe I should look for another approach.Wait, perhaps the problem is considering that the bar is open for 12 hours on non-game days, and the number of customers is 80% of the total capacity per hour. So, 150 seats * 80% = 120 customers per hour. Over 12 hours, that's 120 * 12 = 1,440 customers. Each contributing 25, so total revenue is 1,440 * 25 = 36,000 per day.But then, weekly non-game revenue is 4 * 36,000 = 144,000.But that seems high.Alternatively, maybe the problem is considering that the bar is open for 12 hours, and the number of customers is 80% of the total capacity for the day. So, if on game days, the bar serves 150 customers in 3 hours, then on non-game days, it serves 120 customers in 12 hours.But that would mean 10 customers per hour on non-game days, which seems low.Wait, maybe the problem is considering that the bar's capacity is 150 customers per day, regardless of the hours. So, on game days, they serve 150 customers in 3 hours, and on non-game days, they serve 120 customers in 12 hours.But that seems inconsistent because the bar is open longer on non-game days, so they should serve more customers.Wait, perhaps the problem is considering that the capacity is 150 customers per hour. So, on game days, full capacity is 150 per hour, over 3 hours, so 450 customers. On non-game days, 80% capacity is 120 per hour, over 12 hours, so 1,440 customers.But then, total revenue on game days would be 450 * 60 = 27,000 per day, and on non-game days, 1,440 * 25 = 36,000 per day.But that seems high, and the problem didn't specify that capacity is per hour.Wait, the problem says "the bar has 150 seats," so that's the maximum number of customers at a time. So, capacity is 150 customers at a time, not per hour.Therefore, on game days, the bar is full for 3 hours, so 150 customers for 3 hours, total revenue is 150 * 60 = 9,000.On non-game days, the bar operates at 80% capacity, so 120 customers at a time, for 12 hours. So, the total number of customers would be 120 * 12 = 1,440, assuming customers come in and out every hour.But that's a big assumption because the problem doesn't specify the turnover rate.Alternatively, maybe the problem is considering that the bar serves 120 customers in total for the day, so total revenue is 120 * 25 = 3,000.Given the ambiguity, I think the problem expects us to assume that the number of customers is 80% of capacity, which is 120, and that's the total for the day, regardless of the hours. So, total revenue is 120 * 25 = 3,000 per day, and weekly non-game revenue is 4 * 3,000 = 12,000.But I'm not entirely sure. Maybe I should go with that interpretation.So, to summarize:1. On game days, the bar operates at full capacity (150 customers), each contributing 60, so total revenue is 150 * 60 = 9,000.2. On non-game days, the bar operates at 80% capacity (120 customers), each contributing 25, so total revenue is 120 * 25 = 3,000 per day. Weekly non-game revenue is 4 * 3,000 = 12,000.Therefore, the answers are:1. 9,0002. 12,000But wait, let me double-check the calculations.For part 1:( R_g(3) = 50 + 10sin(pi/2) = 50 + 10*1 = 60 )Total revenue: 150 * 60 = 9,000. Correct.For part 2:( R_n(12) = 30 + 5cos(pi) = 30 - 5 = 25 )Number of customers: 80% of 150 = 120Total revenue per day: 120 * 25 = 3,000Weekly: 4 * 3,000 = 12,000Yes, that seems correct.</think>

🧠:<think>Alright, so I have this problem to solve, and it's about analyzing a system of linear equations and then checking the stability of a quadratic function. Let me try to break this down step by step.First, the system of equations is:1. ( 3x_1 + 2x_2 - x_3 = 5 )2. ( 2x_1 - 2x_2 + 4x_3 = 2 )3. ( x_1 + x_2 + x_3 = 7 )I need to check if there's a unique solution where all ( x_i ) are non-negative. If such a solution exists, I have to find the values of ( x_1, x_2, ) and ( x_3 ).Okay, so starting with the system of equations. I remember that for a system of linear equations, if the determinant of the coefficient matrix is non-zero, there's a unique solution. So, maybe I should write the coefficient matrix and compute its determinant.The coefficient matrix ( A ) is:[A = begin{bmatrix}3 & 2 & -1 2 & -2 & 4 1 & 1 & 1 end{bmatrix}]To find the determinant, I can use the rule of Sarrus or cofactor expansion. Let me do cofactor expansion along the first row.The determinant ( |A| ) is:( 3 times begin{vmatrix} -2 & 4 1 & 1 end{vmatrix} - 2 times begin{vmatrix} 2 & 4 1 & 1 end{vmatrix} + (-1) times begin{vmatrix} 2 & -2 1 & 1 end{vmatrix} )Calculating each minor:First minor: ( (-2)(1) - (4)(1) = -2 - 4 = -6 )Second minor: ( (2)(1) - (4)(1) = 2 - 4 = -2 )Third minor: ( (2)(1) - (-2)(1) = 2 + 2 = 4 )So, determinant is:( 3 times (-6) - 2 times (-2) + (-1) times 4 = -18 + 4 - 4 = -18 )Wait, that's -18. So determinant is -18, which is not zero. That means the system has a unique solution. So, that's good news.Now, I need to solve the system. I can use substitution, elimination, or matrix methods. Maybe Gaussian elimination would be straightforward.Let me write the augmented matrix:[left[begin{array}{ccc|c}3 & 2 & -1 & 5 2 & -2 & 4 & 2 1 & 1 & 1 & 7 end{array}right]]I can perform row operations to get it into row-echelon form.First, let me make the element under the first pivot (3) zero. So, I can use the first row to eliminate the 2 in the second row, first column, and the 1 in the third row, first column.Let me denote rows as R1, R2, R3.First, let's make the third row's first element zero. So, R3 = R3 - (1/3)R1.Calculating R3:( x_1: 1 - (1/3)*3 = 1 - 1 = 0 )( x_2: 1 - (1/3)*2 = 1 - 2/3 = 1/3 )( x_3: 1 - (1/3)*(-1) = 1 + 1/3 = 4/3 )Constant term: ( 7 - (1/3)*5 = 7 - 5/3 = 16/3 )So, new R3 is [0, 1/3, 4/3 | 16/3]Similarly, eliminate the 2 in R2, first column. So, R2 = R2 - (2/3)R1.Calculating R2:( x_1: 2 - (2/3)*3 = 2 - 2 = 0 )( x_2: -2 - (2/3)*2 = -2 - 4/3 = -10/3 )( x_3: 4 - (2/3)*(-1) = 4 + 2/3 = 14/3 )Constant term: ( 2 - (2/3)*5 = 2 - 10/3 = -4/3 )So, new R2 is [0, -10/3, 14/3 | -4/3]Now, the augmented matrix looks like:[left[begin{array}{ccc|c}3 & 2 & -1 & 5 0 & -10/3 & 14/3 & -4/3 0 & 1/3 & 4/3 & 16/3 end{array}right]]Now, let's focus on the second column. The pivot is now in R2, which is -10/3. Let me make the element below it zero. So, I can use R2 to eliminate the 1/3 in R3.So, R3 = R3 + (1/3)/(-10/3) R2. Let me compute the multiplier: (1/3) / (-10/3) = -1/10.So, R3 = R3 - (1/10) R2.Calculating R3:( x_2: 1/3 - (1/10)*(-10/3) = 1/3 + 1/3 = 2/3 )( x_3: 4/3 - (1/10)*(14/3) = 4/3 - 14/30 = 40/30 - 14/30 = 26/30 = 13/15 )Constant term: ( 16/3 - (1/10)*(-4/3) = 16/3 + 4/30 = 16/3 + 2/15 = 80/15 + 2/15 = 82/15 )So, new R3 is [0, 2/3, 13/15 | 82/15]Now, the matrix is:[left[begin{array}{ccc|c}3 & 2 & -1 & 5 0 & -10/3 & 14/3 & -4/3 0 & 0 & 13/15 & 82/15 end{array}right]]Now, we can back-substitute.Starting from the last equation:( (13/15)x_3 = 82/15 )Multiply both sides by 15:( 13x_3 = 82 )So, ( x_3 = 82 / 13 ≈ 6.3077 )Wait, 82 divided by 13 is 6.3077? Let me compute 13*6=78, 82-78=4, so 6 and 4/13, which is approximately 6.3077. Okay.Now, moving up to the second equation:( (-10/3)x_2 + (14/3)x_3 = -4/3 )We know ( x_3 = 82/13 ), so plug that in:( (-10/3)x_2 + (14/3)(82/13) = -4/3 )Compute ( (14/3)(82/13) ):14*82 = 1148, and 3*13=39, so 1148/39.So, equation becomes:( (-10/3)x_2 + 1148/39 = -4/3 )Multiply all terms by 39 to eliminate denominators:( (-10/3)x_2 * 39 + 1148 = (-4/3)*39 )Simplify:( -10*13 x_2 + 1148 = -52 )Which is:( -130x_2 + 1148 = -52 )Subtract 1148:( -130x_2 = -52 - 1148 = -1200 )So, ( x_2 = (-1200)/(-130) = 1200/130 = 120/13 ≈ 9.2308 )Wait, 120 divided by 13 is approximately 9.2308.Now, moving to the first equation:( 3x_1 + 2x_2 - x_3 = 5 )We have ( x_2 = 120/13 ) and ( x_3 = 82/13 ). Plug them in:( 3x_1 + 2*(120/13) - (82/13) = 5 )Compute each term:2*(120/13) = 240/13So, equation becomes:( 3x_1 + 240/13 - 82/13 = 5 )Combine constants:240/13 - 82/13 = (240 - 82)/13 = 158/13So, ( 3x_1 + 158/13 = 5 )Subtract 158/13:( 3x_1 = 5 - 158/13 = (65/13 - 158/13) = (-93)/13 )Thus, ( x_1 = (-93)/(13*3) = (-93)/39 = -31/13 ≈ -2.3846 )Wait, that's negative. But the problem states that all ( x_i ) should be non-negative. So, ( x_1 ) is negative here, which violates the condition.Hmm, so does that mean there's no solution with all non-negative ( x_i )? But the determinant was non-zero, so there is a unique solution, but it's not all non-negative.But wait, maybe I made a mistake in calculations. Let me double-check.Starting from the beginning:First, the determinant was calculated as -18, which is correct.Then, using Gaussian elimination:Original augmented matrix:3 2 -1 |52 -2 4 |21 1 1 |7First, I did R3 = R3 - (1/3)R1:1 - (1/3)*3 = 01 - (1/3)*2 = 1 - 2/3 = 1/31 - (1/3)*(-1) = 1 + 1/3 = 4/37 - (1/3)*5 = 7 - 5/3 = 16/3So, R3 becomes [0, 1/3, 4/3 | 16/3]Then, R2 = R2 - (2/3)R1:2 - (2/3)*3 = 0-2 - (2/3)*2 = -2 - 4/3 = -10/34 - (2/3)*(-1) = 4 + 2/3 = 14/32 - (2/3)*5 = 2 - 10/3 = -4/3So, R2 becomes [0, -10/3, 14/3 | -4/3]Then, moving to eliminate R3's second column. The multiplier was -1/10.So, R3 = R3 - (1/10)R2.Calculating R3:x2: 1/3 - (1/10)*(-10/3) = 1/3 + 1/3 = 2/3x3: 4/3 - (1/10)*(14/3) = 4/3 - 14/30 = 40/30 - 14/30 = 26/30 = 13/15Constant term: 16/3 - (1/10)*(-4/3) = 16/3 + 4/30 = 16/3 + 2/15 = 80/15 + 2/15 = 82/15So, R3 is [0, 2/3, 13/15 | 82/15]Then, solving for x3:13/15 x3 = 82/15 => x3 = 82/13 ≈6.3077Then, back to R2:-10/3 x2 +14/3 x3 = -4/3Plug x3 =82/13:-10/3 x2 +14/3*(82/13) = -4/3Compute 14/3 *82/13 = (14*82)/(3*13)=1148/39So, equation:-10/3 x2 +1148/39 = -4/3Multiply all terms by 39:-10*13 x2 +1148 = -4*13-130x2 +1148 = -52-130x2 = -52 -1148 = -1200x2= (-1200)/(-130)=1200/130=120/13≈9.2308Then, R1:3x1 +2x2 -x3=5Plug x2=120/13, x3=82/13:3x1 +2*(120/13) -82/13=5Compute 2*(120/13)=240/13So, 3x1 +240/13 -82/13=5240/13 -82/13=158/13So, 3x1 +158/13=53x1=5 -158/13=65/13 -158/13= -93/13x1= (-93)/(13*3)= -31/13≈-2.3846So, calculations seem correct. So, x1 is negative. Therefore, the unique solution has x1 negative, which is not allowed as per the problem's condition.Therefore, there is no solution where all x_i are non-negative.Wait, but the problem says "verify whether there is a unique solution that maintains balance and transparency, i.e., all x_i values are non-negative." So, since the unique solution has x1 negative, such a solution does not exist.But wait, maybe I should check if there's another solution? But since determinant is non-zero, there's only one solution. So, no, there's no other solution.Therefore, the answer to part 1 is that there is no unique solution with all x_i non-negative.Wait, but the problem says "verify whether there is a unique solution that maintains balance and transparency, i.e., all x_i values are non-negative. If such a solution exists, find the values of x_1, x_2, and x_3."So, since the unique solution has x1 negative, such a solution doesn't exist. So, the answer is that there is no solution with all x_i non-negative.Alternatively, maybe I made a mistake in interpreting the problem. Maybe the system is supposed to have a solution with non-negative x_i, but in reality, it doesn't. So, the conclusion is that no such solution exists.Moving on to part 2.The student explores the dynamical stability by considering a quadratic function ( f(x_1, x_2, x_3) = 3x_1^2 + x_2^2 + 2x_3^2 -10 ). The Hessian matrix must be positive definite for stability.First, the Hessian matrix of a quadratic function is simply the matrix of coefficients of the squared terms. So, for ( f(x) = 3x_1^2 + x_2^2 + 2x_3^2 -10 ), the Hessian H is:[H = begin{bmatrix}6 & 0 & 0 0 & 2 & 0 0 & 0 & 4 end{bmatrix}]Wait, because the second derivative of f with respect to x1 is 6, x2 is 2, x3 is 4, and the cross partials are zero.So, Hessian is diagonal with entries 6, 2, 4.To check if H is positive definite, we can check if all leading principal minors are positive.First minor: 6 >0Second leading minor: determinant of top-left 2x2 matrix:|6 0||0 2| = 6*2 -0=12>0Third leading minor: determinant of H, which is 6*2*4=48>0Since all leading principal minors are positive, the Hessian is positive definite.Therefore, the function is convex, and the critical point is a minimum, so the system is stable.Alternatively, since all eigenvalues of H are positive (6,2,4), it's positive definite.So, the Hessian is positive definite, so the model is stable.Wait, but the function is f(x1,x2,x3)=3x1² +x2² +2x3² -10. So, the Hessian is indeed diagonal with 6,2,4.Yes, so positive definite.So, summarizing:1. The system has a unique solution, but x1 is negative, so no solution with all x_i non-negative.2. The Hessian is positive definite, so the model is stable.But wait, in part 1, the question is whether there is a unique solution that maintains balance and transparency, i.e., all x_i non-negative. Since the unique solution doesn't satisfy the non-negativity, such a solution doesn't exist.So, the answer is that there is no such solution with all x_i non-negative.But let me just think again: is there a way to have a solution with all x_i non-negative? Maybe if I use another method, like considering the system as a linear program with non-negativity constraints.But since the system has a unique solution, and that solution doesn't satisfy the constraints, there's no other solution. So, the answer is no.Final Answer1. There is no unique solution with all ( x_i ) non-negative. 2. The Hessian matrix is positive definite, indicating stability.The final answers are:1. boxed{text{No solution with all } x_i geq 0}2. boxed{text{The Hessian is positive definite}}</think>

🧠:<think>Okay, so I have these two problems to solve, both related to computer systems from the 1970s. I remember that back then, computers were much simpler and had limited resources, so the algorithms and memory management were quite different from today. Let me try to tackle each problem step by step.Starting with the first question: It's about an algorithm that processes data in blocks of 256 bytes. The system is a 16-bit architecture, meaning each memory address can store 2 bytes of data. The task is to calculate how many such 256-byte blocks can be stored in the memory if the maximum address space is fully utilized.Hmm, okay. So, first, I need to figure out the total memory available. In a 16-bit system, the maximum addressable memory is 2^16 bytes. Wait, no, actually, each memory address can store 2 bytes because it's 16-bit. So, each address holds a word of 2 bytes. Therefore, the total number of addresses is 2^16, and each address holds 2 bytes. So, total memory would be 2^16 addresses multiplied by 2 bytes per address.Let me write that down:Total memory = Number of addresses × Bytes per addressTotal memory = 2^16 × 2 bytesCalculating 2^16: I know that 2^10 is 1024, so 2^16 is 65536. Therefore, 65536 × 2 = 131072 bytes. So, the total memory is 131072 bytes.Now, each block is 256 bytes. To find out how many such blocks can fit into the total memory, I need to divide the total memory by the size of each block.Number of blocks = Total memory / Block sizeNumber of blocks = 131072 bytes / 256 bytesLet me compute that. 131072 divided by 256. Hmm, 256 × 500 is 128000, which is less than 131072. The difference is 131072 - 128000 = 3072. Now, 256 × 12 = 3072. So, 500 + 12 = 512. Therefore, 256 × 512 = 131072.So, the number of blocks is 512.Wait, let me double-check that. 256 × 512. 256 × 500 = 128,000 and 256 × 12 = 3,072. Adding them gives 131,072, which matches the total memory. Yep, that seems correct.So, the first answer is 512 blocks.Moving on to the second question: It's about a sorting algorithm with a time complexity of O(n log n). In the 1970s, the computer processed 10,000 data entries in 1 hour. We need to estimate the time it would take to process 100,000 data entries, assuming the time complexity remains the same and the processing speed is constant.Alright, so time complexity is O(n log n). That means the time taken is proportional to n multiplied by the logarithm of n. So, if we have two different input sizes, n1 and n2, the ratio of their times should be approximately (n2 log n2) / (n1 log n1).Given:n1 = 10,000Time1 = 1 hourn2 = 100,000Time2 = ?So, Time2 / Time1 = (n2 log n2) / (n1 log n1)Let me compute log n1 and log n2. Since the base of the logarithm isn't specified, I think it's safe to assume it's base 2, as that's common in computer science. But actually, since we're dealing with ratios, the base won't matter because it will cancel out. Let me verify that.Wait, actually, let's see. Suppose log is base 10. Then log10(10,000) is 4, and log10(100,000) is 5. If it's base 2, log2(10,000) is approximately 13.2877, and log2(100,000) is approximately 16.6096. But since the base is consistent in both, the ratio will be the same regardless of the base. So, actually, the base doesn't affect the ratio, so we can proceed without worrying about the base.But to be precise, let's compute it using base 10, since the numbers are powers of 10, which might make it easier.So, log(n1) = log10(10,000) = 4log(n2) = log10(100,000) = 5Therefore, the ratio becomes:(n2 log n2) / (n1 log n1) = (100,000 × 5) / (10,000 × 4) = (500,000) / (40,000) = 12.5So, Time2 = Time1 × 12.5 = 1 hour × 12.5 = 12.5 hours.Wait, that seems like a lot. Let me think again. If n increases by a factor of 10, how does the time change? For O(n log n), the time increases by a factor of 10 × log(10). Since log(10) is approximately 1 (in base 10), so it's 10 × 1 = 10. But wait, in our calculation, it's 12.5. Hmm, that's because the log(n) increases by 1 when n increases by a factor of 10 in base 10.Wait, let me clarify. If n1 = 10,000 and n2 = 100,000, which is 10 times larger. So, the ratio is (10 × log(10n1)) / log(n1). If log is base 10, log(10n1) = log(n1) + 1. So, the ratio becomes (10 × (log(n1) + 1)) / log(n1) = 10 × (1 + 1/log(n1)).Since log(n1) is 4, this becomes 10 × (1 + 1/4) = 10 × 1.25 = 12.5. So, that's consistent with our previous result.Alternatively, if we use natural logarithm, which is more common in some contexts, the ratio would be:(n2 ln n2) / (n1 ln n1)n1 = 10,000, ln(10,000) ≈ 9.2103n2 = 100,000, ln(100,000) ≈ 11.5129So, the ratio is (100,000 × 11.5129) / (10,000 × 9.2103) ≈ (1,151,290) / (92,103) ≈ 12.5Same result. So, regardless of the logarithm base, the ratio is 12.5. Therefore, Time2 is 12.5 hours.But wait, is that realistic? Processing 10,000 entries in 1 hour, and 100,000 in 12.5 hours? That seems like a significant increase, but given the O(n log n) complexity, it's expected. Let me think about it in terms of operations.For n=10,000, the number of operations is roughly 10,000 × log(10,000). If log is base 2, that's about 10,000 × 13.2877 ≈ 132,877 operations.For n=100,000, it's 100,000 × log(100,000) ≈ 100,000 × 16.6096 ≈ 1,660,960 operations.So, the number of operations increases by a factor of approximately 1,660,960 / 132,877 ≈ 12.5. So, yes, that's consistent. Therefore, the time increases by the same factor, assuming processing speed is constant.So, the second answer is 12.5 hours.Wait, but 12.5 hours is quite a long time. In the 1970s, computers were much slower, so processing 10,000 entries in an hour might have been a big task. So, scaling up by a factor of 12.5 would indeed take over a day, which seems plausible.Alternatively, if we consider that 10,000 entries take 1 hour, then 100,000 entries would take 10 times longer if it were O(n). But since it's O(n log n), it's a bit more than 10 times. Specifically, 10 × (log(100,000)/log(10,000)) = 10 × (5/4) = 12.5. So, that makes sense.Therefore, I think my calculations are correct.Final Answer1. The total number of 256-byte blocks is boxed{512}.2. The estimated time to process 100,000 data entries is boxed{12.5} hours.</think>