🧠:<think>Alright, so I have this problem about a two-vehicle collision, and I need to figure out the final velocity after the collision and the percentage of kinetic energy lost. Hmm, okay, let me try to break this down step by step.First, the problem says it's a perfectly inelastic collision, which means the two vehicles stick together after the collision and move with the same velocity. That makes sense because they became entangled. So, I remember that in perfectly inelastic collisions, momentum is conserved, but kinetic energy isn't. So, I should use the conservation of momentum principle here.Alright, let's note down the given information:- Mass of Vehicle A, ( m_A = 1500 ) kg- Velocity of Vehicle A, ( v_A = 20 ) m/s (east)- Mass of Vehicle B, ( m_B = 2000 ) kg- Velocity of Vehicle B, ( v_B = 15 ) m/s (north)Since the vehicles are moving in perpendicular directions (east and north), their velocities are perpendicular to each other. That means I can treat the momentum in the east direction and the north direction separately. So, I can calculate the momentum in each direction before the collision and then find the final velocity components in each direction after the collision.Let me recall the formula for momentum: ( p = m times v ). So, the momentum of each vehicle before the collision is:- Momentum of A, ( p_A = m_A times v_A = 1500 times 20 = 30,000 ) kg·m/s (east)- Momentum of B, ( p_B = m_B times v_B = 2000 times 15 = 30,000 ) kg·m/s (north)Wait, both have the same momentum? Interesting, that might make things symmetric. But let me not get ahead of myself.Since momentum is conserved, the total momentum before the collision equals the total momentum after the collision. After the collision, both vehicles move together with a common velocity ( v ). So, the total momentum after the collision will be the combined mass times the final velocity.Let me denote the combined mass as ( m_{total} = m_A + m_B = 1500 + 2000 = 3500 ) kg.So, the total momentum after the collision is ( p_{total} = m_{total} times v ).But since momentum is a vector quantity, I need to consider both the east and north components separately.So, the total momentum in the east direction before collision is ( p_{A_east} = 30,000 ) kg·m/s, and the total momentum in the north direction is ( p_{B_north} = 30,000 ) kg·m/s.After the collision, the combined mass will have components of momentum in both the east and north directions. Therefore, the final velocity will have both east and north components.Let me denote the final velocity as ( vec{v} = (v_east, v_north) ).So, the momentum in the east direction after collision is ( m_{total} times v_east ), and similarly, the momentum in the north direction is ( m_{total} times v_north ).Since momentum is conserved in each direction, we have:- East direction: ( m_A times v_A = m_{total} times v_east )- North direction: ( m_B times v_B = m_{total} times v_north )Plugging in the numbers:For east direction:( 1500 times 20 = 3500 times v_east )( 30,000 = 3500 times v_east )So, ( v_east = 30,000 / 3500 )Let me compute that: 30,000 divided by 3500. Hmm, 3500 goes into 30,000 eight times because 3500 x 8 = 28,000, which leaves 2,000. Then, 2,000 / 3500 is approximately 0.571. So, total is approximately 8.571 m/s east.Similarly, for north direction:( 2000 times 15 = 3500 times v_north )( 30,000 = 3500 times v_north )So, ( v_north = 30,000 / 3500 )Same as above, which is approximately 8.571 m/s north.Wait, that's interesting. Both components are the same? So, the final velocity has equal east and north components. That means the direction of the final velocity is at a 45-degree angle from both east and north. Because if the components are equal, the angle is 45 degrees.So, the magnitude of the final velocity can be found using the Pythagorean theorem because the components are perpendicular.So, ( v = sqrt{v_east^2 + v_north^2} )Since both components are 8.571 m/s, plugging in:( v = sqrt{(8.571)^2 + (8.571)^2} )( v = sqrt{2 times (8.571)^2} )( v = 8.571 times sqrt{2} )Calculating ( sqrt{2} ) is approximately 1.4142, so:( v ≈ 8.571 times 1.4142 ≈ 12.124 ) m/sSo, the magnitude of the final velocity is approximately 12.124 m/s, and the direction is 45 degrees north of east.Wait, is that right? Let me double-check my calculations.First, the momentum in each direction: both 30,000 kg·m/s. Divided by 3500 kg gives 8.571 m/s in each direction. So, yes, that's correct.Then, the magnitude is sqrt(8.571² + 8.571²) = 8.571 * sqrt(2) ≈ 12.124 m/s. That seems correct.So, that answers the first part: the final velocity is approximately 12.124 m/s at a 45-degree angle north of east.Now, moving on to the second part: calculating the kinetic energy before and after the collision and finding the percentage lost.I remember that kinetic energy is given by ( KE = frac{1}{2} m v^2 ). Since kinetic energy is scalar, I can calculate it separately for each vehicle before the collision and sum them up.So, let's compute the initial kinetic energy.For Vehicle A:( KE_A = frac{1}{2} m_A v_A^2 = 0.5 times 1500 times (20)^2 )Calculating that:20 squared is 400.0.5 x 1500 = 750.750 x 400 = 300,000 J.For Vehicle B:( KE_B = frac{1}{2} m_B v_B^2 = 0.5 times 2000 times (15)^2 )Calculating that:15 squared is 225.0.5 x 2000 = 1000.1000 x 225 = 225,000 J.So, total initial kinetic energy is ( KE_{initial} = 300,000 + 225,000 = 525,000 ) J.Now, the kinetic energy after the collision. Since both vehicles are moving together with velocity v, which we found to be approximately 12.124 m/s.So, ( KE_{final} = frac{1}{2} m_{total} v^2 )Plugging in the numbers:0.5 x 3500 x (12.124)^2First, compute (12.124)^2:12.124 x 12.124. Let me compute that.12 x 12 = 14412 x 0.124 = 1.4880.124 x 12 = 1.4880.124 x 0.124 ≈ 0.015376Adding them up:144 + 1.488 + 1.488 + 0.015376 ≈ 144 + 2.976 + 0.015376 ≈ 146.991376So, approximately 147 m²/s².Wait, that can't be right because 12.124 squared is actually:Let me compute 12.124 squared more accurately.12.124 x 12.124:First, 12 x 12 = 14412 x 0.124 = 1.4880.124 x 12 = 1.4880.124 x 0.124 ≈ 0.015376So, adding all four terms:144 + 1.488 + 1.488 + 0.015376 = 144 + 2.976 + 0.015376 = 146.991376So, approximately 146.991 m²/s².Therefore, ( KE_{final} = 0.5 x 3500 x 146.991 )Compute 0.5 x 3500 first: that's 1750.Then, 1750 x 146.991.Let me compute that:First, 1750 x 100 = 175,0001750 x 40 = 70,0001750 x 6.991 ≈ 1750 x 7 = 12,250, but subtract 1750 x 0.009 ≈ 15.75So, 12,250 - 15.75 ≈ 12,234.25Adding all together:175,000 + 70,000 = 245,000245,000 + 12,234.25 ≈ 257,234.25 JSo, approximately 257,234 J.Wait, let me verify that multiplication another way.Alternatively, 1750 x 146.991:Breakdown 146.991 into 100 + 40 + 6.991So, 1750 x 100 = 175,0001750 x 40 = 70,0001750 x 6.991 ≈ 1750 x 7 = 12,250 minus 1750 x 0.009 ≈ 15.75, so 12,250 - 15.75 = 12,234.25Adding up: 175,000 + 70,000 = 245,000; 245,000 + 12,234.25 = 257,234.25 J.Yes, that seems correct.So, the final kinetic energy is approximately 257,234 J.Now, the initial kinetic energy was 525,000 J, and the final is approximately 257,234 J.So, the kinetic energy lost is ( KE_{lost} = KE_{initial} - KE_{final} = 525,000 - 257,234 ≈ 267,766 ) J.To find the percentage lost, we can use:( text{Percentage lost} = left( frac{KE_{lost}}{KE_{initial}} right) times 100 )Plugging in the numbers:( text{Percentage lost} = left( frac{267,766}{525,000} right) times 100 )Calculating the division:267,766 ÷ 525,000 ≈ 0.510So, 0.510 x 100 ≈ 51.0%Therefore, approximately 51% of the kinetic energy was lost in the collision.Wait, let me verify that division more accurately.267,766 ÷ 525,000.Let me compute 267,766 ÷ 525,000.First, note that 525,000 x 0.5 = 262,500.Subtract that from 267,766: 267,766 - 262,500 = 5,266.So, 0.5 + (5,266 / 525,000)Compute 5,266 ÷ 525,000.525,000 goes into 5,266 approximately 0.01003 times.So, total is approximately 0.51003, which is 51.003%.So, approximately 51.0%.Therefore, about 51% of the kinetic energy was lost.Hmm, that seems a bit high, but considering it's a perfectly inelastic collision, which is a type of inelastic collision where maximum kinetic energy is lost, so that makes sense.Let me just recap:1. Calculated the final velocity by conserving momentum in both east and north directions, found each component to be approximately 8.571 m/s, leading to a magnitude of about 12.124 m/s at a 45-degree angle.2. Calculated initial KE as 525,000 J, final KE as approximately 257,234 J, leading to a loss of about 267,766 J, which is roughly 51% of the initial KE.I think that's solid. I don't see any calculation errors upon reviewing, so I feel confident with these results.Final Answer1. The final velocity is boxed{12.1 text{ m/s}} at a direction of boxed{45^circ} north of east.2. The percentage of kinetic energy lost is boxed{51%}.</think>

🧠:<think>Alright, so I'm trying to help the CEO evaluate whether switching to renewable energy for their manufacturing plants is a good idea. They provided some data, and I need to figure out two things: the total cost over 20 years and when the cumulative energy from renewables equals that from non-renewables.First, let's tackle the cost analysis. There are two options: renewable and non-renewable. The renewable setup has a big upfront cost of 50 million, but lower annual operational costs of 500,000. The non-renewable setup has no upfront cost but much higher annual costs of 5 million. We need to calculate the present value of these costs over 20 years with a 5% discount rate.I remember that present value calculations use the formula PV = FV / (1 + r)^n, where r is the discount rate and n is the number of years. For the renewable setup, the initial investment is 50 million right now, so its present value is just 50 million. Then, each year for 20 years, there's an operational cost of 500,000. I need to find the present value of this annuity.The formula for the present value of an annuity is PV = PMT * [1 - (1 + r)^-n] / r. Plugging in the numbers: PMT is 500,000, r is 5% or 0.05, and n is 20. So, PV = 500,000 * [1 - (1.05)^-20] / 0.05. Let me calculate that. First, (1.05)^-20 is approximately 0.3769. So, 1 - 0.3769 is 0.6231. Dividing that by 0.05 gives about 12.462. Multiply by 500,000, that's 12.462 * 500,000 = 6,231,000. So, the present value of the operational costs for renewables is about 6.231 million. Adding the initial investment, the total present value is 50 million + 6.231 million = 56.231 million.Now, for the non-renewable setup, there's no initial cost, but each year we have 5 million in operational costs. Using the same present value of annuity formula: PV = 5,000,000 * [1 - (1.05)^-20] / 0.05. We already know [1 - (1.05)^-20] / 0.05 is 12.462. So, 5,000,000 * 12.462 = 62,310,000. So, the present value for non-renewables is about 62.31 million.Comparing the two, the renewable setup has a present value of about 56.23 million, while non-renewable is 62.31 million. So, over 20 years, the renewable setup is more cost-effective.Next, the second sub-problem is determining when the cumulative energy from renewables equals that from non-renewables. The renewable setup starts at 100 GWh and degrades 1% each year, so it's a decreasing geometric series. The non-renewable setup is constant at 120 GWh each year.The cumulative energy for renewables after n years is the sum of a geometric series: S = a1 * (1 - r^n) / (1 - r), where a1 is 100 GWh, r is 0.99 (since it decreases by 1% each year). For non-renewables, it's just 120 * n.We need to find n such that 100*(1 - 0.99^n)/0.01 = 120n. Simplifying, 100*(1 - 0.99^n)/0.01 is 10,000*(1 - 0.99^n). So, 10,000*(1 - 0.99^n) = 120n.This equation is a bit tricky. Maybe I can approximate it. Let's try plugging in some numbers. Let's start with n=10: 10,000*(1 - 0.99^10) ≈ 10,000*(1 - 0.9044) ≈ 10,000*0.0956 ≈ 956 GWh. Non-renewables would be 120*10=1200 GWh. So, renewables are less.n=15: 10,000*(1 - 0.99^15) ≈ 10,000*(1 - 0.8609) ≈ 10,000*0.1391 ≈ 1391 GWh. Non-renewables: 120*15=1800 GWh. Still less.n=20: 10,000*(1 - 0.99^20) ≈ 10,000*(1 - 0.8171) ≈ 10,000*0.1829 ≈ 1829 GWh. Non-renewables: 120*20=2400 GWh. Still less.Wait, so even at 20 years, renewables haven't caught up. Maybe I need to go beyond 20 years? But the question is whether it's within the 20-year period. Since at 20 years, renewables have 1829 GWh and non-renewables have 2400 GWh, the cumulative energy from renewables hasn't equaled non-renewables yet. So, the year when they are equal is beyond 20 years.But wait, maybe I made a mistake. Let me check the formula again. The cumulative energy for renewables is the sum of 100 + 99 + 98.01 + ... for n years. That's a geometric series with a common ratio of 0.99. The sum is 100*(1 - 0.99^n)/0.01, which is correct.Alternatively, maybe I can set up the equation 100*(1 - 0.99^n)/0.01 = 120n and solve for n numerically. Let's try n=25: 100*(1 - 0.99^25)/0.01 ≈ 100*(1 - 0.7778)/0.01 ≈ 100*0.2222/0.01 ≈ 2222 GWh. Non-renewables: 120*25=3000 GWh. Still less.n=30: 100*(1 - 0.99^30)/0.01 ≈ 100*(1 - 0.7418)/0.01 ≈ 100*0.2582/0.01 ≈ 2582 GWh. Non-renewables: 120*30=3600 GWh.Hmm, it's increasing but not catching up. Wait, maybe I need to consider that the renewable energy is decreasing each year, so the total cumulative will approach a limit. The sum to infinity for the renewable energy is 100 / 0.01 = 10,000 GWh. So, it will never reach 10,000 GWh, but the non-renewables will keep increasing without bound. So, actually, the cumulative energy from renewables will never equal that of non-renewables because non-renewables keep adding 120 each year, while renewables approach 10,000. Wait, but 10,000 is more than 120*83=9960, so at n=83, non-renewables would be 9960, and renewables would be approaching 10,000. So, the cumulative energy from renewables would surpass non-renewables at some point, but it's way beyond 20 years.Wait, but in the problem, the non-renewable setup maintains a constant energy yield, so each year it's 120 GWh. So, the cumulative energy for non-renewables is 120n. For renewables, it's 100*(1 - 0.99^n)/0.01. We need to find n where 100*(1 - 0.99^n)/0.01 = 120n.Let me rearrange the equation: 100*(1 - 0.99^n) = 1.2n. So, 100 - 100*0.99^n = 1.2n. Let's write it as 100 - 1.2n = 100*0.99^n.This is a transcendental equation and can't be solved algebraically. We need to use numerical methods. Let's try n=100: 100 - 1.2*100= -20. 100*0.99^100≈100*0.366≈36.6. So, -20 ≈36.6? No.Wait, maybe I need to try smaller n. Let's try n=50: 100 - 1.2*50=100-60=40. 100*0.99^50≈100*0.6065≈60.65. So, 40≈60.65? No.n=60: 100 - 72=28. 100*0.99^60≈100*0.5488≈54.88. 28≈54.88? No.n=70: 100 - 84=16. 100*0.99^70≈100*0.496≈49.6. 16≈49.6? No.n=80: 100 - 96=4. 100*0.99^80≈100*0.449≈44.9. 4≈44.9? No.n=90: 100 - 108= -8. 100*0.99^90≈100*0.411≈41.1. -8≈41.1? No.Wait, this isn't working. Maybe I need to set up the equation differently. Let's consider the function f(n) = 100*(1 - 0.99^n)/0.01 - 120n. We need to find n where f(n)=0.f(20)=1829 - 2400= -571.f(25)=2222 - 3000= -778.f(30)=2582 - 3600= -1018.f(40)=100*(1 - 0.99^40)/0.01 - 4800≈100*(1 - 0.6676)/0.01 -4800≈100*0.3324/0.01 -4800≈3324 -4800= -1476.Wait, it's getting more negative. Maybe I need to go higher.Wait, but as n increases, the left side approaches 10,000, and the right side is 120n. So, 120n will eventually surpass 10,000 when n=83.33. So, at n=83, 120*83=9960. At n=84, 120*84=10,080. So, the cumulative energy for non-renewables will surpass 10,000 GWh at n=84, while renewables approach 10,000 as n approaches infinity. Therefore, the cumulative energy from renewables will never equal that of non-renewables because non-renewables will keep increasing beyond 10,000, while renewables approach 10,000 asymptotically.Wait, that can't be right because the sum of the geometric series for renewables is 100 / 0.01 = 10,000 GWh as n approaches infinity. So, the cumulative energy for renewables will approach 10,000, while non-renewables will keep increasing without bound. Therefore, the cumulative energy from renewables will never equal that of non-renewables because non-renewables will always be adding more each year, surpassing 10,000 at n=84, while renewables never reach 10,000.Wait, but that contradicts the earlier thought that they might meet at some point. Maybe I made a mistake in the setup. Let me think again.The cumulative energy for renewables is S = 100*(1 - 0.99^n)/0.01. For non-renewables, it's S = 120n. We need to find n where 100*(1 - 0.99^n)/0.01 = 120n.Let me rearrange: 100*(1 - 0.99^n) = 1.2n.So, 100 - 100*0.99^n = 1.2n.Let me define f(n) = 100 - 100*0.99^n - 1.2n. We need to find n where f(n)=0.At n=0: f(0)=100 -100 -0=0. So, n=0 is a solution, but that's trivial.Looking for n>0.Let's try n=10: f(10)=100 -100*0.9044 -12≈100 -90.44 -12≈-2.44.n=9: f(9)=100 -100*0.9135 -10.8≈100 -91.35 -10.8≈-2.15.n=8: f(8)=100 -100*0.9228 -9.6≈100 -92.28 -9.6≈-1.88.n=7: f(7)=100 -100*0.9321 -8.4≈100 -93.21 -8.4≈-1.61.n=6: f(6)=100 -100*0.9415 -7.2≈100 -94.15 -7.2≈-1.35.n=5: f(5)=100 -100*0.9510 -6≈100 -95.1 -6≈-1.1.n=4: f(4)=100 -100*0.9606 -4.8≈100 -96.06 -4.8≈-0.86.n=3: f(3)=100 -100*0.9703 -3.6≈100 -97.03 -3.6≈-0.63.n=2: f(2)=100 -100*0.9801 -2.4≈100 -98.01 -2.4≈-0.41.n=1: f(1)=100 -100*0.99 -1.2≈100 -99 -1.2≈-0.2.n=0.5: f(0.5)=100 -100*sqrt(0.99) -0.6≈100 -100*0.99499 -0.6≈100 -99.499 -0.6≈-0.1.Wait, so f(n) is negative for n>0. At n=0, it's zero, but for any positive n, f(n) is negative. That suggests that the cumulative energy from renewables is always less than non-renewables for n>0. Therefore, they never equal again after n=0.But that can't be right because the sum of renewables approaches 10,000, and non-renewables will pass 10,000 at n=84. So, at n=84, non-renewables have 10,080, while renewables have just under 10,000. So, they don't meet at any point beyond n=0.Wait, but that contradicts the idea that the cumulative energy from renewables would ever catch up. It seems like the non-renewables will always have more cumulative energy because their total keeps increasing without bound, while renewables approach a limit.Therefore, the cumulative energy from renewables will never equal that of non-renewables after n=0. So, within the 20-year period, they don't meet, and in fact, they never meet again.But wait, that seems counterintuitive because the renewable setup starts at 100 GWh and decreases, while non-renewables are constant. So, the total for non-renewables is linear, while renewables are a converging series. So, the total for non-renewables will always be higher after the first year.Wait, let's check n=1: renewables=100, non-renewables=120. So, non-renewables are higher.n=2: renewables=100+99=199, non-renewables=240. Still higher.n=3: 100+99+98.01=297.01 vs 360. Still higher.So, it's clear that non-renewables are always ahead in cumulative energy. Therefore, the cumulative energy from renewables never equals that of non-renewables after the first year. So, the answer is that they never equal within the 20-year period, and in fact, never do so.Wait, but the problem says "determine the year in which the cumulative energy yield from the renewable energy infrastructure will be equal to the cumulative energy yield from the non-renewable energy setup." If they never equal, then the answer is that it never happens within the 20-year period, and actually, never happens at all.But maybe I made a mistake in the setup. Let me double-check.Renewables: each year, energy decreases by 1%, so year 1:100, year2:99, year3:98.01, etc. Cumulative is the sum of this series.Non-renewables: each year, 120 GWh. Cumulative is 120n.We need to find n where sum_{k=1}^n 100*(0.99)^{k-1} = 120n.The sum is 100*(1 - 0.99^n)/0.01.So, 100*(1 - 0.99^n)/0.01 = 120n.As n increases, the left side approaches 10,000, while the right side grows without bound. Therefore, the equation 10,000 = 120n would imply n=83.33, but the left side never actually reaches 10,000; it approaches it asymptotically. Therefore, the cumulative energy from renewables will never equal that of non-renewables because non-renewables will surpass 10,000 at n=84, while renewables never reach 10,000.Therefore, the answer is that the cumulative energy from renewables never equals that of non-renewables within the 20-year period, and in fact, never does so at all.</think>

🧠:<think>Alright, so I've got these two problems about calculating popularity scores for an artist's album. Let me try to break them down step by step.Starting with the first problem:1. Given: - Number of tracks, ( n = 10 ) - Total popularity score for the album, ( P_{text{total}} = 1000 ) - Number of dedicated listeners, ( m = 5 ) - Average days of engagement per listener, ( bar{k} = 30 ) The formula for the popularity score of each track is: [ P_i = a_i cdot left(1 + sum_{j=1}^{m} frac{1}{k_j}right) ] Since each listener has been engaging for an average of 30 days, I can assume that each ( k_j = 30 ). That simplifies the sum because each term is ( frac{1}{30} ). So, the sum ( sum_{j=1}^{m} frac{1}{k_j} ) becomes ( 5 times frac{1}{30} = frac{5}{30} = frac{1}{6} ). Therefore, the popularity score for each track is: [ P_i = a_i cdot left(1 + frac{1}{6}right) = a_i cdot frac{7}{6} ] Since there are 10 tracks, the total popularity score is the sum of all individual track scores: [ P_{text{total}} = sum_{i=1}^{10} P_i = sum_{i=1}^{10} left(a_i cdot frac{7}{6}right) ] Let me denote the total unique streams as ( A = sum_{i=1}^{10} a_i ). Then: [ P_{text{total}} = frac{7}{6} cdot A ] Plugging in the total popularity score: [ 1000 = frac{7}{6} cdot A ] Solving for ( A ): [ A = 1000 times frac{6}{7} ] [ A = frac{6000}{7} ] [ A approx 857.14 ] Since the number of streams should be a whole number, I might need to round this. But the problem doesn't specify, so I can leave it as ( frac{6000}{7} ) or approximately 857.14.Wait, let me double-check my steps.- I assumed each ( k_j = 30 ) because the average is 30. Is that correct? The problem says "each dedicated listener has been actively engaging for an average of 30 days." So, does that mean each listener has exactly 30 days, or is it an average? Hmm, the wording is a bit ambiguous. If it's an average, then the sum ( sum frac{1}{k_j} ) would be ( frac{m}{bar{k}} ) only if all ( k_j ) are equal. But since it's an average, the sum would be ( m times frac{1}{bar{k}} ) only if all ( k_j ) are equal. Otherwise, it's not necessarily true.Wait, actually, the average ( bar{k} = frac{sum k_j}{m} ), so ( sum k_j = m times bar{k} ). But we need ( sum frac{1}{k_j} ), which isn't directly related to ( bar{k} ) unless all ( k_j ) are equal. Since the problem says "each dedicated listener has been actively engaging for an average of 30 days," it might mean that each listener has exactly 30 days. Otherwise, if it's an average, we can't compute the exact sum without more information.Given that, I think it's safe to assume each ( k_j = 30 ), so the sum is ( 5 times frac{1}{30} = frac{1}{6} ). So my initial calculation is correct.Therefore, the total unique streams ( A = frac{6000}{7} approx 857.14 ). Since streams are whole numbers, maybe they expect an exact fraction or rounded to the nearest whole number. The problem doesn't specify, so I'll go with the exact value.Moving on to the second problem:2. Given: - New album has ( n' ) tracks (unknown, but we might not need it explicitly) - Each dedicated listener's engagement increases by 20%, so new ( k_j' = k_j times 1.2 ) - Total popularity score remains the same, ( P_{text{total}}' = 1000 ) - Need to find the average unique streams per track, ( bar{a}' ) First, let's figure out the new sum ( sum_{j=1}^{m} frac{1}{k_j'} ). Since each ( k_j ) increases by 20%, the new ( k_j' = 1.2 times k_j ). Therefore, ( frac{1}{k_j'} = frac{1}{1.2 times k_j} = frac{1}{1.2} times frac{1}{k_j} ). So, the new sum is: [ sum_{j=1}^{m} frac{1}{k_j'} = frac{1}{1.2} times sum_{j=1}^{m} frac{1}{k_j} ] From the first problem, we know that ( sum_{j=1}^{m} frac{1}{k_j} = frac{1}{6} ). So: [ sum_{j=1}^{m} frac{1}{k_j'} = frac{1}{1.2} times frac{1}{6} = frac{1}{7.2} ] Therefore, the new popularity score per track is: [ P_i' = a_i' cdot left(1 + frac{1}{7.2}right) ] [ P_i' = a_i' cdot left(frac{8.2}{7.2}right) ] [ P_i' = a_i' cdot frac{41}{36} ] (since 8.2/7.2 = 41/36) The total popularity score is the sum over all tracks: [ P_{text{total}}' = sum_{i=1}^{n'} P_i' = sum_{i=1}^{n'} left(a_i' cdot frac{41}{36}right) ] Let ( A' = sum_{i=1}^{n'} a_i' ) be the total unique streams for the new album. Then: [ 1000 = frac{41}{36} cdot A' ] Solving for ( A' ): [ A' = 1000 times frac{36}{41} ] [ A' approx 1000 times 0.8780 ] [ A' approx 878.05 ] Now, the problem asks for the average unique streams per track, which is ( bar{a}' = frac{A'}{n'} ). However, we don't know ( n' ). Wait, the first album had 10 tracks, but the second album has ( n' ) tracks. The problem doesn't specify ( n' ), but it says "another album with ( n' ) tracks." So, unless ( n' ) is given or can be inferred, we might need to express the answer in terms of ( n' ). Wait, let me check the problem statement again: "Suppose the artist plans to release another album with ( n' ) tracks, and they expect each dedicated listener's engagement to increase by 20%. If they wish to maintain the same total popularity score of 1000, how many unique streams per track, on average, should they aim for in the new album?" So, they don't specify ( n' ), so perhaps we need to assume that the number of tracks remains the same? Or maybe ( n' ) is different. Hmm. Wait, in the first problem, ( n = 10 ). In the second problem, it's a new album with ( n' ) tracks. So, unless specified, ( n' ) could be different. But since the problem doesn't give ( n' ), perhaps it's the same number of tracks? Or maybe it's a different number, but we need to express the answer in terms of ( n' ). Wait, the question is asking for "how many unique streams per track, on average, should they aim for in the new album?" So, it's ( bar{a}' = frac{A'}{n'} ). We have ( A' approx 878.05 ), so ( bar{a}' = frac{878.05}{n'} ). But without knowing ( n' ), we can't compute a numerical answer. Hmm, maybe I missed something. Wait, perhaps the number of tracks is the same? The first album had 10 tracks, so maybe the new album also has 10 tracks? The problem says "another album with ( n' ) tracks," so unless specified, ( n' ) could be different. But since it's not given, maybe we need to assume ( n' = 10 ) as well? Or perhaps it's a different number, but the problem expects an answer in terms of ( n' ). Wait, let me read the problem again: "Suppose the artist plans to release another album with ( n' ) tracks, and they expect each dedicated listener's engagement to increase by 20%. If they wish to maintain the same total popularity score of 1000, how many unique streams per track, on average, should they aim for in the new album?" It doesn't specify ( n' ), so perhaps the answer is expressed in terms of ( n' ). But the first problem had 10 tracks, so maybe ( n' ) is also 10? Or maybe it's a different number, but we need to express it in terms of ( n' ). Wait, the first problem had 10 tracks, but the second problem is about a new album with ( n' ) tracks. So, unless told otherwise, ( n' ) is a variable, so the answer should be in terms of ( n' ). But the problem asks for "how many unique streams per track, on average," which is a numerical value. So, perhaps ( n' ) is the same as before, 10? Or maybe it's a different number, but the problem expects us to assume it's the same. Hmm, this is a bit ambiguous. Let me think. In the first problem, ( n = 10 ). In the second problem, it's a new album with ( n' ) tracks. Since it's a different album, ( n' ) could be different, but since it's not specified, maybe we need to assume it's the same, 10 tracks. Otherwise, we can't compute a numerical answer. Alternatively, maybe the number of tracks doesn't matter because the total popularity score is the same, so the average per track would adjust accordingly. Wait, but the total streams ( A' ) is fixed at approximately 878.05, so the average per track would be ( frac{878.05}{n'} ). But without ( n' ), we can't compute it numerically. Wait, perhaps I made a mistake earlier. Let me re-examine the second problem. The formula for each track's popularity is: [ P_i' = a_i' cdot left(1 + sum_{j=1}^{m} frac{1}{k_j'}right) ] We found that ( sum frac{1}{k_j'} = frac{1}{7.2} ), so: [ P_i' = a_i' cdot left(1 + frac{1}{7.2}right) = a_i' cdot frac{8.2}{7.2} = a_i' cdot frac{41}{36} ] Therefore, the total popularity score is: [ P_{text{total}}' = sum P_i' = frac{41}{36} cdot A' = 1000 ] So, ( A' = 1000 times frac{36}{41} approx 878.05 ) Now, the average unique streams per track is ( bar{a}' = frac{A'}{n'} ). But since ( n' ) isn't given, perhaps the problem assumes ( n' = n = 10 )? Let me check the problem statement again. It says, "another album with ( n' ) tracks." So, it's a different album, so ( n' ) could be different. But since it's not specified, maybe we need to assume it's the same number of tracks, 10. Otherwise, we can't compute a numerical answer. Alternatively, maybe the problem expects us to express the answer in terms of ( n' ), but the way it's phrased, it's asking for a numerical value. So, perhaps ( n' = 10 ) as well. Let me proceed with that assumption, that ( n' = 10 ). Then: ( bar{a}' = frac{878.05}{10} approx 87.805 ) So, approximately 87.81 streams per track on average. But wait, let me think again. If the number of tracks is different, the average would change. But since the problem doesn't specify, maybe it's safer to assume ( n' = 10 ). Alternatively, perhaps the number of tracks is irrelevant because the total streams are fixed, so the average per track is just ( A' / n' ), but without ( n' ), we can't compute it. Hmm. Wait, maybe I misread the problem. Let me check: "how many unique streams per track, on average, should they aim for in the new album?" It doesn't specify whether the new album has the same number of tracks or not. So, perhaps the answer is expressed in terms of ( n' ), but the problem expects a numerical value. Therefore, maybe ( n' ) is the same as before, 10 tracks. Alternatively, perhaps the number of tracks is not needed because the formula is per track, and the total is fixed. Wait, no, because the total streams ( A' ) is fixed, so the average per track depends on ( n' ). Hmm, this is a bit confusing. Maybe I should proceed with the assumption that ( n' = 10 ), as in the first problem, unless stated otherwise. So, with ( n' = 10 ), the average streams per track would be approximately 87.81. But let me think again. If the number of tracks is different, say ( n' ), then the average would be ( frac{878.05}{n'} ). But since the problem doesn't specify ( n' ), maybe it's expecting an answer in terms of ( n' ), but the way it's phrased, it's asking for a numerical value. Therefore, perhaps ( n' ) is the same as before, 10. Alternatively, maybe the problem expects us to realize that the number of tracks doesn't affect the average streams per track because the total streams are fixed. Wait, no, because the total streams are fixed, so the average per track is total streams divided by number of tracks. So, without knowing ( n' ), we can't compute it numerically. Hmm, this is a bit of a conundrum. Maybe the problem expects us to assume ( n' = n = 10 ), so I'll proceed with that. Therefore, the average streams per track would be approximately 87.81. But let me check my calculations again. From the second problem: - Each ( k_j' = 1.2 times 30 = 36 ) days. - So, ( sum frac{1}{k_j'} = 5 times frac{1}{36} = frac{5}{36} approx 0.1389 ) - Therefore, ( 1 + sum frac{1}{k_j'} = 1 + 0.1389 = 1.1389 ) - So, each track's popularity is ( P_i' = a_i' times 1.1389 ) - Total popularity is ( 1000 = 1.1389 times A' ) - So, ( A' = 1000 / 1.1389 approx 878.05 ) - If ( n' = 10 ), then average per track is ( 878.05 / 10 approx 87.81 ) Yes, that seems correct. Alternatively, if ( n' ) is different, say, for example, ( n' = 12 ), then the average would be ( 878.05 / 12 approx 73.17 ). But since ( n' ) isn't given, I think the problem expects us to assume it's the same as before, 10 tracks. Therefore, the average unique streams per track should be approximately 87.81. But let me think again. The problem says "another album with ( n' ) tracks," so perhaps ( n' ) is a variable, and the answer should be expressed in terms of ( n' ). But the problem asks for "how many unique streams per track, on average," which is a numerical value. Therefore, perhaps ( n' ) is the same as before, 10. Alternatively, maybe the problem expects us to realize that the number of tracks doesn't matter because the total streams are fixed, but that's not the case because the average per track depends on the number of tracks. Hmm, I'm a bit stuck here. Maybe I should proceed with the assumption that ( n' = 10 ), as in the first problem, unless stated otherwise. So, final answers: 1. Total unique streams ( A = frac{6000}{7} approx 857.14 ) 2. Average unique streams per track ( bar{a}' approx 87.81 ) But let me check if I can express the second answer as a fraction. From ( A' = 1000 times frac{36}{41} = frac{36000}{41} approx 878.05 ) So, if ( n' = 10 ), then ( bar{a}' = frac{36000}{41 times 10} = frac{3600}{41} approx 87.80 ) Alternatively, if ( n' ) is different, say, ( n' = 12 ), then ( bar{a}' = frac{36000}{41 times 12} = frac{3000}{41} approx 73.17 ) But since ( n' ) isn't given, I think the problem expects us to assume it's the same as before, 10 tracks. Therefore, the average unique streams per track should be ( frac{3600}{41} approx 87.80 ) So, rounding to two decimal places, approximately 87.80. But let me think again. If the number of tracks is different, the average would change. But since the problem doesn't specify, maybe it's expecting an answer in terms of ( n' ). But the problem asks for a numerical value, so perhaps ( n' = 10 ). Alternatively, maybe the problem expects us to realize that the number of tracks doesn't affect the total streams, so the average per track is just ( A' / n' ), but without ( n' ), we can't compute it. Therefore, perhaps the problem expects us to assume ( n' = 10 ). I think I'll go with that assumption.</think>

🧠:<think>Alright, so my sibling is trying to figure out which healthcare plan is better between Company A and Company B. They’re looking at the total annual cost, which includes both the premiums and the out-of-pocket expenses. Let me try to break this down step by step.First, let me understand what each plan offers.Company A:- Base premium: 500 per month- Deductible: 1,000 annually- Coverage: 80% of medical expenses after deductibleCompany B:- Base premium: 450 per month- Deductible: 1,200 annually- Coverage: 75% of medical expenses after deductibleSo, the employee expects to have x in medical expenses annually. We need to model the total cost for each plan as a function of x.Let me start with Company A.Company A's Total Annual Cost:1. Base Premium: This is straightforward. It's 500 per month, so annually that's 500 * 12 = 6,000.2. Out-of-Pocket Expenses: This is a bit more complex. The plan has a deductible of 1,000, which means the employee has to pay the first 1,000 out of pocket. After that, the plan covers 80% of the remaining expenses. So, if the total medical expenses are x, the out-of-pocket expenses would be: - If x ≤ 1,000: The employee pays the full x. - If x > 1,000: The employee pays 1,000 plus 20% of (x - 1,000). Because the plan covers 80%, so the employee covers the remaining 20%.So, mathematically, the out-of-pocket expenses for Company A can be written as:- If x ≤ 1000: OA(x) = x- If x > 1000: OA(x) = 1000 + 0.20*(x - 1000)Therefore, the total annual cost for Company A, TA(x), is:- If x ≤ 1000: TA(x) = 6000 + x- If x > 1000: TA(x) = 6000 + 1000 + 0.20*(x - 1000) = 7000 + 0.20x - 200 = 6800 + 0.20xWait, let me check that math again. 6000 + 1000 is 7000, and 0.20*(x - 1000) is 0.20x - 200. So, 7000 - 200 is 6800. So yes, 6800 + 0.20x for x > 1000.Company B's Total Annual Cost:Similarly, let's break it down.1. Base Premium: 450 per month, so annually that's 450 * 12 = 5,400.2. Out-of-Pocket Expenses: Deductible is 1,200, so: - If x ≤ 1200: The employee pays the full x. - If x > 1200: The employee pays 1,200 plus 25% of (x - 1,200). Because the plan covers 75%, so the employee covers 25%.So, the out-of-pocket expenses for Company B, OB(x), can be written as:- If x ≤ 1200: OB(x) = x- If x > 1200: OB(x) = 1200 + 0.25*(x - 1200)Therefore, the total annual cost for Company B, TB(x), is:- If x ≤ 1200: TB(x) = 5400 + x- If x > 1200: TB(x) = 5400 + 1200 + 0.25*(x - 1200) = 6600 + 0.25x - 300 = 6300 + 0.25xWait, let me verify that calculation. 5400 + 1200 is 6600, and 0.25*(x - 1200) is 0.25x - 300. So, 6600 - 300 is 6300. So, yes, 6300 + 0.25x for x > 1200.So now, we have the total cost functions for both companies.Summary of Total Annual Costs:- Company A: - TA(x) = 6000 + x, for x ≤ 1000 - TA(x) = 6800 + 0.20x, for x > 1000- Company B: - TB(x) = 5400 + x, for x ≤ 1200 - TB(x) = 6300 + 0.25x, for x > 1200Now, the next part is to find the value of x where TA(x) = TB(x). So, we need to set the two total cost functions equal and solve for x.But before that, let's consider the different ranges of x because the functions change at different points.The functions change at x = 1000 for Company A and x = 1200 for Company B.So, the possible ranges where the functions are linear are:1. x ≤ 10002. 1000 < x ≤ 12003. x > 1200We need to check for equality in each of these ranges.Case 1: x ≤ 1000Here, TA(x) = 6000 + x and TB(x) = 5400 + x.Set them equal:6000 + x = 5400 + xSubtract x from both sides:6000 = 5400Which is not possible. So, no solution in this range.Case 2: 1000 < x ≤ 1200In this range, TA(x) = 6800 + 0.20x and TB(x) = 5400 + x.Set them equal:6800 + 0.20x = 5400 + xLet me solve for x.Subtract 5400 from both sides:1400 + 0.20x = xSubtract 0.20x from both sides:1400 = 0.80xDivide both sides by 0.80:x = 1400 / 0.80 = 1750But wait, in this case, we're considering 1000 < x ≤ 1200. But x = 1750 is outside this range. So, no solution in this range either.Case 3: x > 1200Here, TA(x) = 6800 + 0.20x and TB(x) = 6300 + 0.25x.Set them equal:6800 + 0.20x = 6300 + 0.25xSubtract 6300 from both sides:500 + 0.20x = 0.25xSubtract 0.20x from both sides:500 = 0.05xDivide both sides by 0.05:x = 500 / 0.05 = 10,000So, x = 10,000 is the point where both plans cost the same.But let me verify that. If x = 10,000,For Company A: TA(10,000) = 6800 + 0.20*10,000 = 6800 + 2000 = 8,800For Company B: TB(10,000) = 6300 + 0.25*10,000 = 6300 + 2500 = 8,800Yes, that checks out.So, the total cost is equal at x = 10,000.Now, the question is, for values of x below this threshold, which plan is cheaper?So, for x < 10,000, which plan is cheaper?We can analyze this by testing a value less than 10,000, say x = 5,000.Compute TA(5,000) and TB(5,000).But wait, let's consider the different ranges.Since x = 5,000 is greater than 1200, we'll use the respective formulas for x > 1200.TA(5,000) = 6800 + 0.20*5,000 = 6800 + 1,000 = 7,800TB(5,000) = 6300 + 0.25*5,000 = 6300 + 1,250 = 7,550So, TB(x) is cheaper at x = 5,000.Wait, but let me check another value, say x = 1,500, which is in the range 1000 < x ≤ 1200.TA(1,500) = 6800 + 0.20*1,500 = 6800 + 300 = 7,100TB(1,500) = 5400 + 1,500 = 6,900So, TB(x) is still cheaper.Wait, but what about x just above 1000, say x = 1,100.TA(1,100) = 6800 + 0.20*1,100 = 6800 + 220 = 7,020TB(1,100) = 5400 + 1,100 = 6,500Again, TB(x) is cheaper.Wait, but what about x = 900, which is in the first range.TA(900) = 6000 + 900 = 6,900TB(900) = 5400 + 900 = 6,300So, TB(x) is cheaper here as well.Wait, so is TB(x) always cheaper for x < 10,000?Wait, let me check x = 0.TA(0) = 6000 + 0 = 6,000TB(0) = 5400 + 0 = 5,400So, TB(x) is cheaper.Wait, so actually, for all x < 10,000, TB(x) is cheaper.But let me confirm with another value, say x = 1,200.TA(1,200) = 6800 + 0.20*1,200 = 6800 + 240 = 7,040TB(1,200) = 5400 + 1,200 = 6,600Still, TB(x) is cheaper.Wait, so it seems that for all x < 10,000, TB(x) is cheaper than TA(x). At x = 10,000, they are equal.Therefore, the value of x where the total costs are equal is 10,000. For x below this, Company B is cheaper.But let me think again. Is there a point where Company A becomes cheaper? Because sometimes, depending on the slopes, one plan might become cheaper beyond a certain point.Wait, let's check x = 20,000.TA(20,000) = 6800 + 0.20*20,000 = 6800 + 4,000 = 10,800TB(20,000) = 6300 + 0.25*20,000 = 6300 + 5,000 = 11,300So, at x = 20,000, TA(x) is cheaper.Wait, so actually, beyond x = 10,000, Company A becomes cheaper.So, the crossover point is at x = 10,000. For x < 10,000, Company B is cheaper, and for x > 10,000, Company A is cheaper.Therefore, the answer is that the total annual cost is equal at x = 10,000, and for x below this, Company B is cheaper.But let me just make sure I didn't make any calculation errors.Let me recast the equations:For x > 1200:TA(x) = 6800 + 0.20xTB(x) = 6300 + 0.25xSet equal:6800 + 0.20x = 6300 + 0.25xSubtract 6300:500 + 0.20x = 0.25xSubtract 0.20x:500 = 0.05xx = 500 / 0.05 = 10,000Yes, that's correct.So, the conclusion is that at x = 10,000, both plans cost the same. For x below 10,000, Company B is cheaper, and for x above 10,000, Company A is cheaper.Therefore, the value of x where the total costs are equal is 10,000, and for x below this, Company B offers the cheaper plan.</think>