🧠:<think>Okay, so I have this problem about a business owner who is auctioning off paintings to raise funds for cancer charities. The owner is planning to hold these auctions annually for the next T years. Each painting has an initial value V_i and appreciates in value by a factor A_i each year, compounded continuously. The first part of the problem asks me to determine the total expected value of the paintings after T years. The formula given is V_i(t) = V_i * e^(A_i * t). So, I need to figure out how to calculate the total value after T years for all N paintings.Let me start by understanding the formula. For each painting i, its value at time t is V_i multiplied by e raised to the power of A_i times t. Since the appreciation is compounded continuously, this exponential growth model makes sense. So, if I have N paintings, each with their own V_i and A_i, then after T years, each painting's value will be V_i * e^(A_i * T). To find the total expected value, I need to sum up the values of all N paintings after T years.Therefore, the total value, let's call it V_total(T), should be the sum from i=1 to N of V_i * e^(A_i * T). That seems straightforward. Let me write that out:V_total(T) = Σ (from i=1 to N) [V_i * e^(A_i * T)]Yes, that makes sense. Each painting's value grows exponentially based on its own appreciation factor, and then we just add them all up for the total.Now, moving on to the second part of the problem. The business owner wants to ensure that the total funds raised from these paintings over T years is at least F, which is the target amount for the charity. I need to express the necessary condition in terms of V_i, A_i, T, and F.Hmm, so the total funds raised would be the total value after T years, right? So, if the owner auctions off all the paintings after T years, the proceeds would be V_total(T). Therefore, to ensure that this amount is at least F, we need V_total(T) ≥ F.So, substituting the expression we found earlier:Σ (from i=1 to N) [V_i * e^(A_i * T)] ≥ FIs that the necessary condition? It seems so. The sum of each painting's appreciated value must be greater than or equal to F.Wait, but let me think again. The problem says "over T years." Does that mean the total funds raised each year, or the total after T years? The wording is a bit ambiguous. It says "the total funds raised from these paintings over T years." Hmm.If it's over T years, does that mean each year they auction off the paintings, and each year the paintings appreciate? Or do they auction them all at once after T years?Looking back at the problem statement: "the owner plans to hold these auctions annually for the next T years." So, it's an annual auction. So, does that mean each year, they auction off the paintings, and the proceeds from each year's auction contribute to the total funds?Wait, but if they auction off the paintings each year, then the paintings are sold each year, so they wouldn't be around for the next year's auction. That seems conflicting.Wait, maybe I misinterpret. Maybe the owner is auctioning off the same collection each year, but the paintings appreciate each year. So, each year, the paintings are sold at their current appreciated value, and then the proceeds are donated. So, over T years, they have T auctions, each time selling the paintings at their current value, which has appreciated since the previous year.But that would mean that the paintings are sold each year, so they can't be sold again in the next year. That doesn't make much sense because once you sell a painting, it's no longer in your possession.Alternatively, maybe the owner is auctioning off the paintings each year, but not necessarily selling them all each time. Maybe they are just putting them up for auction each year, but the paintings remain with the owner, and each year, their value increases. So, the total funds raised would be the sum of the proceeds from each year's auction, but since the paintings are not sold, their value continues to appreciate each year.Wait, that seems a bit more plausible. So, each year, the paintings are auctioned, but not necessarily sold, so their value increases each year, and the proceeds from each auction contribute to the total funds. But if they are not sold, the value just keeps appreciating.But the problem says "auction off a collection of N paintings." So, if they auction them off, that implies selling them. So, perhaps each year, they auction off the same collection, but the collection is replenished each year? That seems a bit odd.Wait, maybe the owner is auctioning off the same collection each year, but the paintings are not sold; instead, they are just exhibited and auctioned, but the owner keeps them. So, the value of each painting increases each year, and each year, they auction them again, but the proceeds are added to the total.But that seems a bit unclear. The problem says "auction off a collection of N paintings." So, maybe each year, they auction off the same collection, but the paintings are sold each year, which would mean that after the first year, the owner no longer has the paintings to auction in subsequent years.But that contradicts the idea of holding auctions annually for T years. So, perhaps the owner is auctioning off the same paintings each year, but the paintings are not sold; instead, they are just exhibited, and the auction is more of a fundraising event where people can bid, but the owner keeps the paintings. So, the value of the paintings is just used to calculate the funds raised each year.Wait, that might make more sense. So, each year, the paintings are appraised at their current value, and the total value is donated to charity. So, the funds raised each year are the total value of the paintings at that time, which is then donated. So, over T years, the total funds raised would be the sum of the total value each year.But in that case, the total funds raised would be the sum from t=1 to T of V_total(t), where V_total(t) is the total value at year t.But the problem says "the total funds raised from these paintings over T years." So, if each year, the paintings are auctioned, meaning their value is realized and donated, then the total funds would be the sum of each year's total value.But if the paintings are sold each year, then after the first year, the owner no longer has the paintings to auction in the subsequent years. So, that seems contradictory.Alternatively, perhaps the owner is auctioning off the paintings, but not necessarily selling them. So, each year, the paintings are auctioned, but if they are not sold, they remain with the owner, and their value continues to appreciate. So, each year, the owner can choose to sell some or all of the paintings, but in order to maximize the funds, they might wait until the value is higher.But the problem doesn't specify whether the paintings are sold each year or not. It just says the owner auctions them off annually for the next T years. So, perhaps the total funds raised is the sum of the proceeds from each year's auction, regardless of whether the paintings are sold or not.Wait, but if they are auctioned off, that usually implies that they are sold. So, if they are sold each year, then after the first year, the owner doesn't have the paintings anymore. So, the funds raised would just be the proceeds from the first year's auction.But that contradicts the idea of holding auctions annually for T years. So, maybe the owner is auctioning off the same collection each year, but the paintings are not sold; instead, the auction is just a way to raise funds, perhaps through donations or pledges, based on the appreciated value of the paintings.In that case, the total funds raised over T years would be the sum of the total value of the paintings each year, which is V_total(t) for t from 1 to T.So, V_total(t) = Σ (from i=1 to N) [V_i * e^(A_i * t)]Therefore, the total funds raised over T years would be Σ (from t=1 to T) [Σ (from i=1 to N) [V_i * e^(A_i * t)]]Which can be rewritten as Σ (from i=1 to N) [V_i * Σ (from t=1 to T) e^(A_i * t)]That's a double summation. So, for each painting, we sum its value over each year, and then sum across all paintings.But this seems more complicated. However, the problem says "the total funds raised from these paintings over T years." If it's the total after T years, then it's just V_total(T). But if it's the total over the years, meaning each year's proceeds, then it's the sum over t=1 to T of V_total(t).But the problem statement isn't entirely clear. Let me re-read it."A local business owner sponsors art exhibitions to raise funds for cancer charities. During one of these events, the owner decides to auction off a collection of N paintings. Each painting i (where i ranges from 1 to N) has a unique value V_i and is expected to appreciate in value by a factor of A_i each year. The owner plans to hold these auctions annually for the next T years, and all proceeds will be donated to cancer charities."So, the owner is auctioning off the same collection each year for T years. Each year, the paintings appreciate, so their value increases. The proceeds from each auction are donated. So, each year, the owner auctions the paintings, which have appreciated since the last auction, and the proceeds from each year's auction are added to the total funds.Therefore, the total funds raised over T years would be the sum of the total value of the paintings each year, from year 1 to year T.So, in that case, the total funds F_total would be:F_total = Σ (from t=1 to T) [Σ (from i=1 to N) V_i * e^(A_i * t)]Which can be written as:F_total = Σ (from i=1 to N) [V_i * Σ (from t=1 to T) e^(A_i * t)]Yes, that seems to be the case.But wait, if the owner auctions the paintings each year, doesn't that mean that after the first year, the paintings are sold and the owner no longer has them for the next year's auction? That would mean that the funds raised would only be from the first year's auction. So, perhaps the initial interpretation is incorrect.Alternatively, maybe the owner is not selling the paintings but just using their appreciated value each year as a way to raise funds, perhaps through donations or pledges based on the value. So, each year, the value of the paintings is higher, and the owner can raise more funds based on that value without actually selling the paintings.In that case, the total funds raised over T years would indeed be the sum of the total value each year.But the problem says "auction off a collection of N paintings." Auctioning off usually implies selling them. So, if they are sold each year, the owner can't auction them again in subsequent years. Therefore, perhaps the owner is auctioning off the same collection each year, but the paintings are not sold; instead, the auction is a fundraising event where people can bid, but the owner keeps the paintings. So, the proceeds are based on the value of the paintings each year, which appreciates.In that case, the total funds raised would be the sum of the total value each year.Alternatively, maybe the owner is auctioning off the paintings each year, but only selling them if the bids meet a certain threshold. If not, the paintings remain with the owner and continue to appreciate. So, the owner might choose to sell some or all of the paintings in a given year, but the problem doesn't specify that. It just says the owner plans to hold these auctions annually for the next T years.Given the ambiguity, I think the most straightforward interpretation is that the owner is auctioning off the paintings each year, meaning selling them, so the funds raised each year are the total value of the paintings at that time. However, since the paintings are sold, the owner can't auction them again in subsequent years. Therefore, the total funds raised would just be the total value after T years, because the paintings are sold in the last year.Wait, but that doesn't make sense either because if they are sold each year, the owner would have sold them in the first year and wouldn't have them for the next years.Therefore, perhaps the owner is auctioning off the paintings each year, but not selling them; instead, the auction is a way to raise funds based on the appreciated value. So, each year, the value of the paintings is higher, and the owner can raise more funds based on that value without actually selling the paintings.In that case, the total funds raised over T years would be the sum of the total value each year.But the problem says "auction off a collection of N paintings." So, if they are auctioned off, that implies selling them. So, perhaps the owner is auctioning off the same collection each year, but the paintings are not sold; instead, the auction is a fundraising event where the value of the paintings is used to determine the amount donated. So, each year, the paintings are appraised, and the total value is donated, even though they are not sold.In that case, the total funds raised over T years would be the sum of the total value each year.Alternatively, maybe the owner is auctioning off the paintings each year, but the proceeds are the amount raised from the auction, which could be the total value of the paintings at that time. So, each year, the owner auctions the paintings, and the proceeds are added to the total funds, but the paintings remain with the owner, so they can be auctioned again next year.But in reality, if you auction something, you usually sell it, so the owner wouldn't have the paintings anymore after the first auction. Therefore, perhaps the problem is that the owner is auctioning off the paintings after T years, meaning that the total funds raised is the total value after T years.But the problem says "annually for the next T years," so it's an annual event. So, maybe each year, the owner auctions off the paintings, but the paintings are not sold; instead, the value is used to calculate the funds raised, and the paintings continue to appreciate.Given the ambiguity, I think the safest assumption is that the total funds raised after T years is the total value of the paintings at time T, which is the sum of V_i * e^(A_i * T) for all i.Therefore, for part 1, the total expected value after T years is Σ V_i * e^(A_i * T).For part 2, the necessary condition is that this total value must be at least F, so Σ V_i * e^(A_i * T) ≥ F.But wait, if the auctions are held annually, and the owner is raising funds each year, then perhaps the total funds raised is the sum over each year's total value. So, each year t, the total value is Σ V_i * e^(A_i * t), and the total funds over T years would be Σ (from t=1 to T) [Σ V_i * e^(A_i * t)].But again, if the owner is auctioning off the paintings each year, meaning selling them, then after the first year, the owner doesn't have the paintings anymore, so the funds raised would only be from the first year.This is confusing. Let me think again.The problem says: "the owner decides to auction off a collection of N paintings. Each painting i... The owner plans to hold these auctions annually for the next T years..."So, the owner is auctioning off the same collection each year for T years. So, each year, the paintings are auctioned, but since they are not sold (because the owner continues to hold auctions each year), the value of the paintings continues to appreciate.Therefore, each year, the owner can raise funds based on the current value of the paintings, which has appreciated since the last auction.So, the total funds raised over T years would be the sum of the total value each year.Therefore, the total funds F_total = Σ (from t=1 to T) [Σ (from i=1 to N) V_i * e^(A_i * t)]So, for part 2, the necessary condition is that F_total ≥ F, which would be:Σ (from i=1 to N) [V_i * Σ (from t=1 to T) e^(A_i * t)] ≥ FAlternatively, since the inner sum is a geometric series, we can express it as:Σ (from i=1 to N) [V_i * (e^(A_i) * (e^(A_i * T) - 1) / (e^(A_i) - 1))] ≥ FBut maybe that's complicating it. The problem just asks to express the necessary condition in terms of V_i, A_i, T, and F, so perhaps leaving it as a double summation is acceptable.However, given the ambiguity, I think the first interpretation is more likely intended, where the total funds raised after T years is the total value at time T, which is Σ V_i * e^(A_i * T). Therefore, the necessary condition is Σ V_i * e^(A_i * T) ≥ F.But to be thorough, let me consider both interpretations.If the owner auctions off the paintings each year, selling them, then the total funds raised would only be from the first year, because after that, the owner doesn't have the paintings anymore. So, that can't be right because the problem mentions T years.Alternatively, if the owner auctions off the paintings each year but doesn't sell them, then the total funds raised over T years would be the sum of the total value each year.But since the problem says "auction off a collection of N paintings," which implies selling them, I think the intended interpretation is that the owner is auctioning off the paintings after T years, meaning the total funds raised is the total value after T years.Therefore, part 1 is Σ V_i * e^(A_i * T), and part 2 is Σ V_i * e^(A_i * T) ≥ F.But wait, the problem says "annually for the next T years," so it's an annual event. So, perhaps the owner is auctioning off the paintings each year, but not selling them, so the total funds raised each year is the total value at that time, and the total over T years is the sum of those.But again, the problem says "auction off a collection of N paintings," which usually means selling them. So, perhaps the owner is auctioning off the paintings each year, selling them, but then replenishing the collection each year. That is, each year, the owner has the same collection of paintings, which have appreciated since the last year.But that seems a bit unrealistic, unless the owner is buying back the paintings each year after the auction, which would complicate things.Alternatively, maybe the owner is auctioning off the paintings each year, but the paintings are not sold; instead, the auction is a way to raise funds based on the value of the paintings, which are kept by the owner. So, each year, the value increases, and the funds raised each year are the total value at that time.In that case, the total funds over T years would be the sum from t=1 to T of the total value at time t.Given the ambiguity, I think the problem is more likely asking for the total value after T years, as the auctions are held annually, but the total funds raised is the final value after T years. Because if it were the sum over each year, the problem would probably specify that.Therefore, I think the answers are:1. Total expected value after T years: Σ (from i=1 to N) V_i * e^(A_i * T)2. Necessary condition: Σ (from i=1 to N) V_i * e^(A_i * T) ≥ FSo, I'll go with that.</think>

🧠:<think>Okay, so I have this problem where a single mother is creating a customized educational plan for her child with developmental delays. She wants to use visual aids, interactive activities, and structured learning sessions. The child does best when there's a balance between these three components. The first part of the problem says she dedicates a total of 15 hours per week to these activities. She wants the time on visual aids to be twice the time on interactive activities, and the time on structured learning to be three times the time on interactive activities. I need to figure out how many hours she should allocate to each component.Alright, let's break this down. Let me assign variables to each component to make it easier. Let me denote:- Let ( I ) be the hours spent on interactive activities.- Then, since visual aids should be twice that, visual aids ( V = 2I ).- Similarly, structured learning sessions ( S ) should be three times interactive activities, so ( S = 3I ).Now, the total time spent on all three components is 15 hours. So, the sum of ( V + I + S ) should equal 15.Substituting the expressions for ( V ) and ( S ) in terms of ( I ), we get:( V + I + S = 2I + I + 3I = 6I )So, ( 6I = 15 ) hours.To find ( I ), I can divide both sides by 6:( I = 15 / 6 )Simplifying that, ( 15 ÷ 6 = 2.5 ). So, ( I = 2.5 ) hours.Now, let's find ( V ) and ( S ):- ( V = 2I = 2 * 2.5 = 5 ) hours.- ( S = 3I = 3 * 2.5 = 7.5 ) hours.Let me double-check to make sure these add up to 15:5 (Visual) + 2.5 (Interactive) + 7.5 (Structured) = 15 hours. Yep, that works.So, the allocations are 5 hours for visual aids, 2.5 hours for interactive activities, and 7.5 hours for structured learning sessions.Now, moving on to the second part. The effectiveness ( E ) of the plan is given by the function ( E = 4V^2 + 5I^2 + 3S^2 ). I need to plug in the values of ( V ), ( I ), and ( S ) from the first part into this function to find ( E ).Let me write down the values again:- ( V = 5 ) hours- ( I = 2.5 ) hours- ( S = 7.5 ) hoursSo, plugging these into the formula:( E = 4*(5)^2 + 5*(2.5)^2 + 3*(7.5)^2 )Let me compute each term step by step.First term: ( 4*(5)^2 )( 5^2 = 25 ), so ( 4*25 = 100 ).Second term: ( 5*(2.5)^2 )( 2.5^2 = 6.25 ), so ( 5*6.25 = 31.25 ).Third term: ( 3*(7.5)^2 )( 7.5^2 = 56.25 ), so ( 3*56.25 = 168.75 ).Now, adding all these together:100 (from visual) + 31.25 (from interactive) + 168.75 (from structured) = Let me add 100 and 31.25 first: 100 + 31.25 = 131.25Then, add 168.75 to that: 131.25 + 168.75 = 300.So, the effectiveness ( E ) is 300.Wait, let me verify the calculations again to make sure I didn't make a mistake.First term: 4*(5)^2 = 4*25 = 100. Correct.Second term: 5*(2.5)^2 = 5*6.25 = 31.25. Correct.Third term: 3*(7.5)^2 = 3*56.25 = 168.75. Correct.Adding them up: 100 + 31.25 is 131.25, plus 168.75 is indeed 300. So, that seems right.So, the effectiveness is 300.I think that's all. Let me just recap:1. Set up the variables with ( I ) as the base, expressed ( V ) and ( S ) in terms of ( I ), summed them up to 15, solved for ( I ), then found ( V ) and ( S ).2. Plugged those values into the effectiveness formula, calculated each term, and summed them to get the effectiveness score.Everything seems to check out. I don't see any errors in my calculations or reasoning.Final Answer1. The mother should allocate boxed{5} hours to visual aids, boxed{2.5} hours to interactive activities, and boxed{7.5} hours to structured learning sessions.2. The effectiveness ( E ) of the plan is boxed{300}.</think>

🧠:<think>Okay, so I need to solve this problem about Alex organizing a community event at an octagonal landmark. The first part is about deriving the maximum allowable area for temporary structures in terms of the radius ( r ) of the circumscribed circle. The second part is about relating the areas of the stage and tents to this maximum area. Let me tackle them one by one.Starting with the first problem. The landmark is an octagon inscribed in a circle of radius ( r ). I remember that for regular polygons, the area can be calculated using the formula involving the number of sides and the radius. Since it's an octagon, it has 8 sides. I think the formula for the area of a regular polygon is ( frac{1}{2} times perimeter times apothem ), but I might be mixing things up. Alternatively, another formula I recall is ( frac{1}{2} n r^2 sinleft(frac{2pi}{n}right) ), where ( n ) is the number of sides. Let me verify this.Yes, that formula makes sense because each of the ( n ) triangles that make up the polygon has an area of ( frac{1}{2} r^2 sinleft(frac{2pi}{n}right) ), so multiplying by ( n ) gives the total area. So for an octagon, ( n = 8 ), so the area ( A ) is ( frac{1}{2} times 8 times r^2 times sinleft(frac{2pi}{8}right) ). Simplifying that, it becomes ( 4 r^2 sinleft(frac{pi}{4}right) ). Since ( sinleft(frac{pi}{4}right) = frac{sqrt{2}}{2} ), substituting that in gives ( 4 r^2 times frac{sqrt{2}}{2} = 2 sqrt{2} r^2 ). So the area of the octagon is ( 2 sqrt{2} r^2 ).But wait, the problem mentions that no more than 30% of the total area can be covered by temporary structures. So the maximum allowable area is 30% of the octagon's area. Therefore, the maximum area ( A_{max} ) is ( 0.3 times 2 sqrt{2} r^2 ). Calculating that, it's ( 0.6 sqrt{2} r^2 ). Alternatively, ( 0.6 sqrt{2} ) is approximately 0.8485, but since they want the formula in terms of ( r ), I should keep it exact. So ( A_{max} = 0.3 times 2 sqrt{2} r^2 = 0.6 sqrt{2} r^2 ). Alternatively, I can write 0.6 as 3/5, so ( A_{max} = frac{3}{5} times 2 sqrt{2} r^2 = frac{6 sqrt{2}}{5} r^2 ). Either way is correct, but perhaps expressing it as ( frac{3 sqrt{2}}{5} r^2 ) is simpler? Wait, no, ( 0.6 sqrt{2} ) is ( frac{3}{5} sqrt{2} ), which is the same as ( frac{3 sqrt{2}}{5} r^2 ). So that's the maximum allowable area.Wait, hold on. Let me double-check the area of the octagon. I used the formula ( frac{1}{2} n r^2 sinleft(frac{2pi}{n}right) ). Plugging in ( n = 8 ), we get ( frac{1}{2} times 8 times r^2 times sinleft(frac{pi}{4}right) ). That's correct because ( frac{2pi}{8} = frac{pi}{4} ). So, ( 4 r^2 times frac{sqrt{2}}{2} = 2 sqrt{2} r^2 ). So that part is right. Then 30% of that is ( 0.3 times 2 sqrt{2} r^2 = 0.6 sqrt{2} r^2 ). So that's correct.Alternatively, another way to compute the area of a regular octagon is using the formula ( 2(1 + sqrt{2}) a^2 ), where ( a ) is the side length. But since we have the radius, not the side length, maybe it's better to stick with the formula I used earlier because it directly relates to the radius.So, yes, the maximum allowable area is ( 0.6 sqrt{2} r^2 ), which can also be written as ( frac{3 sqrt{2}}{5} r^2 ). Both are equivalent, so either form is acceptable.Moving on to the second problem. Alex wants to set up a central stage with area ( S ) and ( n ) tents each with area ( T ). The total area used should not exceed the maximum allowable area from part 1. So, the total area ( A_{total} ) is ( S + nT ). This must be less than or equal to ( A_{max} ), which is ( 0.6 sqrt{2} r^2 ). Therefore, the relationship is ( S + nT leq 0.6 sqrt{2} r^2 ).Alternatively, since ( A_{max} ) can also be expressed as ( frac{3 sqrt{2}}{5} r^2 ), the inequality can be written as ( S + nT leq frac{3 sqrt{2}}{5} r^2 ). Either way is correct, but perhaps using the decimal coefficient is more straightforward.So, summarizing my thoughts:1. The area of the octagon is ( 2 sqrt{2} r^2 ). 30% of this is ( 0.6 sqrt{2} r^2 ), which is the maximum allowable area for temporary structures.2. The total area used by the stage and tents is ( S + nT ), which must be less than or equal to ( 0.6 sqrt{2} r^2 ).I think that covers both parts. I should make sure I didn't make any calculation errors. Let me verify the area of the octagon again. Using the formula ( frac{1}{2} n r^2 sinleft(frac{2pi}{n}right) ), plugging in ( n = 8 ), we get ( frac{1}{2} times 8 times r^2 times sinleft(frac{pi}{4}right) = 4 r^2 times frac{sqrt{2}}{2} = 2 sqrt{2} r^2 ). That's correct. So 30% is indeed ( 0.6 sqrt{2} r^2 ). Another way to think about it is that the area of the octagon is approximately ( 2.828 r^2 ) (since ( sqrt{2} approx 1.414 )), so 30% of that is about ( 0.848 r^2 ). But since the problem asks for the exact formula, I should keep it in terms of radicals.Therefore, the final expressions are:1. ( A_{max} = 0.6 sqrt{2} r^2 ) or ( frac{3 sqrt{2}}{5} r^2 )2. ( S + nT leq 0.6 sqrt{2} r^2 ) or ( S + nT leq frac{3 sqrt{2}}{5} r^2 )I think that's solid. I don't see any mistakes in my reasoning.Final Answer1. The maximum allowable area is boxed{dfrac{3sqrt{2}}{5} r^2}.2. The relationship is boxed{S + nT leq dfrac{3sqrt{2}}{5} r^2}.</think>

🧠:<think>Okay, so I have these two sub-problems to solve about gears and pulleys. Let me take them one at a time.Starting with Sub-problem 1. It involves three gears connected in series: A, B, and C. Their radii are given as 5 cm, 8 cm, and 12 cm respectively. Gear A is rotating at 120 RPM, and I need to find the rotational speed of Gear C in RPM. Hmm, gears connected in series... I remember that when two gears are connected, their rotational speeds are inversely proportional to their radii. So, if one gear is bigger, it will rotate slower compared to a smaller gear it's connected to.Wait, actually, it's the ratio of their radii that affects the RPM. The formula I think is (RPM of first gear) / (RPM of second gear) = (radius of second gear) / (radius of first gear). So, RPM1/RPM2 = r2/r1. That way, if the second gear is larger, its RPM is smaller.But since there are three gears here, A, B, and C, connected in series, I need to consider the ratios step by step. So, first, from Gear A to Gear B, and then from Gear B to Gear C.Let me write down the given values:- r_A = 5 cm- r_B = 8 cm- r_C = 12 cm- RPM_A = 120 RPMFirst, find RPM_B. Using the ratio:RPM_A / RPM_B = r_B / r_ASo, RPM_B = RPM_A * (r_A / r_B)Plugging in the numbers:RPM_B = 120 * (5 / 8) = 120 * 0.625 = 75 RPMOkay, so Gear B is rotating at 75 RPM.Now, moving on to Gear C. Using the same ratio between Gear B and Gear C:RPM_B / RPM_C = r_C / r_BSo, RPM_C = RPM_B * (r_B / r_C)Plugging in the numbers:RPM_C = 75 * (8 / 12) = 75 * (2/3) = 50 RPMSo, Gear C is rotating at 50 RPM.Wait, let me double-check that. So, starting from Gear A at 120 RPM, moving to Gear B which is larger, so it should slow down. 120 * (5/8) is indeed 75. Then, Gear C is even larger, so it should slow down further. 75 * (8/12) is 50. That seems correct.Alternatively, I could have thought about the total ratio from A to C. Since A is connected to B, and B is connected to C, the overall ratio from A to C is (r_A / r_B) * (r_B / r_C) = r_A / r_C. So, RPM_C = RPM_A * (r_A / r_C) = 120 * (5 / 12) = 50 RPM. Yep, same answer. So that's consistent.Alright, Sub-problem 1 seems solved. Gear C is rotating at 50 RPM.Moving on to Sub-problem 2. This involves a pulley system connected to Gear C. The pulley has a wheel with radius r_W = 6 cm. Alex wants the outer edge of the wheel to move at a linear speed of 150 cm/min. I need to find the necessary angular velocity of the wheel in radians per second.Hmm, okay. So, linear speed at the edge of the wheel is related to angular velocity by the formula:v = ω * rWhere v is linear speed, ω is angular velocity in radians per unit time, and r is the radius.Given that v is 150 cm/min, and r is 6 cm, I can solve for ω.But wait, the units are in cm/min, and I need ω in radians per second. So, I need to convert the units appropriately.First, let's write down the formula:v = ω * rSo, ω = v / rGiven:v = 150 cm/minr = 6 cmSo, ω = 150 / 6 = 25 radians per minute.But the question asks for radians per second, so I need to convert minutes to seconds.There are 60 seconds in a minute, so:ω = 25 / 60 radians per second.Simplify that:25 divided by 60 is equal to 5/12.So, ω = 5/12 radians per second.Wait, let me double-check. 150 divided by 6 is indeed 25. 25 radians per minute. To convert to per second, divide by 60, so 25/60 simplifies to 5/12. Yep, that's correct.Alternatively, I could have converted the linear speed to cm per second first.150 cm/min is equal to 150 / 60 = 2.5 cm/s.Then, ω = v / r = 2.5 / 6 = 0.416666... radians per second, which is 5/12. So, same result.Therefore, the angular velocity needed is 5/12 radians per second.Wait, just to make sure, let me recap:- Linear speed v = 150 cm/min- Convert to cm/s: 150 / 60 = 2.5 cm/s- Angular velocity ω = v / r = 2.5 / 6 = 0.416666... rad/s- 0.416666... is equal to 5/12.Yes, that's correct.Alternatively, if I didn't convert units first:v = 150 cm/min = 150 cm/min * (1 min / 60 s) = 2.5 cm/sThen, ω = 2.5 cm/s / 6 cm = 0.416666... rad/s = 5/12 rad/s.Same answer.So, I think that's solid.Just to recap both problems:1. For the gears, the RPM of Gear C is 50 RPM.2. For the pulley, the angular velocity needed is 5/12 radians per second.I don't see any mistakes in my reasoning. The key was remembering that for connected gears, the RPM ratio is inverse to the radius ratio, and for the pulley, linear speed relates to angular velocity via the radius, with proper unit conversions.Final AnswerSub-problem 1: The rotational speed of Gear C is boxed{50} RPM.Sub-problem 2: The necessary angular velocity of the wheel is boxed{dfrac{5}{12}} radians per second.</think>