💡Alright, so I've got this problem here where I need to calculate the sum of the digits of the result when a number made up of 47 nines multiplied by a number made up of 47 fours, and then add 100,000 to that product. Then, I have to find the sum of the digits of the final result. The answer choices are given, so that might help if I get stuck or want to verify my answer.First, let me try to understand the problem step by step. I have two very large numbers: one is 47 nines in a row, which is like 999...99 with 47 digits, and the other is 47 fours in a row, which is 444...44 with 47 digits. I need to multiply these two numbers together and then add 100,000 to the result. After that, I have to find the sum of all the digits in this final number.Okay, so breaking it down:1. **Multiplying 47 nines by 47 fours**: This seems like a massive multiplication problem. I wonder if there's a pattern or a shortcut to figure out the product without actually performing the entire multiplication, which would be time-consuming and error-prone.2. **Adding 100,000 to the product**: Once I have the product, I just need to add 100,000. That seems straightforward, but I need to make sure that adding 100,000 doesn't cause any carries or changes in the digits that might affect the sum.3. **Summing the digits of the final result**: After the multiplication and addition, I have to sum all the digits of the resulting number. This is a common type of problem where sometimes patterns emerge in the digits that can help simplify the calculation.Let me start by thinking about the multiplication part. Multiplying a number with all nines by a number with all fours. I remember that multiplying numbers with repeating digits can sometimes result in numbers with repeating patterns or specific digit sums.Let me consider smaller cases to see if I can spot a pattern.**Case 1: 9 × 4**That's simple: 9 × 4 = 36. The sum of the digits is 3 + 6 = 9.**Case 2: 99 × 44**Let's compute this:99 × 44 = (100 - 1) × 44 = 4400 - 44 = 4356.Sum of digits: 4 + 3 + 5 + 6 = 18.Hmm, interesting. The sum is 18, which is 9 × 2. Wait, 2 is the number of digits in each original number.**Case 3: 999 × 444**Let me compute this:999 × 444 = (1000 - 1) × 444 = 444000 - 444 = 443556.Sum of digits: 4 + 4 + 3 + 5 + 5 + 6 = 27.Again, 27 is 9 × 3, where 3 is the number of digits in each original number.Wait a second, so in each case, the sum of the digits of the product is 9 multiplied by the number of digits in the original numbers. So, for 1 digit, it was 9 × 1 = 9; for 2 digits, 9 × 2 = 18; for 3 digits, 9 × 3 = 27.If this pattern holds, then for 47 nines multiplied by 47 fours, the sum of the digits of the product should be 9 × 47.Let me calculate that: 9 × 47 = 423.Okay, so the sum of the digits of the product is 423.Now, I need to add 100,000 to this product. But wait, adding 100,000 is just adding 1 followed by five zeros. So, if I think about the number, it's going to be a very large number, but adding 100,000 will only affect the last few digits.But how does this affect the sum of the digits? Well, adding 100,000 is equivalent to adding 1 to the digit in the hundred thousands place and adding zeros to the other places. However, since 100,000 is a 6-digit number, and our original product is a number with 47 + 47 - 1 = 93 digits (since multiplying two n-digit numbers gives a number with up to 2n digits). So, adding 100,000 to a 93-digit number will only affect the last six digits.But wait, let me think about this more carefully. When we add 100,000 to the product, we are essentially adding 1 to the sixth digit from the end and zeros to the rest. However, depending on the last few digits of the product, adding 100,000 might cause a carry-over if the sixth digit from the end is a 9. But in our case, the product is 47 nines multiplied by 47 fours, which, as we saw in the smaller cases, results in a number where the digits are in a specific pattern.Wait, in the smaller cases, like 99 × 44 = 4356, the digits are 4, 3, 5, 6. Similarly, 999 × 444 = 443556. So, the pattern seems to be that the digits are a series of 4s, followed by a 3, then a series of 5s, and ending with a 6.So, for 47 nines multiplied by 47 fours, the product would be a number that starts with 4s, then has a 3, followed by 5s, and ends with a 6. The number of 4s would be 46, then a 3, then 46 5s, and ending with a 6. So, the total number of digits would be 46 + 1 + 46 + 1 = 94 digits. Wait, but earlier I thought it would be 93 digits. Hmm, maybe I was wrong.Wait, let's think about multiplication of two n-digit numbers. The maximum number of digits in the product is 2n. So, for n=47, it's 94 digits. So, the product is a 94-digit number.So, the structure is 46 4s, followed by a 3, followed by 46 5s, and ending with a 6.So, when we add 100,000 to this number, which is 1 followed by five zeros, we need to see how this affects the digits.But 100,000 is a 6-digit number, so adding it to a 94-digit number will affect the last six digits.So, let's consider the last six digits of the product. The product ends with 46 5s and a 6. So, the last six digits would be 555556.Adding 100,000 to 555556:555556 + 100000 = 655556.Wait, no, that's not correct. Wait, 555556 + 100000 = 655556? Wait, 555556 + 100000 is actually 655556. Wait, but 555556 + 100000 is 655556. Wait, but 555556 + 100000 is 655556. Wait, but 555556 + 100000 is 655556. Wait, but 555556 + 100000 is 655556.Wait, that seems correct. So, the last six digits change from 555556 to 655556. So, the sixth digit from the end increases by 1, and the rest remain the same.Therefore, the sum of the digits of the product was 423. Now, after adding 100,000, the last six digits change from 555556 to 655556. So, the sixth digit from the end was a 5, and now it's a 6. So, the sum of the digits increases by 1 (from 5 to 6). Therefore, the new sum of the digits is 423 + 1 = 424.Wait, but let me double-check that. The last six digits were 555556, which sum to 5+5+5+5+5+6 = 31. After adding 100,000, they become 655556, which sums to 6+5+5+5+5+6 = 32. So, the total sum increases by 1, from 31 to 32. Therefore, the overall sum of the digits increases by 1, from 423 to 424.Therefore, the final answer should be 424.But wait, let me make sure I didn't make any mistakes in my reasoning.First, I assumed that the product of 47 nines and 47 fours has a digit sum of 9 × 47 = 423. This was based on smaller cases where the digit sum was 9 × n, where n is the number of digits. That seems consistent.Then, I considered the structure of the product, which for smaller n is a series of 4s, a 3, a series of 5s, and ending with a 6. So, for n=2, it's 4356; for n=3, it's 443556. So, for n=47, it should be 46 4s, a 3, 46 5s, and a 6.Therefore, the last six digits are 555556. Adding 100,000 to this would change the sixth digit from the end from 5 to 6, making it 655556. The sum of these six digits increases by 1, so the total digit sum increases by 1.Therefore, the final digit sum is 423 + 1 = 424.So, the answer should be 424, which is option C.Wait, but let me think again about the structure of the product. When I multiply 99...99 (47 nines) by 44...44 (47 fours), does it really result in a number that starts with 46 4s, a 3, 46 5s, and ends with a 6?Let me test this with n=4:9999 × 4444.Let me compute this:9999 × 4444.I can compute this as (10000 - 1) × 4444 = 44440000 - 4444 = 44435556.So, the product is 44435556.Sum of digits: 4+4+4+3+5+5+5+6 = 36.Which is 9 × 4 = 36. So, the pattern holds.Looking at the digits: 44435556.So, for n=4, it's 3 4s, a 3, 3 5s, and a 6.Wait, so for n=4, it's (n-1) 4s, a 3, (n-1) 5s, and a 6.Similarly, for n=3: 443556, which is 2 4s, a 3, 2 5s, and a 6.So, for n=47, it should be 46 4s, a 3, 46 5s, and a 6.Therefore, the last six digits are 555556.Adding 100,000 to this would change the sixth digit from the end from 5 to 6, making it 655556.Therefore, the sum of the digits increases by 1.So, the total digit sum is 423 + 1 = 424.Therefore, the answer is 424, which is option C.I think this makes sense. I don't see any flaws in my reasoning. The pattern holds for smaller n, and the addition of 100,000 only affects the last six digits, increasing the digit sum by 1.

💡First, I need to understand what the question is asking. It involves two teams, Team A and Team B, each with 6 members. Both teams have the same average height of 160 cm. The question provides the variances of their heights: Team A has a variance of 1.2, and Team B has a variance of 2.0. The goal is to determine which team has more uniform heights.I recall that variance is a statistical measure that indicates how much the values in a dataset differ from the mean. A lower variance means the data points are closer to the mean, indicating more uniformity. Conversely, a higher variance means the data points are more spread out, indicating less uniformity.Given that Team A has a variance of 1.2 and Team B has a variance of 2.0, I can compare these two values directly. Since 1.2 is less than 2.0, it means that the heights in Team A are more tightly clustered around the average height of 160 cm compared to Team B.Therefore, based on the provided variances, Team A has more uniform heights than Team B.

💡First, let's denote the breadth of the first carpet as ( b ) meters. The length of the first carpet is given as ( 1.44b ) meters.The area of the first carpet is:[text{Area}_{text{first}} = b times 1.44b = 1.44b^2]The second carpet has dimensions that are 40% more in length and 25% more in breadth than the first carpet. Therefore, the length of the second carpet is ( 1.40 times 1.44b ) and the breadth is ( 1.25b ).The area of the second carpet is:[text{Area}_{text{second}} = (1.40 times 1.44b) times (1.25b)][text{Area}_{text{second}} = (1.40 times 1.44) times (1.25) times b^2][text{Area}_{text{second}} = (2.016) times (1.25) times b^2][text{Area}_{text{second}} = 2.52 times b^2]The cost of the second carpet is Rs. 4082.4, and the rate of the carpet is Rs. 45 per sq. m. So, the area of the second carpet can also be calculated by dividing the total cost by the rate per sq. m.[text{Area}_{text{second}} = frac{text{Cost}_{text{second}}}{text{Rate}}][text{Area}_{text{second}} = frac{4082.4}{45}][text{Area}_{text{second}} = 90.72 text{ sq. m}]Now, we can equate the two expressions for the area of the second carpet:[2.52 times b^2 = 90.72][b^2 = frac{90.72}{2.52}][b^2 = 36]Taking the square root of both sides, we get:[b = sqrt{36}][b = 6]So, the breadth of the first carpet is (boxed{6}) meters.

💡Okay, so I have this geometry problem here about a unit circle with five points on it and a point inside the circle. The problem is asking for the product of certain segment lengths, specifically ( A_1 Q_1 cdot A_2 Q_2 cdot cdots cdot A_5 Q_5 ). Hmm, let me try to visualize this.First, the setup: there's a unit circle centered at point ( O ). On this circle, five points ( A_1, A_2, A_3, A_4, A_5 ) are placed sequentially. There's another point ( P ) inside the circle. For each ( i ) from 1 to 5, we're supposed to find the intersection point ( Q_i ) of the line segment ( A_i A_{i+2} ) with the line segment ( P A_{i+1} ). Here, ( A_6 ) is the same as ( A_1 ) to make it cyclic. Then, ( O Q_i = d_i ) for each ( i ). The goal is to find the product of all ( A_i Q_i ).Alright, so let me break this down. Each ( Q_i ) is an intersection point between two lines: one connecting ( A_i ) to ( A_{i+2} ) and the other connecting ( P ) to ( A_{i+1} ). Since all these points are on a unit circle, their distances from ( O ) are all 1. But ( P ) is inside the circle, so its distance from ( O ) is less than 1.I think the key here might be to use some properties of circles and intersecting chords. Maybe the Power of a Point theorem could be helpful here. The Power of a Point theorem states that for a point ( Q ) inside a circle, the product of the lengths of the segments from ( Q ) to the points where a line through ( Q ) intersects the circle is constant. That is, if two chords intersect at ( Q ), then ( A_i Q cdot Q A_{i+2} = text{constant} ).Wait, in this case, each ( Q_i ) is the intersection of ( A_i A_{i+2} ) and ( P A_{i+1} ). So, for each ( Q_i ), we can apply the Power of a Point theorem. Let's try that.For each ( i ), the Power of a Point theorem tells us that:[A_i Q_i cdot Q_i A_{i+2} = P Q_i cdot Q_i A_{i+1}]But I'm not sure if that's directly applicable here because ( P ) is another point inside the circle, not necessarily on the circle. Maybe I need to think differently.Alternatively, since all the points ( A_i ) are on the unit circle, their distances from ( O ) are 1. The point ( Q_i ) is somewhere inside the circle, so its distance from ( O ) is ( d_i ). Perhaps I can relate ( A_i Q_i ) to ( d_i ) using coordinates or vectors.Wait, another thought: maybe using coordinate geometry. If I assign coordinates to each ( A_i ) and ( P ), I can find the equations of the lines ( A_i A_{i+2} ) and ( P A_{i+1} ), find their intersection ( Q_i ), and then compute ( A_i Q_i ). But that seems complicated because there are five points and five intersections, and the product might not simplify easily.Alternatively, maybe there's a symmetry or a pattern here because the points are equally spaced on the circle. If the five points are equally spaced, the angles between them are equal. But the problem doesn't specify that they're equally spaced, just that they're taken sequentially. Hmm.Wait, maybe I can use Ceva's Theorem. Ceva's Theorem relates the ratios of segments created by concurrent lines in a triangle. But in this case, we have a pentagon, not a triangle. Maybe Ceva's Theorem can be generalized or applied in some way here.Alternatively, maybe using Menelaus' Theorem, which relates the lengths of segments created by a transversal cutting through a triangle. But again, we have a pentagon, so it's not straightforward.Wait, another idea: since all the points ( A_i ) are on the unit circle, and ( O ) is the center, maybe we can use vectors or complex numbers to represent these points. That might make the calculations more manageable.Let me try to think in terms of complex numbers. Let me represent each point ( A_i ) as a complex number on the unit circle, so ( |A_i| = 1 ). Point ( P ) is also a complex number inside the unit circle, so ( |P| < 1 ). The line ( A_i A_{i+2} ) can be represented parametrically, and similarly, the line ( P A_{i+1} ) can be represented parametrically. Then, solving for their intersection ( Q_i ) would give me the point in terms of ( A_i, A_{i+1}, A_{i+2} ), and ( P ).But this seems like a lot of algebra, and I'm not sure if it will lead me directly to the product ( A_1 Q_1 cdot A_2 Q_2 cdot cdots cdot A_5 Q_5 ). Maybe there's a smarter way.Wait, going back to the Power of a Point theorem. For each ( Q_i ), since it's the intersection of ( A_i A_{i+2} ) and ( P A_{i+1} ), we can say that:[A_i Q_i cdot Q_i A_{i+2} = P Q_i cdot Q_i A_{i+1}]But I'm not sure if that's correct because ( P ) is not on the circle. Wait, actually, the Power of a Point theorem states that for a point ( Q ) outside the circle, the product of the lengths from ( Q ) to the points of intersection with the circle is equal for any two lines through ( Q ). But here, ( Q_i ) is inside the circle, so the theorem still applies but in a different form.Yes, for a point inside the circle, the Power of a Point theorem says that for any chord passing through ( Q_i ), the product ( A_i Q_i cdot Q_i A_{i+2} ) is equal to ( OP^2 - OQ_i^2 ) or something like that. Wait, no, let me recall.Actually, the Power of a Point theorem for a point inside the circle states that for any chord through ( Q_i ), the product ( A_i Q_i cdot Q_i A_{i+2} ) is equal to ( R^2 - OQ_i^2 ), where ( R ) is the radius of the circle. Since it's a unit circle, ( R = 1 ), so:[A_i Q_i cdot Q_i A_{i+2} = 1 - d_i^2]where ( d_i = OQ_i ).Ah, that seems useful! So for each ( i ), we have:[A_i Q_i cdot Q_i A_{i+2} = 1 - d_i^2]So, if I can express ( A_i Q_i ) in terms of ( Q_i A_{i+2} ), or vice versa, maybe I can find a relationship.But I need the product of all ( A_i Q_i ). Let's think about multiplying all these equations together for ( i = 1 ) to ( 5 ):[prod_{i=1}^{5} (A_i Q_i cdot Q_i A_{i+2}) = prod_{i=1}^{5} (1 - d_i^2)]Simplifying the left side, we have:[prod_{i=1}^{5} A_i Q_i cdot prod_{i=1}^{5} Q_i A_{i+2} = prod_{i=1}^{5} (1 - d_i^2)]But notice that ( A_{i+2} ) cycles through all the points as ( i ) goes from 1 to 5. Specifically, ( A_{i+2} ) for ( i = 1 ) is ( A_3 ), for ( i = 2 ) is ( A_4 ), and so on, until ( i = 5 ), which gives ( A_7 = A_2 ) (since ( A_6 = A_1 ) and ( A_7 = A_2 )). So, the product ( prod_{i=1}^{5} Q_i A_{i+2} ) is actually the same as ( prod_{i=1}^{5} A_i Q_i ), just shifted by two indices.Wait, let me check that. If I write out the terms:- For ( i = 1 ): ( Q_1 A_3 )- For ( i = 2 ): ( Q_2 A_4 )- For ( i = 3 ): ( Q_3 A_5 )- For ( i = 4 ): ( Q_4 A_1 )- For ( i = 5 ): ( Q_5 A_2 )So, the product is ( Q_1 A_3 cdot Q_2 A_4 cdot Q_3 A_5 cdot Q_4 A_1 cdot Q_5 A_2 ). If I rearrange the terms, it's ( (Q_1 A_3 cdot Q_2 A_4 cdot Q_3 A_5 cdot Q_4 A_1 cdot Q_5 A_2) ). Hmm, this isn't exactly the same as ( prod_{i=1}^{5} A_i Q_i ), but it's related.Wait, actually, each ( A_i ) appears exactly once in the product, but multiplied by a different ( Q_j ). So, it's not a direct product of ( A_i Q_i ), but rather a cyclic permutation. Maybe I can relate this to another product.Alternatively, perhaps I can consider the ratio of the two products. Let me denote ( P = prod_{i=1}^{5} A_i Q_i ) and ( Q = prod_{i=1}^{5} Q_i A_{i+2} ). Then, from the Power of a Point theorem, we have ( P cdot Q = prod_{i=1}^{5} (1 - d_i^2) ).But I need to find ( P ), which is ( prod_{i=1}^{5} A_i Q_i ). So, if I can express ( Q ) in terms of ( P ), maybe I can solve for ( P ).Looking at ( Q = prod_{i=1}^{5} Q_i A_{i+2} ), as I wrote earlier, this is ( Q_1 A_3 cdot Q_2 A_4 cdot Q_3 A_5 cdot Q_4 A_1 cdot Q_5 A_2 ). Notice that each ( A_i ) is multiplied by a different ( Q_j ). If I factor out the ( A_i )'s, I get ( (A_1 A_2 A_3 A_4 A_5) cdot (Q_4 Q_5 Q_1 Q_2 Q_3) ). Wait, that's not quite right because the indices don't align.Wait, actually, each ( A_i ) is multiplied by a ( Q_j ) where ( j ) is ( i - 2 ) modulo 5. So, ( A_3 ) is multiplied by ( Q_1 ), ( A_4 ) by ( Q_2 ), etc. So, the product ( Q ) can be written as ( prod_{i=1}^{5} A_{i+2} Q_i ), which is the same as ( prod_{i=1}^{5} A_i Q_{i-2} ) where indices are modulo 5.But this seems a bit convoluted. Maybe there's another approach. Let me think about the areas involved.Alternatively, perhaps using Ceva's Theorem in a cyclic manner. Ceva's Theorem states that for concurrent lines in a triangle, the product of certain ratios equals 1. But since we have a pentagon, maybe a generalized version applies.Wait, another idea: since all the ( Q_i ) are defined similarly, maybe there's a relationship between their distances ( d_i ) and the product we're trying to find.Wait, going back to the Power of a Point result:[A_i Q_i cdot Q_i A_{i+2} = 1 - d_i^2]So, for each ( i ), ( A_i Q_i = frac{1 - d_i^2}{Q_i A_{i+2}} ). If I take the product over all ( i ), I get:[prod_{i=1}^{5} A_i Q_i = prod_{i=1}^{5} frac{1 - d_i^2}{Q_i A_{i+2}}]But the denominator is ( prod_{i=1}^{5} Q_i A_{i+2} ), which, as I thought earlier, is a cyclic permutation of the ( A_i Q_i ) terms. Specifically, ( prod_{i=1}^{5} Q_i A_{i+2} = prod_{i=1}^{5} A_{i+2} Q_i = prod_{i=1}^{5} A_i Q_{i-2} ) (with indices modulo 5).So, if I denote ( P = prod_{i=1}^{5} A_i Q_i ), then the denominator is ( prod_{i=1}^{5} A_i Q_{i-2} ). But ( Q_{i-2} ) is just another index, so the denominator is also ( P ). Therefore, we have:[P = frac{prod_{i=1}^{5} (1 - d_i^2)}{P}]Multiplying both sides by ( P ), we get:[P^2 = prod_{i=1}^{5} (1 - d_i^2)]Taking the square root of both sides, we find:[P = sqrt{prod_{i=1}^{5} (1 - d_i^2)}]So, the product ( A_1 Q_1 cdot A_2 Q_2 cdot cdots cdot A_5 Q_5 ) is equal to the square root of the product of ( (1 - d_i^2) ) for ( i = 1 ) to ( 5 ).Wait, let me double-check this reasoning. We started with the Power of a Point theorem for each ( Q_i ), leading to ( A_i Q_i cdot Q_i A_{i+2} = 1 - d_i^2 ). Then, multiplying all these equations together gives ( P cdot Q = prod (1 - d_i^2) ), where ( Q ) is the product of ( Q_i A_{i+2} ). We then realized that ( Q ) is a cyclic permutation of ( P ), so ( Q = P ). Therefore, ( P^2 = prod (1 - d_i^2) ), leading to ( P = sqrt{prod (1 - d_i^2)} ).Yes, that seems consistent. So, the final answer is the square root of the product of ( (1 - d_i^2) ) for all five points.**Final Answer**boxed{sqrt{(1 - d_1^2)(1 - d_2^2)(1 - d_3^2)(1 - d_4^2)(1 - d_5^2)}}

💡Alright, so I've got this problem here about distributing bonuses among four people—A, B, C, and D—using something called "shuai fen ratio." I'm not entirely sure what that is, but from the example given, it seems like it's a way to distribute bonuses in decreasing proportions. The problem gives me the total reward amount and tells me that A and C together receive a certain amount. I need to find out what the shuai fen ratio is and how much D gets.First, let me try to understand what "shuai fen ratio" means. The example says that if the ratio is 20%, and A gets 1000 yuan, then B gets 800, and C gets 640. So, it looks like each subsequent person gets 80% of what the previous person got. That makes sense—it's a decreasing proportion. So, if the ratio is 20%, each next person gets 80% of the previous one's amount.In this problem, there are four people: A, B, C, and D. The total reward is 68780 yuan, and A and C together get 36200 yuan. I need to find the shuai fen ratio and D's bonus.Let me think about how to model this. If it's a decreasing proportion, it's like a geometric sequence where each term is a certain ratio less than the previous one. So, if I denote the shuai fen ratio as 'm', then each subsequent person gets (1 - m) times the previous person's amount.Let me denote A's bonus as 'a'. Then B's bonus would be a*(1 - m), C's bonus would be a*(1 - m)^2, and D's bonus would be a*(1 - m)^3.Given that, the total bonus is a + a*(1 - m) + a*(1 - m)^2 + a*(1 - m)^3 = 68780.Also, A and C together get 36200 yuan, so a + a*(1 - m)^2 = 36200.I need to solve these equations to find 'm' and then find D's bonus, which is a*(1 - m)^3.Hmm, that seems a bit complicated. Maybe I can simplify it. Let me denote r = (1 - m), so the equations become:a + a*r + a*r^2 + a*r^3 = 68780anda + a*r^2 = 36200So, from the second equation, I can factor out 'a':a*(1 + r^2) = 36200Which means a = 36200 / (1 + r^2)Then, plugging this into the first equation:(36200 / (1 + r^2)) * (1 + r + r^2 + r^3) = 68780Simplify this:36200 * (1 + r + r^2 + r^3) / (1 + r^2) = 68780Let me compute (1 + r + r^2 + r^3) / (1 + r^2). Maybe I can factor the numerator:1 + r + r^2 + r^3 = (1 + r) + r^2*(1 + r) = (1 + r)*(1 + r^2)So, (1 + r + r^2 + r^3) / (1 + r^2) = (1 + r)*(1 + r^2) / (1 + r^2) = 1 + rTherefore, the equation simplifies to:36200 * (1 + r) = 68780So, 1 + r = 68780 / 36200Let me compute that:68780 divided by 36200. Let's see, 36200 times 1.9 is 68780 because 36200*1=36200, 36200*0.9=32580, so 36200+32580=68780. So, 1 + r = 1.9Therefore, r = 0.9Since r = (1 - m), then m = 1 - r = 1 - 0.9 = 0.1, which is 10%.So, the shuai fen ratio is 10%.Now, to find D's bonus, which is a*r^3.From earlier, a = 36200 / (1 + r^2) = 36200 / (1 + 0.81) = 36200 / 1.81Let me compute that:36200 divided by 1.81. Let's see, 1.81 times 20000 is 36200. So, 36200 / 1.81 = 20000Wait, that can't be right because 1.81 times 20000 is 36200, yes, but that would mean a = 20000.But if a is 20000, then B is a*r = 20000*0.9 = 18000C is a*r^2 = 20000*0.81 = 16200D is a*r^3 = 20000*0.729 = 14580Let me check the total:A: 20000B: 18000C: 16200D: 14580Adding them up: 20000 + 18000 = 38000; 38000 + 16200 = 54200; 54200 + 14580 = 68780Yes, that matches the total.Also, A + C = 20000 + 16200 = 36200, which is correct.So, D's bonus is 14580 yuan.Therefore, the shuai fen ratio is 10%, and D gets 14580 yuan.

💡Okay, so I have this problem about finding the eccentricity of a hyperbola given by the equation (dfrac{x^2}{m^2 - 4} + dfrac{y^2}{m^2} = 1), where (m) is an integer. The options are A: 3, B: 2, C: (sqrt{5}), and D: (sqrt{3}). Hmm, let me try to figure this out step by step.First, I remember that the general form of a hyperbola is either (dfrac{x^2}{a^2} - dfrac{y^2}{b^2} = 1) (which opens left and right) or (dfrac{y^2}{a^2} - dfrac{x^2}{b^2} = 1) (which opens up and down). The given equation is (dfrac{x^2}{m^2 - 4} + dfrac{y^2}{m^2} = 1). Wait, that looks more like an ellipse equation because of the plus sign. But since it's a hyperbola, maybe I need to rearrange it.Let me check the denominators. For a hyperbola, one term should be positive and the other negative. So, if I move one term to the other side, it should become a hyperbola. Let me try that.Starting with (dfrac{x^2}{m^2 - 4} + dfrac{y^2}{m^2} = 1), if I subtract (dfrac{x^2}{m^2 - 4}) from both sides, I get (dfrac{y^2}{m^2} = 1 - dfrac{x^2}{m^2 - 4}). Hmm, that doesn't seem helpful. Maybe I should factor out the negative sign. Let me rewrite the equation as (dfrac{y^2}{m^2} - dfrac{x^2}{4 - m^2} = 1). Ah, that looks better because now it's in the standard hyperbola form.So, comparing this to the standard hyperbola equation (dfrac{y^2}{a^2} - dfrac{x^2}{b^2} = 1), I can see that (a^2 = m^2) and (b^2 = 4 - m^2). Since (a^2) and (b^2) must be positive, I need to ensure that both denominators are positive. Starting with (a^2 = m^2), since (m) is an integer, (m^2) is always non-negative. But for it to be positive, (m) can't be zero. So, (m neq 0). Next, (b^2 = 4 - m^2) must also be positive. So, (4 - m^2 > 0), which implies (m^2 < 4). Since (m) is an integer, the possible values for (m) are -1, 0, 1. But we already established that (m) can't be zero because (a^2) would be zero, which isn't allowed. So, (m) can be either 1 or -1. However, since (m^2) is the same for both 1 and -1, we can just consider (m = 1) without loss of generality.Now, with (m = 1), let's find (a^2) and (b^2). (a^2 = m^2 = 1^2 = 1), so (a = 1).(b^2 = 4 - m^2 = 4 - 1 = 3), so (b = sqrt{3}).For hyperbolas, the relationship between (a), (b), and (c) (the distance from the center to each focus) is given by (c^2 = a^2 + b^2). So, let's compute (c^2):(c^2 = a^2 + b^2 = 1 + 3 = 4), so (c = 2).The eccentricity (e) of a hyperbola is defined as (e = dfrac{c}{a}). Plugging in the values we have:(e = dfrac{c}{a} = dfrac{2}{1} = 2).So, the eccentricity is 2, which corresponds to option B.Wait, let me double-check my steps to make sure I didn't make a mistake. 1. I recognized the equation as a hyperbola by rearranging the terms. That seems correct because the original equation had a plus sign, but by moving one term to the other side, it became a hyperbola.2. I identified (a^2 = m^2) and (b^2 = 4 - m^2). That makes sense because in the standard hyperbola equation, the positive term is under (y^2), indicating it opens up and down.3. I found the constraints on (m): (m) must be an integer, (m^2 < 4), so (m) can be -1, 0, or 1. Excluding 0 because (a^2) can't be zero, so (m = pm1).4. Calculated (a = 1) and (b = sqrt{3}), then (c = 2), and eccentricity (e = 2). That all checks out.I think I did everything correctly. So, the answer should be B: 2.**Final Answer**boxed{B}