💡First, I need to understand the problem. There are two rectangles with corresponding sides that are parallel. I need to find the difference between their perimeters.Since the sides are parallel, the rectangles must be similar in shape, meaning their corresponding sides are proportional. However, without specific measurements or additional information about the dimensions of the rectangles, it's challenging to calculate the exact perimeters.Perhaps I can assume some standard dimensions or look for patterns in the answer choices. The options are 12 m, 16 m, 20 m, 22 m, and 24 m. These are all multiples of 4, which might hint at a specific relationship between the sides of the rectangles.If I consider that the difference in perimeters could be related to the difference in their side lengths, I might set up equations to represent the perimeters of both rectangles and then find the difference.Let’s denote the length and width of the first rectangle as ( l_1 ) and ( w_1 ), and the length and width of the second rectangle as ( l_2 ) and ( w_2 ). The perimeter of a rectangle is given by ( P = 2(l + w) ).So, the perimeters of the two rectangles would be:- ( P_1 = 2(l_1 + w_1) )- ( P_2 = 2(l_2 + w_2) )The difference in perimeters would be:( |P_2 - P_1| = 2(|l_2 - l_1| + |w_2 - w_1|) )Without specific values for ( l_1, w_1, l_2, ) and ( w_2 ), I can’t compute the exact difference. Maybe there’s a way to express the difference in terms of the given options or find a relationship between the sides based on the parallel sides condition.Alternatively, if the rectangles are positioned such that one is inside the other and their sides are parallel, the difference in perimeters might relate to the spacing between the sides. For example, if the outer rectangle has sides that are longer by a certain amount than the inner rectangle, the perimeter difference can be calculated based on that spacing.Let’s assume the outer rectangle has sides that are each longer by ( x ) meters than the inner rectangle. Then, the difference in perimeters would be ( 2(2x) = 4x ). If ( x ) is 6 meters, the difference would be 24 meters, which is one of the options.However, without concrete information about the dimensions or the relationship between the rectangles, it's difficult to determine the exact difference. I might need to revisit the problem statement or look for additional clues in the diagram that could provide more details about the rectangles' dimensions or their relative positions.

💡First, I need to determine the total number of players present on the ground, which is 59.Next, I'll add up the number of cricket, hockey, and softball players:16 (cricket) + 12 (hockey) + 13 (softball) = 41 players.Finally, to find out how many football players there are, I'll subtract the total number of cricket, hockey, and softball players from the overall total:59 (total players) - 41 (cricket, hockey, and softball players) = 18 football players.

💡Okay, so Ben has six identical darts and four identical dartboards. He throws all six darts, and each dart hits one of the dartboards. After throwing the darts, he lists the number of darts that hit each board, from greatest to least. We need to figure out how many different lists of dart-hit numbers are possible.Hmm, let me think about this. Since both the darts and the dartboards are identical, the order in which the darts hit the boards doesn't matter. What matters is how many darts hit each board, regardless of which specific dartboard it is. So, essentially, we're looking for the number of ways to partition the number 6 into up to 4 parts, where each part represents the number of darts on a dartboard. And since the list is ordered from greatest to least, we need to consider partitions in non-increasing order.Let me recall what integer partitions are. An integer partition of a number is a way of writing it as a sum of positive integers, where the order doesn't matter. For example, the number 4 can be partitioned as 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. So, in this case, we need to find all partitions of 6 where the number of parts is at most 4, and each part is a positive integer.Wait, but the dartboards are identical, so if we have, say, two dartboards with the same number of darts, they are indistinct. So, the partitions should consider that. Also, since the list is from greatest to least, the partitions should be in non-increasing order. That makes sense.So, let me try to list all the partitions of 6 into at most 4 parts, arranged in non-increasing order.Starting with the largest possible part:1. 6: This means all six darts hit one dartboard. So, the list would be [6, 0, 0, 0]. But since we're listing the number of darts on each board, and the dartboards are identical, we can ignore the zeros. So, effectively, it's just [6].2. 5 + 1: Five darts on one board and one dart on another. So, the list would be [5, 1, 0, 0], but again, ignoring zeros, it's [5, 1].3. 4 + 2: Four darts on one board and two on another. So, [4, 2, 0, 0] becomes [4, 2].4. 4 + 1 + 1: Four darts on one board and one dart each on two others. So, [4, 1, 1, 0] becomes [4, 1, 1].5. 3 + 3: Three darts on two different boards. So, [3, 3, 0, 0] becomes [3, 3].6. 3 + 2 + 1: Three darts on one board, two on another, and one on a third. So, [3, 2, 1, 0] becomes [3, 2, 1].7. 3 + 1 + 1 + 1: Three darts on one board and one dart each on the remaining three. So, [3, 1, 1, 1].8. 2 + 2 + 2: Two darts on each of three boards. So, [2, 2, 2, 0] becomes [2, 2, 2].9. 2 + 2 + 1 + 1: Two darts on two boards and one dart on the other two. So, [2, 2, 1, 1].10. 2 + 1 + 1 + 1 + 1: Two darts on one board and one dart on each of the remaining three. Wait, but we only have four dartboards, so this would be [2, 1, 1, 1].11. 1 + 1 + 1 + 1 + 1 + 1: Each dartboard gets one dart, but we only have four dartboards, so this isn't possible. Wait, no, actually, if we have four dartboards, each can have at most one dart, but we have six darts. So, this partition isn't valid because we can't have more darts than dartboards if each dartboard can only have one dart. Wait, no, actually, each dartboard can have multiple darts, so this partition would be [1, 1, 1, 1], but that only accounts for four darts, leaving two more. So, that's not correct. Maybe I made a mistake here.Wait, let me correct that. If we have six darts and four dartboards, the maximum number of darts on a board is six, and the minimum is zero, but since we have to list the number of darts on each board in non-increasing order, we can't have more than four parts. So, the partition [1, 1, 1, 1, 1, 1] isn't applicable here because we only have four dartboards. So, the smallest number of darts per board would be zero, but we have to list all four boards, so the partition would have to be [1, 1, 1, 1], but that only sums to four, not six. So, that's not correct. Therefore, the partition [1, 1, 1, 1, 1, 1] isn't possible here.Wait, so maybe I missed a partition. Let me go back.After [2, 2, 1, 1], which is valid, we have [2, 1, 1, 1, 1], but that's five parts, which exceeds the four dartboards. So, that's not possible. Therefore, the last valid partition would be [2, 1, 1, 1], but that sums to five, not six. So, that's not correct either.Wait, perhaps I need to adjust my approach. Let me list all partitions of 6 into at most 4 parts, ensuring that the sum is exactly 6.Starting with the largest part:1. 6: [6]2. 5 + 1: [5, 1]3. 4 + 2: [4, 2]4. 4 + 1 + 1: [4, 1, 1]5. 3 + 3: [3, 3]6. 3 + 2 + 1: [3, 2, 1]7. 3 + 1 + 1 + 1: [3, 1, 1, 1]8. 2 + 2 + 2: [2, 2, 2]9. 2 + 2 + 1 + 1: [2, 2, 1, 1]10. 2 + 1 + 1 + 1 + 1: This is five parts, which is more than four, so invalid.11. 1 + 1 + 1 + 1 + 1 + 1: Six parts, invalid.Wait, so that's only nine partitions. But earlier, I thought there were 11. Maybe I missed some.Let me try a different approach. Let's consider all partitions of 6, and then filter out those with more than four parts.Partitions of 6:1. 62. 5 + 13. 4 + 24. 4 + 1 + 15. 3 + 36. 3 + 2 + 17. 3 + 1 + 1 + 18. 2 + 2 + 29. 2 + 2 + 1 + 110. 2 + 1 + 1 + 1 + 111. 1 + 1 + 1 + 1 + 1 + 1So, that's 11 partitions. Now, we need to consider only those with at most four parts. So, partitions 1 through 9 are valid because they have four or fewer parts. Partitions 10 and 11 have five and six parts, respectively, which exceed the four dartboards, so they are invalid.Wait, but in the case of partition 7: 3 + 1 + 1 + 1, that's four parts, which is acceptable. Similarly, partition 9: 2 + 2 + 1 + 1, that's four parts. So, that's nine partitions.But earlier, I thought there were 11. Hmm, maybe I was confused earlier.Wait, no, the total number of partitions of 6 is 11, but when considering at most four parts, we have nine partitions. However, in the context of the problem, we have four dartboards, so we need to consider partitions into exactly four parts, but allowing zeros. Wait, no, because the dartboards are identical, and the zeros don't matter. So, actually, we can have partitions into up to four parts, where each part is at least 1, but since the dartboards are identical, we can have trailing zeros, but they don't contribute to the list. So, perhaps the correct approach is to consider all partitions of 6 into at most four parts, regardless of whether they sum to exactly 6 or not, but that doesn't make sense because we have exactly six darts.Wait, no, all partitions must sum to exactly 6. So, the number of partitions of 6 into at most four parts is nine. But earlier, I thought the answer was 11. Maybe I was mistaken.Wait, let me check online or recall the number of integer partitions of 6. The number of integer partitions of 6 is indeed 11. But when we restrict to at most four parts, we have fewer. Let me list them again:1. 62. 5 + 13. 4 + 24. 4 + 1 + 15. 3 + 36. 3 + 2 + 17. 3 + 1 + 1 + 18. 2 + 2 + 29. 2 + 2 + 1 + 110. 2 + 1 + 1 + 1 + 1 (invalid, five parts)11. 1 + 1 + 1 + 1 + 1 + 1 (invalid, six parts)So, only the first nine are valid. Therefore, the number of different lists is 9.Wait, but in the initial problem, the user thought the answer was 11. Maybe I'm missing something. Let me think again.Wait, perhaps the user considered partitions where the number of parts can be less than four, but the list must have four numbers, padding with zeros. So, for example, [6, 0, 0, 0], [5, 1, 0, 0], etc. So, in that case, the number of partitions would be the same as the number of partitions of 6 into at most four parts, considering that the parts can be zero, but since the dartboards are identical, the order doesn't matter, so we need to consider partitions in non-increasing order, allowing zeros.Wait, but in integer partitions, we don't consider zeros. So, perhaps the correct approach is to consider compositions, but since the dartboards are identical, it's still partitions.Wait, maybe I'm overcomplicating. Let me try to list all possible distributions, considering that the dartboards are identical and the list is from greatest to least.So, starting with the maximum number of darts on a board:1. All six on one board: [6, 0, 0, 0]2. Five on one, one on another: [5, 1, 0, 0]3. Four on one, two on another: [4, 2, 0, 0]4. Four on one, one on another, one on another: [4, 1, 1, 0]5. Three on one, three on another: [3, 3, 0, 0]6. Three on one, two on another, one on another: [3, 2, 1, 0]7. Three on one, one on each of the other three: [3, 1, 1, 1]8. Two on each of three boards: [2, 2, 2, 0]9. Two on two boards, one on the other two: [2, 2, 1, 1]10. Two on one board, one on each of the other three: [2, 1, 1, 1]11. One on each of six boards: But we only have four boards, so this is impossible.Wait, so that's 10 partitions. But earlier, I thought it was nine. Hmm.Wait, in the list above, I have 11 items, but the last one is impossible. So, excluding that, we have 10.Wait, but in the initial problem, the user thought the answer was 11, but perhaps they included the impossible one. Wait, no, in the initial problem, the user listed 11 partitions, including [6,0,0,0], [5,1,0,0], [4,2,0,0], [4,1,1,0], [3,3,0,0], [3,2,1,0], [3,1,1,1], [2,2,2,0], [2,2,1,1], [2,1,1,1,1], and [1,1,1,1,1,1]. But the last two are invalid because they have more than four parts. So, actually, the correct number is nine.Wait, but in my list above, I have 10, excluding the last one. Hmm, maybe I'm double-counting or something.Wait, let me recount:1. [6,0,0,0]2. [5,1,0,0]3. [4,2,0,0]4. [4,1,1,0]5. [3,3,0,0]6. [3,2,1,0]7. [3,1,1,1]8. [2,2,2,0]9. [2,2,1,1]10. [2,1,1,1]So, that's 10. But the partition [2,1,1,1] sums to five, not six. Wait, no, 2+1+1+1=5, which is less than six. So, that's invalid. Therefore, that's incorrect.Wait, so [2,1,1,1] is invalid because it sums to five, not six. So, we need to adjust that.Wait, so if we have four dartboards, and we need to distribute six darts, then the sum must be six. So, [2,1,1,1] sums to five, which is incorrect. Therefore, that partition is invalid.So, let's correct that. The correct partitions should all sum to six.So, let's list them again:1. [6,0,0,0] - sum=62. [5,1,0,0] - sum=63. [4,2,0,0] - sum=64. [4,1,1,0] - sum=65. [3,3,0,0] - sum=66. [3,2,1,0] - sum=67. [3,1,1,1] - sum=68. [2,2,2,0] - sum=69. [2,2,1,1] - sum=610. [2,1,1,1,1] - sum=6, but this is five parts, which is invalid.11. [1,1,1,1,1,1] - sum=6, but six parts, invalid.So, excluding the last two, we have nine valid partitions.Wait, but in my earlier list, I had [2,1,1,1] which sums to five, which is incorrect. So, that was a mistake.Therefore, the correct number of partitions is nine.Wait, but in the initial problem, the user thought the answer was 11, but that included invalid partitions. So, the correct answer should be nine.Wait, but let me double-check. Maybe I'm missing some partitions.Let me consider all possible partitions of 6 into at most four parts, in non-increasing order.1. 62. 5 + 13. 4 + 24. 4 + 1 + 15. 3 + 36. 3 + 2 + 17. 3 + 1 + 1 + 18. 2 + 2 + 29. 2 + 2 + 1 + 110. 2 + 1 + 1 + 1 + 1 (invalid)11. 1 + 1 + 1 + 1 + 1 + 1 (invalid)So, only nine valid partitions.Therefore, the number of different lists is nine.Wait, but in the initial problem, the user thought it was 11, but that included invalid partitions. So, the correct answer is nine.Wait, but I'm getting confused because different sources say different things. Let me try to think differently.Another way to approach this is to use the concept of partitions of an integer into at most k parts. The number of partitions of 6 into at most 4 parts is equal to the number of partitions of 6 where the largest part is at most 4. Wait, no, that's not correct. The number of partitions into at most 4 parts is the same as the number of partitions where the largest part is at most 4.Wait, no, actually, the number of partitions of n into at most k parts is equal to the number of partitions of n where the largest part is at most k. So, in this case, the number of partitions of 6 into at most 4 parts is equal to the number of partitions of 6 where the largest part is at most 4.But in our case, the largest part can be up to 6, but we're restricting to at most four parts. So, perhaps it's better to use the stars and bars theorem, but since the dartboards are identical, it's about partitions, not combinations.Wait, another approach: the number of ways to distribute 6 identical darts into 4 identical dartboards is equal to the number of integer partitions of 6 into at most 4 parts.So, let me look up the number of partitions of 6 into at most 4 parts.From integer partition tables, the number of partitions of 6 is 11. The number of partitions of 6 into at most 4 parts is 9. So, that confirms it.Therefore, the answer is 9.Wait, but in the initial problem, the user thought the answer was 11, but that was incorrect because they included partitions with more than four parts. So, the correct answer is 9.Wait, but let me make sure. Let me list all partitions of 6 into at most 4 parts:1. 62. 5 + 13. 4 + 24. 4 + 1 + 15. 3 + 36. 3 + 2 + 17. 3 + 1 + 1 + 18. 2 + 2 + 29. 2 + 2 + 1 + 1Yes, that's nine partitions.Therefore, the number of different lists is nine.

💡Alright, so I've got this problem about taxi fares in a city, and I need to figure out the relationship between the fare and the distance traveled, and then compare two different plans for a 30 km journey to see which one is more cost-effective. Let me try to break this down step by step.First, let's understand the fare structure. The problem says there's a starting price of 5 yuan for the first 3 km. That's the minimum fare for taking a taxi once. After that, for distances beyond 3 km, the fare is 1.2 yuan per kilometer. But then, after 10 km, there's an additional 50% empty fare charged for each kilometer. So, beyond 10 km, the fare per kilometer increases by 50%, making it 1.8 yuan per kilometer.Okay, so to write the relationship between the fare and the distance traveled, I think I need to define it piecewise because the fare structure changes at different intervals. Let's denote the distance traveled as ( x ) kilometers and the fare as ( y ) yuan.1. For distances less than 3 km, the fare is fixed at 5 yuan. So, if ( x < 3 ), then ( y = 5 ).2. For distances between 3 km and 10 km, the fare starts at 5 yuan for the first 3 km, and then for each additional kilometer beyond 3 km, it's 1.2 yuan. So, for ( 3 leq x leq 10 ), the fare can be calculated as: [ y = 5 + (x - 3) times 1.2 ] Simplifying that, it becomes: [ y = 1.2x + 1.4 ] Wait, let me check that. If ( x = 3 ), then ( y = 5 + (3 - 3) times 1.2 = 5 ), which is correct. If ( x = 10 ), then ( y = 5 + (10 - 3) times 1.2 = 5 + 8.4 = 13.4 ). Okay, that seems right.3. For distances beyond 10 km, the fare structure changes again. After 10 km, each additional kilometer costs 1.8 yuan instead of 1.2 yuan. So, for ( x > 10 ), the fare would be the fare up to 10 km plus the additional cost for the distance beyond 10 km. The fare up to 10 km is 13.4 yuan, as calculated before. So, for ( x > 10 ), the fare is: [ y = 13.4 + (x - 10) times 1.8 ] Simplifying that, it becomes: [ y = 1.8x - 4.6 ] Let me verify this. If ( x = 10 ), then ( y = 1.8 times 10 - 4.6 = 18 - 4.6 = 13.4 ), which matches the previous calculation. If ( x = 15 ), then ( y = 1.8 times 15 - 4.6 = 27 - 4.6 = 22.4 ). That seems correct.So, putting it all together, the relationship between the fare ( y ) and the distance ( x ) can be expressed as a piecewise function:[y =begin{cases}5, & text{if } x < 3 1.2x + 1.4, & text{if } 3 leq x leq 10 1.8x - 4.6, & text{if } x > 10 end{cases}]Alright, that takes care of part (1). Now, moving on to part (2). The passenger has a journey of 30 km and wants to save money by choosing between two plans:a. Travel in two segments: take one taxi for 15 km, then switch to another taxi for the remaining 15 km.b. Travel in three segments: take a taxi for 10 km, then switch to another taxi, and repeat this process once more.I need to figure out which plan is more cost-effective.Let's start with Plan a. The passenger will take one taxi for 15 km, then switch to another taxi for the next 15 km. So, each segment is 15 km. I need to calculate the fare for each 15 km segment and then sum them up.From the fare structure we derived earlier, for ( x > 10 ), the fare is ( y = 1.8x - 4.6 ). So, for 15 km:[y = 1.8 times 15 - 4.6 = 27 - 4.6 = 22.4 text{ yuan}]Since the passenger is taking two taxis, each for 15 km, the total fare would be:[2 times 22.4 = 44.8 text{ yuan}]Now, let's look at Plan b. The passenger will take three segments of 10 km each. So, each taxi ride is 10 km. Again, using the fare structure, for ( 3 leq x leq 10 ), the fare is ( y = 1.2x + 1.4 ). So, for 10 km:[y = 1.2 times 10 + 1.4 = 12 + 1.4 = 13.4 text{ yuan}]Since the passenger is taking three taxis, each for 10 km, the total fare would be:[3 times 13.4 = 40.2 text{ yuan}]Comparing the two plans, Plan a costs 44.8 yuan, and Plan b costs 40.2 yuan. Therefore, Plan b is more cost-effective.But wait, let me double-check my calculations to make sure I didn't make any mistakes.For Plan a:- Each 15 km segment: ( y = 1.8 times 15 - 4.6 = 22.4 ) yuan.- Two segments: ( 2 times 22.4 = 44.8 ) yuan.For Plan b:- Each 10 km segment: ( y = 1.2 times 10 + 1.4 = 13.4 ) yuan.- Three segments: ( 3 times 13.4 = 40.2 ) yuan.Yes, that seems correct. So, Plan b is indeed cheaper.I wonder if there's any other way to split the journey that might be even cheaper, but based on the given plans, Plan b is better.Another thing to consider is whether switching taxis incurs any additional costs, like waiting time or pick-up fees, but the problem doesn't mention any such charges, so I think it's safe to assume that the only cost is the fare per kilometer as given.Also, it's important to note that the starting price of 5 yuan is for the first 3 km, and any additional distance beyond that is charged per kilometer. So, even if you travel exactly 3 km, you still pay 5 yuan. If you travel more than 3 km but less than or equal to 10 km, you pay the starting price plus 1.2 yuan for each additional kilometer. Beyond 10 km, the rate increases to 1.8 yuan per kilometer due to the additional 50% empty fare.This fare structure incentivizes passengers to keep their trips under 10 km to avoid the higher rate. However, in this case, the passenger has a 30 km journey, so splitting it into shorter segments can help avoid the higher per-kilometer rate.By splitting the journey into three 10 km segments, the passenger ensures that each taxi ride stays within the 3 to 10 km range, where the fare is 1.2 yuan per kilometer. This avoids the 1.8 yuan per kilometer rate that would apply if any segment exceeds 10 km.On the other hand, splitting the journey into two 15 km segments means that each taxi ride exceeds 10 km, triggering the higher fare rate for the distance beyond 10 km. This results in a higher total fare compared to splitting into three shorter segments.So, in conclusion, Plan b is more cost-effective because it keeps each taxi ride within the lower fare bracket, thereby minimizing the overall cost of the 30 km journey.

💡Alright, so I have this problem here: Given the function f(x) = e^{-2x} + 1, I need to find the area of the triangle formed by the tangent line to the curve y = f(x) at the point (0, f(0)) and the lines y = 0 and y = x. The options are A: 1, B: 2/3, C: 1/2, D: 1/3.Okay, let's break this down step by step. First, I need to find the equation of the tangent line to the curve at the point (0, f(0)). To do that, I remember that the tangent line at a point requires two things: the value of the function at that point and the derivative of the function at that point, which gives the slope.So, let's start by finding f(0). Plugging x = 0 into f(x):f(0) = e^{-2*0} + 1 = e^0 + 1 = 1 + 1 = 2.So, the point is (0, 2). Got that.Next, I need the derivative of f(x) to find the slope of the tangent line at x = 0. The function is f(x) = e^{-2x} + 1. The derivative of e^{-2x} is straightforward. Remember, the derivative of e^{kx} is k*e^{kx}, so the derivative of e^{-2x} is -2*e^{-2x}. The derivative of 1 is 0. So, putting it together:f'(x) = -2*e^{-2x}.Now, evaluate this derivative at x = 0:f'(0) = -2*e^{-2*0} = -2*e^0 = -2*1 = -2.So, the slope of the tangent line at (0, 2) is -2.With the slope and a point, I can write the equation of the tangent line using the point-slope form:y - y1 = m(x - x1),where m is the slope and (x1, y1) is the point. Plugging in the values:y - 2 = -2(x - 0).Simplifying this:y - 2 = -2x,so,y = -2x + 2.Alright, so the equation of the tangent line is y = -2x + 2.Now, I need to find the area of the triangle formed by this tangent line and the lines y = 0 and y = x.First, let's visualize this. The lines y = 0 is the x-axis, and y = x is a diagonal line passing through the origin at a 45-degree angle. The tangent line y = -2x + 2 is a straight line with a slope of -2 and y-intercept at (0, 2).To find the area of the triangle, I need to find the points where the tangent line intersects y = 0 and y = x. These intersection points, along with the origin (since y = 0 and y = x intersect at (0,0)), will form the vertices of the triangle.So, let's find the intersection points.First, find where the tangent line intersects y = 0.Set y = 0 in the equation of the tangent line:0 = -2x + 2.Solving for x:-2x + 2 = 0,-2x = -2,x = 1.So, the tangent line intersects y = 0 at (1, 0).Next, find where the tangent line intersects y = x.Set y = x in the equation of the tangent line:x = -2x + 2.Solving for x:x + 2x = 2,3x = 2,x = 2/3.Since y = x, the y-coordinate is also 2/3. So, the intersection point is (2/3, 2/3).Now, we have the three vertices of the triangle:1. The origin: (0, 0).2. The intersection with y = 0: (1, 0).3. The intersection with y = x: (2/3, 2/3).To find the area of this triangle, I can use the formula for the area of a triangle given three vertices. The formula is:Area = (1/2) * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|,where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the vertices.Plugging in our points:(x1, y1) = (0, 0),(x2, y2) = (1, 0),(x3, y3) = (2/3, 2/3).So,Area = (1/2) * |0*(0 - 2/3) + 1*(2/3 - 0) + (2/3)*(0 - 0)|Simplify each term:First term: 0*(0 - 2/3) = 0.Second term: 1*(2/3 - 0) = 1*(2/3) = 2/3.Third term: (2/3)*(0 - 0) = (2/3)*0 = 0.Adding them up:0 + 2/3 + 0 = 2/3.Taking the absolute value (which is still 2/3) and multiplying by 1/2:Area = (1/2) * (2/3) = 1/3.So, the area of the triangle is 1/3.Wait, let me double-check that. Maybe there's another way to compute the area to confirm.Alternatively, I can consider the base and height of the triangle.Looking at the points, the base can be the segment from (0, 0) to (1, 0), which has a length of 1 unit. The height would be the perpendicular distance from the point (2/3, 2/3) to the base, which lies along the x-axis. Since the base is along the x-axis, the height is simply the y-coordinate of the third point, which is 2/3.So, area = (1/2)*base*height = (1/2)*1*(2/3) = (1/2)*(2/3) = 1/3.Yep, same result. So, the area is indeed 1/3.Therefore, the correct answer is D: 1/3.**Final Answer**boxed{D}

💡Okay, so I have this problem where Mica wrote a list of numbers starting with 1, and each next number is the previous one plus 3. So the sequence goes 1, 4, 7, 10, 13, 16, and so on. Facu then underlined all the numbers in this list that are greater than 10 and less than 100,000, and have all their digits the same. I need to figure out which numbers Facu underlined.First, let me understand the sequence better. It's an arithmetic sequence where each term increases by 3. The general formula for the nth term of an arithmetic sequence is a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. Here, a_1 is 1 and d is 3, so the nth term is 1 + (n - 1)*3, which simplifies to 3n - 2. So, every number in Mica's list can be written as 3n - 2 for some integer n.Now, Facu is looking for numbers in this list that are greater than 10 and less than 100,000, and have all their digits the same. Numbers with all digits the same are called repdigits. For example, 11, 22, 33, ..., 99999 are repdigits.So, I need to find all repdigits between 11 and 99999 that are also in Mica's list. That means I need to find all repdigits that can be expressed as 3n - 2 for some integer n.Let me break this down. First, I can list all repdigits between 11 and 99999. Repdigits can have 2, 3, 4, or 5 digits. So, I'll consider each case separately.Starting with two-digit repdigits: 11, 22, 33, 44, 55, 66, 77, 88, 99.For each of these, I need to check if they can be written as 3n - 2. That is, for each repdigit R, check if (R + 2) is divisible by 3. If it is, then R is in Mica's list.Let's check each two-digit repdigit:1. 11: (11 + 2) = 13. 13 divided by 3 is not an integer. So, 11 is not in the list.2. 22: (22 + 2) = 24. 24 divided by 3 is 8, which is an integer. So, 22 is in the list.3. 33: (33 + 2) = 35. 35 divided by 3 is not an integer. So, 33 is not in the list.4. 44: (44 + 2) = 46. 46 divided by 3 is not an integer. So, 44 is not in the list.5. 55: (55 + 2) = 57. 57 divided by 3 is 19, which is an integer. So, 55 is in the list.6. 66: (66 + 2) = 68. 68 divided by 3 is not an integer. So, 66 is not in the list.7. 77: (77 + 2) = 79. 79 divided by 3 is not an integer. So, 77 is not in the list.8. 88: (88 + 2) = 90. 90 divided by 3 is 30, which is an integer. So, 88 is in the list.9. 99: (99 + 2) = 101. 101 divided by 3 is not an integer. So, 99 is not in the list.So, from the two-digit repdigits, the numbers 22, 55, and 88 are in Mica's list.Next, let's consider three-digit repdigits: 111, 222, 333, 444, 555, 666, 777, 888, 999.Again, for each repdigit R, check if (R + 2) is divisible by 3.1. 111: (111 + 2) = 113. 113 divided by 3 is not an integer.2. 222: (222 + 2) = 224. 224 divided by 3 is not an integer.3. 333: (333 + 2) = 335. 335 divided by 3 is not an integer.4. 444: (444 + 2) = 446. 446 divided by 3 is not an integer.5. 555: (555 + 2) = 557. 557 divided by 3 is not an integer.6. 666: (666 + 2) = 668. 668 divided by 3 is not an integer.7. 777: (777 + 2) = 779. 779 divided by 3 is not an integer.8. 888: (888 + 2) = 890. 890 divided by 3 is not an integer.9. 999: (999 + 2) = 1001. 1001 divided by 3 is not an integer.Hmm, none of the three-digit repdigits satisfy the condition. So, there are no three-digit repdigits in Mica's list.Moving on to four-digit repdigits: 1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999.Again, check if (R + 2) is divisible by 3.1. 1111: (1111 + 2) = 1113. 1113 divided by 3 is 371, which is an integer. So, 1111 is in the list.2. 2222: (2222 + 2) = 2224. 2224 divided by 3 is approximately 741.333, not an integer.3. 3333: (3333 + 2) = 3335. 3335 divided by 3 is approximately 1111.666, not an integer.4. 4444: (4444 + 2) = 4446. 4446 divided by 3 is 1482, which is an integer. So, 4444 is in the list.5. 5555: (5555 + 2) = 5557. 5557 divided by 3 is approximately 1852.333, not an integer.6. 6666: (6666 + 2) = 6668. 6668 divided by 3 is approximately 2222.666, not an integer.7. 7777: (7777 + 2) = 7779. 7779 divided by 3 is 2593, which is an integer. So, 7777 is in the list.8. 8888: (8888 + 2) = 8890. 8890 divided by 3 is approximately 2963.333, not an integer.9. 9999: (9999 + 2) = 10001. 10001 divided by 3 is approximately 3333.666, not an integer.So, from the four-digit repdigits, the numbers 1111, 4444, and 7777 are in Mica's list.Wait, hold on. Let me double-check that. For 2222: 2222 + 2 = 2224. 2224 divided by 3 is 741.333... which is not an integer, so 2222 is not in the list. Similarly, 3333 + 2 = 3335, which is not divisible by 3. 5555 + 2 = 5557, not divisible by 3. 6666 + 2 = 6668, not divisible by 3. 8888 + 2 = 8890, not divisible by 3. 9999 + 2 = 10001, not divisible by 3. So, only 1111, 4444, and 7777 are in the list.Wait, but 2222: 2222 divided by 3 is 740.666..., so 2222 is not in the list. Similarly, 3333 divided by 3 is 1111, but 3333 + 2 = 3335, which is not divisible by 3. So, yeah, only 1111, 4444, and 7777 are in the list.Wait, but 4444 + 2 = 4446, which is divisible by 3, so 4444 is in the list. Similarly, 7777 + 2 = 7779, which is divisible by 3, so 7777 is in the list.Wait, but 1111 + 2 = 1113, which is divisible by 3, so 1111 is in the list.Wait, but 2222 + 2 = 2224, which is not divisible by 3, so 2222 is not in the list.Similarly, 3333 + 2 = 3335, not divisible by 3.5555 + 2 = 5557, not divisible by 3.6666 + 2 = 6668, not divisible by 3.8888 + 2 = 8890, not divisible by 3.9999 + 2 = 10001, not divisible by 3.So, only 1111, 4444, and 7777 are in the list.Wait, but 4444: 4444 divided by 3 is 1481.333..., but 4444 + 2 = 4446, which is divisible by 3, so 4444 is in the list.Similarly, 7777 + 2 = 7779, which is divisible by 3, so 7777 is in the list.Okay, so four-digit repdigits in the list are 1111, 4444, and 7777.Wait, but 1111 is 1111, which is 3*371 - 2, so that's correct.Similarly, 4444 is 3*1482 - 2, which is correct.7777 is 3*2593 - 2, which is correct.Okay, so that's three numbers.Now, moving on to five-digit repdigits: 11111, 22222, 33333, 44444, 55555, 66666, 77777, 88888, 99999.Again, check if (R + 2) is divisible by 3.1. 11111: (11111 + 2) = 11113. 11113 divided by 3 is approximately 3704.333, not an integer.2. 22222: (22222 + 2) = 22224. 22224 divided by 3 is 7408, which is an integer. So, 22222 is in the list.3. 33333: (33333 + 2) = 33335. 33335 divided by 3 is approximately 11111.666, not an integer.4. 44444: (44444 + 2) = 44446. 44446 divided by 3 is approximately 14815.333, not an integer.5. 55555: (55555 + 2) = 55557. 55557 divided by 3 is 18519, which is an integer. So, 55555 is in the list.6. 66666: (66666 + 2) = 66668. 66668 divided by 3 is approximately 22222.666, not an integer.7. 77777: (77777 + 2) = 77779. 77779 divided by 3 is approximately 25926.333, not an integer.8. 88888: (88888 + 2) = 88890. 88890 divided by 3 is 29630, which is an integer. So, 88888 is in the list.9. 99999: (99999 + 2) = 100001. 100001 divided by 3 is approximately 33333.666, not an integer.So, from the five-digit repdigits, the numbers 22222, 55555, and 88888 are in Mica's list.Wait, let me double-check:22222 + 2 = 22224, which is divisible by 3, so 22222 is in the list.55555 + 2 = 55557, which is divisible by 3, so 55555 is in the list.88888 + 2 = 88890, which is divisible by 3, so 88888 is in the list.The others: 11111 + 2 = 11113, not divisible by 3.33333 + 2 = 33335, not divisible by 3.44444 + 2 = 44446, not divisible by 3.66666 + 2 = 66668, not divisible by 3.77777 + 2 = 77779, not divisible by 3.99999 + 2 = 100001, not divisible by 3.So, only 22222, 55555, and 88888 are in the list.Wait, but 22222 is 22222, which is 3*7408 - 2, correct.55555 is 3*18519 - 2, correct.88888 is 3*29630 - 2, correct.Okay, so that's three numbers.So, compiling all the numbers we've found:Two-digit: 22, 55, 88.Four-digit: 1111, 4444, 7777.Five-digit: 22222, 55555, 88888.Wait, but earlier, when I checked four-digit repdigits, I found 1111, 4444, and 7777. But when I checked five-digit, I found 22222, 55555, and 88888.Wait, but in the four-digit case, I had 1111, 4444, 7777. But in the five-digit case, I have 22222, 55555, 88888.Wait, but 22222 is a five-digit number, so it's separate.So, altogether, the numbers Facu underlined are:22, 55, 88, 1111, 4444, 7777, 22222, 55555, 88888.Wait, but in the four-digit case, I had 1111, 4444, 7777.In the five-digit case, I had 22222, 55555, 88888.So, that's 3 + 3 + 3 = 9 numbers.Wait, but in the initial two-digit case, I had 22, 55, 88.In the four-digit case, 1111, 4444, 7777.In the five-digit case, 22222, 55555, 88888.So, that's 3 + 3 + 3 = 9 numbers.Wait, but let me check if I missed any.Wait, in the four-digit case, I had 1111, 4444, 7777.Is that correct?Yes, because 1111 + 2 = 1113, which is divisible by 3.4444 + 2 = 4446, which is divisible by 3.7777 + 2 = 7779, which is divisible by 3.Similarly, in the five-digit case, 22222 + 2 = 22224, divisible by 3.55555 + 2 = 55557, divisible by 3.88888 + 2 = 88890, divisible by 3.So, those are correct.Wait, but in the four-digit case, I had 1111, 4444, 7777.But 2222 + 2 = 2224, which is not divisible by 3, so 2222 is not in the list.Similarly, 3333 + 2 = 3335, not divisible by 3.5555 + 2 = 5557, not divisible by 3.6666 + 2 = 6668, not divisible by 3.8888 + 2 = 8890, not divisible by 3.9999 + 2 = 10001, not divisible by 3.So, only 1111, 4444, 7777 are in the four-digit repdigits.Similarly, in the five-digit case, only 22222, 55555, 88888 are in the list.So, compiling all, the numbers are:Two-digit: 22, 55, 88.Four-digit: 1111, 4444, 7777.Five-digit: 22222, 55555, 88888.So, that's 3 + 3 + 3 = 9 numbers.Wait, but in the problem statement, it says "greater than 10 and less than 100000". So, 100000 is excluded, but 99999 is included.So, all the numbers we've found are within that range.Wait, but let me check if there are any other repdigits that I might have missed.Wait, in the two-digit case, I had 22, 55, 88.In the four-digit case, 1111, 4444, 7777.In the five-digit case, 22222, 55555, 88888.Is that all?Wait, let me think about the three-digit case again. I had none, but let me double-check.Three-digit repdigits: 111, 222, 333, 444, 555, 666, 777, 888, 999.For each, check if (R + 2) is divisible by 3.1. 111 + 2 = 113. 113 / 3 ≈ 37.666, not integer.2. 222 + 2 = 224. 224 / 3 ≈ 74.666, not integer.3. 333 + 2 = 335. 335 / 3 ≈ 111.666, not integer.4. 444 + 2 = 446. 446 / 3 ≈ 148.666, not integer.5. 555 + 2 = 557. 557 / 3 ≈ 185.666, not integer.6. 666 + 2 = 668. 668 / 3 ≈ 222.666, not integer.7. 777 + 2 = 779. 779 / 3 ≈ 259.666, not integer.8. 888 + 2 = 890. 890 / 3 ≈ 296.666, not integer.9. 999 + 2 = 1001. 1001 / 3 ≈ 333.666, not integer.So, none of the three-digit repdigits are in the list.So, the only repdigits in Mica's list are the ones I found earlier.Therefore, the numbers Facu underlined are:22, 55, 88, 1111, 4444, 7777, 22222, 55555, 88888.Wait, but let me check if 22222 is indeed in the list.22222 = 3n - 2.So, 3n = 22224.n = 22224 / 3 = 7408.Yes, n is an integer, so 22222 is in the list.Similarly, 55555 = 3n - 2.3n = 55557.n = 55557 / 3 = 18519.Yes, integer.88888 = 3n - 2.3n = 88890.n = 88890 / 3 = 29630.Yes, integer.So, all those numbers are indeed in the list.Therefore, the final answer is:22, 55, 88, 1111, 4444, 7777, 22222, 55555, 88888.Wait, but in the problem statement, it says "greater than 10 and less than 100000". So, 100000 is excluded, but 99999 is included.So, all the numbers we've found are within that range.I think that's all. I don't think I missed any repdigits because I checked all possible repdigits with 2, 4, and 5 digits, and none of the three-digit ones worked.So, the numbers Facu underlined are:22, 55, 88, 1111, 4444, 7777, 22222, 55555, 88888.