💡Okay, so I have this problem where I need to figure out how many of the sets ( T_0, T_1, T_2, ldots, T_{1999} ) do not contain a perfect square. Each set ( T_i ) is defined as all integers ( n ) such that ( 50i leq n < 50(i + 1) ). For example, ( T_4 ) is the set ( {200, 201, 202, ldots, 249} ). First, I need to understand what each ( T_i ) represents. Each set ( T_i ) contains 50 consecutive integers starting from ( 50i ) up to ( 50(i + 1) - 1 ). So, ( T_0 ) is ( {0, 1, 2, ldots, 49} ), ( T_1 ) is ( {50, 51, 52, ldots, 99} ), and so on, up to ( T_{1999} ), which would be ( {99950, 99951, ldots, 99999} ).The question is asking how many of these 2000 sets do not contain any perfect squares. So, I need to figure out for each ( T_i ), whether there's a perfect square in that range, and then count how many sets don't have any.To approach this, I think I should first determine the range of perfect squares that could lie within these sets. The smallest number in the first set ( T_0 ) is 0, which is a perfect square (0^2). The largest number in the last set ( T_{1999} ) is 99999. So, the perfect squares we're considering range from 0^2 up to the largest integer whose square is less than 100000.Let me calculate the largest integer ( x ) such that ( x^2 < 100000 ). Taking the square root of 100000 gives approximately 316.227. So, the largest integer ( x ) is 316 because ( 316^2 = 99856 ) and ( 317^2 = 100489 ), which is greater than 100000.Therefore, the perfect squares we need to consider are from ( 0^2 ) up to ( 316^2 ). That gives us 317 perfect squares (including 0). Now, each perfect square ( k^2 ) will lie in exactly one set ( T_i ), specifically the set where ( 50i leq k^2 < 50(i + 1) ). So, each perfect square corresponds to a unique ( T_i ). Therefore, the number of sets that contain at least one perfect square is equal to the number of perfect squares, which is 317. But wait, that might not be entirely accurate because some sets might contain more than one perfect square, right? So, if two perfect squares fall into the same ( T_i ), then that set would still only count once.Therefore, I need to find how many distinct ( T_i ) sets contain at least one perfect square. So, perhaps it's better to find the range of ( i ) such that ( T_i ) contains a perfect square.Let me think about how perfect squares are distributed. The difference between consecutive squares is ( (k+1)^2 - k^2 = 2k + 1 ). For smaller ( k ), this difference is smaller, meaning that consecutive squares are closer together. As ( k ) increases, the difference between consecutive squares increases.Given that each ( T_i ) is a block of 50 numbers, I need to find out for each ( k ), which ( T_i ) the square ( k^2 ) falls into. Then, I can count how many unique ( T_i ) sets are covered by these squares.Alternatively, perhaps I can find the smallest ( i ) such that ( T_i ) contains a perfect square and the largest ( i ) such that ( T_i ) contains a perfect square, and then determine how many ( T_i ) sets lie between them, considering overlaps.Wait, but the problem is that for smaller ( k ), the squares are closer together, so multiple squares might fall into the same ( T_i ). For larger ( k ), the squares are spaced more than 50 apart, so each square falls into a unique ( T_i ).So, perhaps I can find a point where the difference between consecutive squares exceeds 50, meaning that beyond that point, each square will lie in a distinct ( T_i ). Let me calculate when ( 2k + 1 > 50 ). Solving for ( k ), we get ( k > (50 - 1)/2 = 24.5 ). So, for ( k geq 25 ), the difference between consecutive squares is more than 50. Therefore, starting from ( k = 25 ), each square will lie in a separate ( T_i ).But wait, let's verify that. For ( k = 24 ), ( 24^2 = 576 ), ( 25^2 = 625 ). The difference is 49, which is less than 50. So, ( 24^2 ) and ( 25^2 ) are 49 apart, meaning they could potentially lie in the same ( T_i ) if they are within the same 50-number block. But actually, 576 is in ( T_{11} ) (since ( 50*11 = 550 ) and ( 50*12 = 600 )), and 625 is in ( T_{12} ) (since ( 50*12 = 600 ) and ( 50*13 = 650 )). So, even though the difference is 49, they are in adjacent sets. So, actually, each square from ( k = 25 ) onwards will be in a unique ( T_i ), because the next square is more than 50 apart, so it can't be in the same ( T_i ).Wait, but let's check ( k = 25 ) and ( k = 26 ). ( 25^2 = 625 ), ( 26^2 = 676 ). The difference is 51, which is more than 50. So, 625 is in ( T_{12} ) and 676 is in ( T_{13} ). So, each square from ( k = 25 ) onwards will be in a unique ( T_i ), because the next square is more than 50 apart, so it can't be in the same ( T_i ).Therefore, for ( k geq 25 ), each square ( k^2 ) lies in a distinct ( T_i ). For ( k < 25 ), the squares are closer together, so some ( T_i ) sets might contain more than one square, but each square still corresponds to a unique ( T_i ).Wait, but actually, even for ( k < 25 ), each square is in a unique ( T_i ), because even though the squares are close, each square is still in its own ( T_i ). For example, ( 24^2 = 576 ) is in ( T_{11} ), ( 25^2 = 625 ) is in ( T_{12} ), so each square is in a separate ( T_i ). So, actually, each square from ( k = 0 ) to ( k = 316 ) is in a unique ( T_i ). Therefore, the number of ( T_i ) sets that contain at least one perfect square is equal to the number of perfect squares, which is 317.But wait, that can't be right because ( T_0 ) contains 0, which is a perfect square, and ( T_{1999} ) contains 99999, which is not a perfect square. So, the number of ( T_i ) sets containing at least one perfect square is 317, but we have to consider that some ( T_i ) sets might not contain any perfect squares, especially the ones beyond the last perfect square.Wait, no, because the last perfect square is ( 316^2 = 99856 ), which is in ( T_{1997} ) because ( 50*1997 = 99850 ) and ( 50*1998 = 99950 ). So, ( 99856 ) is in ( T_{1997} ). Then, ( T_{1998} ) is ( 99950 ) to ( 99999 ), which doesn't contain any perfect squares because the next square after ( 316^2 ) is ( 317^2 = 100489 ), which is beyond 100000.So, the perfect squares go up to ( T_{1997} ). Therefore, the number of ( T_i ) sets containing at least one perfect square is from ( T_0 ) up to ( T_{1997} ), but not all of these sets contain a perfect square. Specifically, each perfect square corresponds to a unique ( T_i ), so the number of ( T_i ) sets with at least one perfect square is 317.Wait, but that seems conflicting with my earlier thought. Let me clarify.Each perfect square ( k^2 ) for ( k = 0 ) to ( 316 ) lies in exactly one ( T_i ). Therefore, there are 317 perfect squares, each in a unique ( T_i ). Therefore, there are 317 ( T_i ) sets that contain at least one perfect square.But the total number of ( T_i ) sets is 2000 (from ( T_0 ) to ( T_{1999} )). Therefore, the number of sets without any perfect squares is ( 2000 - 317 = 1683 ). But wait, that doesn't match the initial thought process where the assistant concluded 1733. So, perhaps my reasoning is flawed.Wait, let's think again. The key is that for ( k geq 25 ), each square ( k^2 ) is in a unique ( T_i ), but for ( k < 25 ), the squares are spaced less than 50 apart, so some ( T_i ) sets might contain multiple squares. However, each square is still in a unique ( T_i ), so the total number of ( T_i ) sets with at least one square is still 317.But wait, let's check for small ( k ). For example, ( k = 0 ), ( 0^2 = 0 ) is in ( T_0 ). ( k = 1 ), ( 1^2 = 1 ) is in ( T_0 ). Wait, no, ( T_0 ) is ( 0 ) to ( 49 ), so ( 1 ) is in ( T_0 ). Similarly, ( 2^2 = 4 ) is in ( T_0 ), and so on up to ( 7^2 = 49 ) is in ( T_0 ). So, ( T_0 ) contains multiple perfect squares, specifically from ( 0^2 ) to ( 7^2 ).Similarly, ( T_1 ) is ( 50 ) to ( 99 ). The perfect squares in this range are ( 8^2 = 64 ) and ( 9^2 = 81 ), so ( T_1 ) contains two perfect squares.Continuing this way, each ( T_i ) for ( i ) such that ( 50i leq k^2 < 50(i+1) ) will contain some perfect squares. However, for ( k geq 25 ), each ( k^2 ) is in a unique ( T_i ), but for ( k < 25 ), multiple ( k^2 ) can fall into the same ( T_i ).Therefore, the number of ( T_i ) sets that contain at least one perfect square is equal to the number of ( T_i ) sets that contain any of the 317 perfect squares. However, since multiple squares can fall into the same ( T_i ), the number of unique ( T_i ) sets is less than or equal to 317.Wait, but actually, each square is in a unique ( T_i ), so the number of ( T_i ) sets that contain at least one perfect square is exactly 317, because each square is in a distinct ( T_i ). Therefore, the number of sets without any perfect squares is ( 2000 - 317 = 1683 ).But this contradicts the initial thought process where the assistant concluded 1733. So, I must be making a mistake here.Wait, perhaps the mistake is that for ( k geq 25 ), each square is in a unique ( T_i ), but for ( k < 25 ), some ( T_i ) sets contain multiple squares, but each square is still in a unique ( T_i ). Therefore, the total number of ( T_i ) sets with at least one square is 317, as each square is in a unique ( T_i ). Therefore, the number of sets without squares is ( 2000 - 317 = 1683 ).But wait, let's think about the distribution of squares. For ( k ) from 0 to 7, ( k^2 ) is in ( T_0 ). So, ( T_0 ) contains 8 perfect squares. Then, ( k = 8 ) to ( k = 9 ), ( k^2 ) is in ( T_1 ). So, ( T_1 ) contains 2 perfect squares. Similarly, ( k = 10 ) to ( k = 14 ), ( k^2 ) is in ( T_2 ) to ( T_3 ). Wait, let me calculate more carefully.Let me list the squares and see which ( T_i ) they fall into.( k = 0 ): 0 → ( T_0 )( k = 1 ): 1 → ( T_0 )( k = 2 ): 4 → ( T_0 )( k = 3 ): 9 → ( T_0 )( k = 4 ): 16 → ( T_0 )( k = 5 ): 25 → ( T_0 )( k = 6 ): 36 → ( T_0 )( k = 7 ): 49 → ( T_0 )( k = 8 ): 64 → ( T_1 ) (since 50 ≤ 64 < 100)( k = 9 ): 81 → ( T_1 )( k = 10 ): 100 → ( T_2 ) (100 is the start of ( T_2 ))( k = 11 ): 121 → ( T_2 )( k = 12 ): 144 → ( T_2 )( k = 13 ): 169 → ( T_3 ) (150 ≤ 169 < 200)( k = 14 ): 196 → ( T_3 )( k = 15 ): 225 → ( T_4 ) (200 ≤ 225 < 250)( k = 16 ): 256 → ( T_5 ) (250 ≤ 256 < 300)( k = 17 ): 289 → ( T_5 )( k = 18 ): 324 → ( T_6 ) (300 ≤ 324 < 350)( k = 19 ): 361 → ( T_7 ) (350 ≤ 361 < 400)( k = 20 ): 400 → ( T_8 ) (400 is the start of ( T_8 ))( k = 21 ): 441 → ( T_8 )( k = 22 ): 484 → ( T_9 ) (450 ≤ 484 < 500)( k = 23 ): 529 → ( T_{10} ) (500 ≤ 529 < 550)( k = 24 ): 576 → ( T_{11} ) (550 ≤ 576 < 600)( k = 25 ): 625 → ( T_{12} ) (600 ≤ 625 < 650)( k = 26 ): 676 → ( T_{13} ) (650 ≤ 676 < 700)( k = 27 ): 729 → ( T_{14} ) (700 ≤ 729 < 750)( k = 28 ): 784 → ( T_{15} ) (750 ≤ 784 < 800)( k = 29 ): 841 → ( T_{16} ) (800 ≤ 841 < 850)( k = 30 ): 900 → ( T_{18} ) (900 is the start of ( T_{18} ))Wait, ( T_{18} ) is 900 to 949, so 900 is in ( T_{18} ).( k = 31 ): 961 → ( T_{19} ) (950 ≤ 961 < 1000)( k = 32 ): 1024 → ( T_{20} ) (1000 ≤ 1024 < 1050)And so on, up to ( k = 316 ): 99856 → ( T_{1997} ) (99850 ≤ 99856 < 99900)Wait, so from ( k = 0 ) to ( k = 24 ), the squares are in ( T_0 ) to ( T_{11} ). Each of these ( T_i ) sets contains multiple squares, but each square is in a unique ( T_i ). So, the number of ( T_i ) sets with at least one square from ( k = 0 ) to ( k = 24 ) is 12 sets (from ( T_0 ) to ( T_{11} )).Then, from ( k = 25 ) to ( k = 316 ), each square is in a unique ( T_i ). The number of these squares is ( 316 - 24 = 292 ). Therefore, the number of ( T_i ) sets with at least one square from ( k = 25 ) to ( k = 316 ) is 292.Therefore, the total number of ( T_i ) sets with at least one perfect square is ( 12 + 292 = 304 ).Wait, but earlier I thought it was 317. So, where is the discrepancy?Wait, from ( k = 0 ) to ( k = 316 ), there are 317 squares. However, some of these squares fall into the same ( T_i ) sets, particularly for ( k < 25 ). So, the number of unique ( T_i ) sets with at least one square is less than 317.From ( k = 0 ) to ( k = 24 ), we have squares in ( T_0 ) to ( T_{11} ), which is 12 sets. Then, from ( k = 25 ) to ( k = 316 ), each square is in a unique ( T_i ), which is ( 316 - 24 = 292 ) sets. So, total unique ( T_i ) sets with squares is ( 12 + 292 = 304 ).Therefore, the number of ( T_i ) sets without any perfect squares is ( 2000 - 304 = 1696 ).But wait, let's verify this with a different approach.Another way to think about it is to find the smallest ( i ) such that ( T_i ) contains a perfect square and the largest ( i ) such that ( T_i ) contains a perfect square, and then count the number of ( T_i ) sets in between, considering overlaps.The smallest ( i ) is ( i = 0 ), since ( T_0 ) contains 0, which is a perfect square.The largest ( i ) is ( i = 1997 ), since ( 316^2 = 99856 ) is in ( T_{1997} ).Now, the number of ( T_i ) sets from ( T_0 ) to ( T_{1997} ) is 1998 sets. However, not all of these sets contain a perfect square. The number of sets that do contain a perfect square is equal to the number of unique ( T_i ) sets that contain any of the 317 perfect squares.But as we saw earlier, from ( k = 0 ) to ( k = 24 ), the squares are in ( T_0 ) to ( T_{11} ), which is 12 sets. Then, from ( k = 25 ) to ( k = 316 ), each square is in a unique ( T_i ), which is 292 sets. So, total unique ( T_i ) sets with squares is 304.Therefore, the number of ( T_i ) sets without any perfect squares is ( 2000 - 304 = 1696 ).But wait, the initial thought process concluded 1733. So, perhaps my calculation is still off.Wait, let's think about the distribution of squares again. For ( k geq 25 ), each square is in a unique ( T_i ), so the number of such ( T_i ) sets is ( 316 - 24 = 292 ). For ( k < 25 ), the squares are in ( T_0 ) to ( T_{11} ), which is 12 sets. Therefore, total unique ( T_i ) sets with squares is 12 + 292 = 304.Therefore, the number of sets without squares is 2000 - 304 = 1696.But wait, let's check for ( k = 25 ) to ( k = 316 ). The number of squares is 316 - 24 = 292, and each corresponds to a unique ( T_i ). So, 292 sets.For ( k = 0 ) to ( k = 24 ), we have 25 squares, but they fall into 12 sets. So, total unique sets with squares is 12 + 292 = 304.Therefore, total sets without squares is 2000 - 304 = 1696.But wait, let's check the initial calculation. The assistant thought that from ( T_{50} ) to ( T_{316} ), each set contains at least one perfect square, which would be 316 - 50 + 1 = 267 sets. Then, subtracting from 2000 gives 1733. But that seems incorrect because the squares start from ( T_0 ) and go up to ( T_{1997} ), and the number of unique ( T_i ) sets with squares is 304, not 267.Wait, perhaps the assistant made a mistake in assuming that from ( T_{50} ) onwards, each set contains a perfect square, but that's not the case. The squares are spread out, and each square from ( k = 25 ) onwards is in a unique ( T_i ), but the number of such ( T_i ) sets is 292, not 267.Wait, let's calculate the number of ( T_i ) sets from ( k = 25 ) to ( k = 316 ). Each square ( k^2 ) is in a unique ( T_i ). The number of such squares is 316 - 24 = 292. Therefore, the number of ( T_i ) sets from ( k = 25 ) to ( k = 316 ) is 292.Additionally, from ( k = 0 ) to ( k = 24 ), the squares are in ( T_0 ) to ( T_{11} ), which is 12 sets.Therefore, total unique ( T_i ) sets with squares is 12 + 292 = 304.Therefore, the number of sets without squares is 2000 - 304 = 1696.But this contradicts the initial thought process. So, perhaps the initial thought process was incorrect.Alternatively, perhaps the initial thought process considered that from ( T_{50} ) onwards, each set contains a perfect square, but that's not accurate. The squares are spread out, and each square from ( k = 25 ) onwards is in a unique ( T_i ), but the number of such ( T_i ) sets is 292, not 267.Wait, let's calculate the number of ( T_i ) sets from ( k = 25 ) to ( k = 316 ). Each square ( k^2 ) is in a unique ( T_i ). The number of such squares is 316 - 24 = 292. Therefore, the number of ( T_i ) sets from ( k = 25 ) to ( k = 316 ) is 292.Additionally, from ( k = 0 ) to ( k = 24 ), the squares are in ( T_0 ) to ( T_{11} ), which is 12 sets.Therefore, total unique ( T_i ) sets with squares is 12 + 292 = 304.Therefore, the number of sets without squares is 2000 - 304 = 1696.But wait, let's think about the initial thought process again. The assistant said:1. Calculate the smallest and largest possible values of ( i ) such that ( T_i ) contains at least one perfect square: - Consider ( (x + 1)^2 - x^2 = 2x + 1 ). For ( x = 50 ), ( (50 + 1)^2 - 50^2 = 101 ), which means that from ( 2500 ) onwards, consecutive squares at least lie more than 50 apart. Thus, the sets starting from ( T_{50} ) will not consecutively overlap single sets.2. Finding the upper limit for ( i ): - Maximum value for ( n ) is ( 99950 ) in ( T_{1999} ). Hence we find the largest ( x ) such that ( x^2 < 99950 ), which we find using an estimate: ( 316^2 = 99856 ) is the last perfect square below ( 99950 ), so ( 316 ) is the largest ( x ). - Thus, from ( T_{50} ) to ( T_{316} ), each set will include at least one perfect square.3. Count sets without any perfect square: - There are ( 1999 + 1 = 2000 ) total sets, and ( 316 - 50 + 1 = 267 ) sets with at least one perfect square (from ( 50 ) to ( 316 ) inclusive). - ( 2000 - 267 = boxed{1733} ) sets do not contain any perfect squares.But this approach is flawed because it assumes that from ( T_{50} ) to ( T_{316} ), each set contains at least one perfect square, which is not true. The squares are spread out, and each square from ( k = 25 ) onwards is in a unique ( T_i ), but the number of such ( T_i ) sets is 292, not 267.Wait, let's calculate ( 316 - 50 + 1 = 267 ). But this is incorrect because the squares from ( k = 25 ) to ( k = 316 ) are in ( T_i ) sets starting from ( T_{12} ) (since ( 25^2 = 625 ) is in ( T_{12} )) up to ( T_{1997} ). Therefore, the number of ( T_i ) sets from ( T_{12} ) to ( T_{1997} ) is 1997 - 12 + 1 = 1986 sets. But this is not correct because each square from ( k = 25 ) to ( k = 316 ) is in a unique ( T_i ), but the ( T_i ) sets are not necessarily consecutive.Wait, no. Each square from ( k = 25 ) to ( k = 316 ) is in a unique ( T_i ), but the ( T_i ) sets are spread out, not necessarily consecutive. Therefore, the number of ( T_i ) sets with squares from ( k = 25 ) to ( k = 316 ) is 292, as calculated earlier.Therefore, the initial thought process was incorrect in assuming that from ( T_{50} ) to ( T_{316} ), each set contains a perfect square. Instead, the correct number of ( T_i ) sets with squares is 304, leading to 2000 - 304 = 1696 sets without squares.But wait, let's think again. The assistant's initial thought process concluded 1733, but my calculation is 1696. Which one is correct?Alternatively, perhaps the correct approach is to calculate the number of ( T_i ) sets that do not contain any perfect squares by considering the gaps between squares.For ( k geq 25 ), each square is in a unique ( T_i ), and the gap between consecutive squares is more than 50, so each square is in a separate ( T_i ). Therefore, the number of ( T_i ) sets with squares from ( k = 25 ) to ( k = 316 ) is 292.Additionally, from ( k = 0 ) to ( k = 24 ), the squares are in ( T_0 ) to ( T_{11} ), which is 12 sets.Therefore, total unique ( T_i ) sets with squares is 12 + 292 = 304.Therefore, the number of sets without squares is 2000 - 304 = 1696.But let's verify this with another method. Let's calculate the number of ( T_i ) sets that do not contain any perfect squares by considering the intervals between squares.For ( k geq 25 ), each square ( k^2 ) is in a unique ( T_i ), and the next square ( (k+1)^2 ) is more than 50 away, so the interval between ( k^2 ) and ( (k+1)^2 ) is more than 50, meaning that the ( T_i ) sets between ( T_i ) containing ( k^2 ) and ( T_i ) containing ( (k+1)^2 ) do not contain any squares.Wait, no. Because the squares are spaced more than 50 apart, the ( T_i ) sets between the ones containing squares will not contain any squares. Therefore, the number of ( T_i ) sets without squares is the total number of sets minus the number of sets containing squares.But we already calculated that the number of sets containing squares is 304, so the number of sets without squares is 2000 - 304 = 1696.But wait, let's think about the initial sets. From ( T_0 ) to ( T_{11} ), we have 12 sets containing squares. Then, from ( T_{12} ) to ( T_{1997} ), we have 292 sets containing squares. The remaining sets from ( T_{1998} ) to ( T_{1999} ) do not contain any squares because the last square is in ( T_{1997} ).So, the number of sets without squares is:- From ( T_{1998} ) to ( T_{1999} ): 2 sets.- Plus, the sets from ( T_{12} ) to ( T_{1997} ) that do not contain any squares.Wait, no. Because from ( T_{12} ) to ( T_{1997} ), we have 292 sets containing squares, but the total number of sets from ( T_{12} ) to ( T_{1997} ) is 1997 - 12 + 1 = 1986 sets. Therefore, the number of sets without squares in this range is 1986 - 292 = 1694 sets.Additionally, from ( T_{1998} ) to ( T_{1999} ), there are 2 sets without squares.Therefore, total sets without squares is 1694 + 2 = 1696.This matches our earlier calculation.Therefore, the correct answer should be 1696.But wait, the initial thought process concluded 1733, which is 2000 - 267. So, perhaps the initial thought process was incorrect because it assumed that from ( T_{50} ) to ( T_{316} ), each set contains a perfect square, which is not the case. Instead, the correct number of sets containing squares is 304, leading to 2000 - 304 = 1696 sets without squares.Therefore, the correct answer is 1696.But wait, let's double-check the calculation of the number of sets containing squares.From ( k = 0 ) to ( k = 24 ), the squares are in ( T_0 ) to ( T_{11} ), which is 12 sets.From ( k = 25 ) to ( k = 316 ), each square is in a unique ( T_i ), so 316 - 24 = 292 sets.Total sets with squares: 12 + 292 = 304.Therefore, sets without squares: 2000 - 304 = 1696.Yes, that seems correct.Therefore, the initial thought process was incorrect because it assumed that from ( T_{50} ) to ( T_{316} ), each set contains a perfect square, which is not the case. Instead, the correct number of sets containing squares is 304, leading to 1696 sets without squares.

💡Okay, so I need to find the area of a sector of a circle. The circle has a radius of 10 meters, and the central angle is 42 degrees. Hmm, I remember that the area of a sector is related to the angle and the radius, but I'm not exactly sure about the formula. Let me think.I think the formula involves the angle divided by 360 degrees multiplied by the area of the whole circle. Yeah, that sounds right. So, the area of the whole circle is π times the radius squared, right? So, for this circle, the area would be π * (10)^2, which is 100π square meters.Now, since the sector is just a portion of the circle, I need to find what fraction of the whole circle the sector represents. The central angle is 42 degrees, and a full circle is 360 degrees. So, the fraction would be 42/360. Let me simplify that. 42 divided by 360... Well, both numbers are divisible by 6. 42 divided by 6 is 7, and 360 divided by 6 is 60. So, the fraction is 7/60.Now, to find the area of the sector, I multiply this fraction by the area of the whole circle. So, that would be (7/60) * 100π. Let me calculate that. 100 divided by 60 is... Hmm, 100 divided by 60 is the same as 5/3. So, 7 times 5/3 is 35/3. Therefore, the area of the sector is (35/3)π square meters.Wait, but the question might want a numerical value instead of an exact fraction with π. Okay, so I'll approximate π as 3.1416. So, 35 divided by 3 is approximately 11.6667. Multiplying that by 3.1416 gives me... Let me do that multiplication. 11.6667 * 3.1416 is approximately 36.6519 square meters.Hmm, that seems reasonable. Let me double-check my steps to make sure I didn't make a mistake. I started with the formula for the area of a sector, which is (θ/360) * πr². Plugged in θ = 42 degrees and r = 10 meters. Calculated the fraction 42/360, simplified it to 7/60. Then multiplied by π * 10², which is 100π. That gave me 700π/60, which simplifies to 35π/3. Converted that to a decimal by multiplying 35/3 by 3.1416, resulting in approximately 36.6519 square meters.I think that makes sense. The area of the sector should be less than the area of the whole circle, which is 100π or about 314.16 square meters. 36.65 is much less, so that seems plausible. Also, 42 degrees is a bit less than a tenth of 360 degrees, and 36.65 is roughly a tenth of 314.16, so the numbers align.Okay, I feel confident with this answer now.

💡Okay, so I have this integral to solve: the definite integral from -2 to 0 of (x + 2) squared times cosine of 3x dx. Hmm, that looks a bit complicated, but I think I can handle it with integration by parts. Let me recall how integration by parts works. The formula is ∫u dv = uv - ∫v du. So I need to choose which part of the integrand to set as u and which as dv.Looking at (x + 2)^2 cos(3x), I think it's best to let u be (x + 2)^2 because when I take its derivative, it will simplify, and dv will be cos(3x) dx. That way, I can integrate dv to get v. Let me write that down:Let u = (x + 2)^2, so du = 2(x + 2) dx.And dv = cos(3x) dx, so v = (1/3) sin(3x).Now, applying the integration by parts formula:∫(x + 2)^2 cos(3x) dx = uv - ∫v duWhich becomes:(x + 2)^2 * (1/3) sin(3x) - ∫(1/3) sin(3x) * 2(x + 2) dxSimplifying this, I get:(1/3)(x + 2)^2 sin(3x) - (2/3) ∫sin(3x)(x + 2) dxOkay, so now I have another integral to solve: ∫sin(3x)(x + 2) dx. This still looks like it requires integration by parts. Let me set up for that.Let me choose u again as the polynomial part, so u = x + 2, which means du = dx.And dv = sin(3x) dx, so v = -(1/3) cos(3x).Applying integration by parts again:∫sin(3x)(x + 2) dx = uv - ∫v duWhich is:-(x + 2)(1/3) cos(3x) - ∫-(1/3) cos(3x) dxSimplifying:-(1/3)(x + 2) cos(3x) + (1/3) ∫cos(3x) dxNow, ∫cos(3x) dx is straightforward. The integral of cos(ax) is (1/a) sin(ax), so:(1/3) * (1/3) sin(3x) = (1/9) sin(3x)Putting it all together:∫sin(3x)(x + 2) dx = -(1/3)(x + 2) cos(3x) + (1/9) sin(3x) + CNow, going back to the earlier expression:(1/3)(x + 2)^2 sin(3x) - (2/3)[ -(1/3)(x + 2) cos(3x) + (1/9) sin(3x) ] + CLet me distribute the (2/3):= (1/3)(x + 2)^2 sin(3x) + (2/9)(x + 2) cos(3x) - (2/27) sin(3x) + CNow, I need to evaluate this from -2 to 0. Let's plug in the limits.First, at x = 0:(1/3)(0 + 2)^2 sin(0) + (2/9)(0 + 2) cos(0) - (2/27) sin(0)Simplify:(1/3)(4)(0) + (2/9)(2)(1) - (2/27)(0) = 0 + 4/9 + 0 = 4/9Next, at x = -2:(1/3)(-2 + 2)^2 sin(-6) + (2/9)(-2 + 2) cos(-6) - (2/27) sin(-6)Simplify:(1/3)(0)^2 sin(-6) + (2/9)(0) cos(-6) - (2/27)(-sin(6)) = 0 + 0 + (2/27) sin(6)So, the integral from -2 to 0 is:[4/9] - [ (2/27) sin(6) ] = 4/9 - (2/27) sin(6)To combine these terms, I can write 4/9 as 12/27:12/27 - 2/27 sin(6) = (12 - 2 sin(6))/27So, the final answer is (12 - 2 sin(6))/27.Wait, let me double-check my steps to make sure I didn't make any mistakes. First, I did integration by parts twice, which seems correct. Then, I evaluated the expression at the limits, making sure to handle the negative signs properly, especially at x = -2. It looks like I accounted for the negative sine correctly. Yeah, I think this is right.

💡Okay, so I have this algebra problem to solve: simplify the expression ((a + b)^2 - 2a(a - b) + (a + 2b)(a - 2b)) and then evaluate it when (a = -1) and (b = 4). Hmm, let me see how to approach this step by step.First, I need to simplify the expression. It looks like there are three main parts: ((a + b)^2), (-2a(a - b)), and ((a + 2b)(a - 2b)). I think I should expand each part separately and then combine like terms.Starting with the first part, ((a + b)^2). I remember that squaring a binomial like this follows the formula ((x + y)^2 = x^2 + 2xy + y^2). Applying that here:[(a + b)^2 = a^2 + 2ab + b^2]Okay, that's straightforward. Next, the second part is (-2a(a - b)). I'll distribute the (-2a) across the terms inside the parentheses:[-2a(a - b) = -2a cdot a + (-2a) cdot (-b) = -2a^2 + 2ab]Wait, let me double-check that. Multiplying (-2a) by (a) gives (-2a^2), and multiplying (-2a) by (-b) gives (+2ab). Yeah, that seems right.Now, the third part is ((a + 2b)(a - 2b)). This looks like a difference of squares, which has the formula ((x + y)(x - y) = x^2 - y^2). Applying that here:[(a + 2b)(a - 2b) = a^2 - (2b)^2 = a^2 - 4b^2]Great, so now I have all three parts expanded:1. (a^2 + 2ab + b^2)2. (-2a^2 + 2ab)3. (a^2 - 4b^2)Now, I need to combine these together:[(a^2 + 2ab + b^2) + (-2a^2 + 2ab) + (a^2 - 4b^2)]Let me remove the parentheses:[a^2 + 2ab + b^2 - 2a^2 + 2ab + a^2 - 4b^2]Now, I'll combine like terms. Let's group the (a^2) terms, the (ab) terms, and the (b^2) terms separately.For the (a^2) terms:[a^2 - 2a^2 + a^2 = (1 - 2 + 1)a^2 = 0a^2]Hmm, that cancels out to zero.For the (ab) terms:[2ab + 2ab = 4ab]For the (b^2) terms:[b^2 - 4b^2 = -3b^2]So, putting it all together, the simplified expression is:[4ab - 3b^2]Okay, that's much simpler! Now, I need to evaluate this simplified expression with (a = -1) and (b = 4).Let's substitute the values into (4ab - 3b^2):First, calculate (4ab):[4 times (-1) times 4 = 4 times (-4) = -16]Next, calculate (3b^2):[3 times (4)^2 = 3 times 16 = 48]Now, subtract the second result from the first:[-16 - 48 = -64]So, after simplifying the expression and substituting the given values, the final result is (-64).Wait, let me just go through that again to make sure I didn't make any mistakes. Expanding each part, combining like terms, and substituting the values step by step seems correct. The (a^2) terms canceled out, which is interesting but seems right. Then, combining the (ab) and (b^2) terms gave me (4ab - 3b^2), which when evaluated with (a = -1) and (b = 4) gives (-16 - 48 = -64). Yeah, that all checks out.I think I'm confident with this answer.

💡Okay, so I have this problem about a sine function, and I need to find its analytical expression and some other properties. Let me try to break it down step by step.First, the function is given as ( f(x) = Asin(wx + varphi) ) with some conditions: ( A > 0 ), ( w > 0 ), and ( |varphi| < frac{pi}{2} ). I also know that one of its highest points is at ( left( frac{pi}{6}, 2 right) ) and an adjacent lowest point is at ( left( frac{2pi}{3}, -2 right) ).Starting with part (1), I need to find the analytical expression of ( f(x) ). Let me recall that for a sine function ( Asin(wx + varphi) ), the amplitude is ( A ), the period is ( frac{2pi}{w} ), the phase shift is ( -frac{varphi}{w} ), and the vertical shift is 0 since there's no constant term added.Given that the highest point is at ( ( frac{pi}{6}, 2 ) ) and the lowest is at ( ( frac{2pi}{3}, -2 ) ), I can figure out the amplitude first. The amplitude ( A ) is the distance from the midline to the highest or lowest point. Since the highest point is 2 and the lowest is -2, the amplitude should be the average of these two extremes. Wait, actually, no. The amplitude is just the maximum value of the function, which is 2 because the sine function oscillates between -A and A. So, ( A = 2 ).Next, I need to find the angular frequency ( w ). To find ( w ), I can use the period. The distance between a highest point and an adjacent lowest point is half the period because from peak to trough is half a cycle. So, let's calculate the distance between ( frac{pi}{6} ) and ( frac{2pi}{3} ).Calculating the difference: ( frac{2pi}{3} - frac{pi}{6} = frac{4pi}{6} - frac{pi}{6} = frac{3pi}{6} = frac{pi}{2} ). So, half the period is ( frac{pi}{2} ), which means the full period ( T ) is ( pi ).Since the period ( T = frac{2pi}{w} ), we can solve for ( w ):( frac{2pi}{w} = pi )Dividing both sides by ( pi ):( frac{2}{w} = 1 )So, ( w = 2 ).Now, the function is ( f(x) = 2sin(2x + varphi) ). I need to find the phase shift ( varphi ). To do this, I can use one of the given points. Let's use the highest point ( left( frac{pi}{6}, 2 right) ).At this point, the sine function reaches its maximum value of 1, so:( 2sinleft(2 cdot frac{pi}{6} + varphiright) = 2 )Dividing both sides by 2:( sinleft(frac{pi}{3} + varphiright) = 1 )The sine function equals 1 at ( frac{pi}{2} + 2kpi ) where ( k ) is an integer. So,( frac{pi}{3} + varphi = frac{pi}{2} + 2kpi )Solving for ( varphi ):( varphi = frac{pi}{2} - frac{pi}{3} + 2kpi = frac{3pi}{6} - frac{2pi}{6} + 2kpi = frac{pi}{6} + 2kpi )Given that ( |varphi| < frac{pi}{2} ), the only value that satisfies this is when ( k = 0 ), so ( varphi = frac{pi}{6} ).Therefore, the function is ( f(x) = 2sinleft(2x + frac{pi}{6}right) ).Moving on to part (2), I need to find the intervals where the function is monotonically increasing and the equation of its axis of symmetry.First, let's find the intervals of monotonic increase. The sine function increases when its derivative is positive. Let's compute the derivative:( f'(x) = 2 cdot 2cosleft(2x + frac{pi}{6}right) = 4cosleft(2x + frac{pi}{6}right) )The function is increasing when ( f'(x) > 0 ), so:( 4cosleft(2x + frac{pi}{6}right) > 0 )Dividing both sides by 4:( cosleft(2x + frac{pi}{6}right) > 0 )The cosine function is positive in the intervals ( (-frac{pi}{2} + 2kpi, frac{pi}{2} + 2kpi) ) for any integer ( k ). So, we set up the inequality:( -frac{pi}{2} + 2kpi < 2x + frac{pi}{6} < frac{pi}{2} + 2kpi )Subtract ( frac{pi}{6} ) from all parts:( -frac{pi}{2} - frac{pi}{6} + 2kpi < 2x < frac{pi}{2} - frac{pi}{6} + 2kpi )Simplify the terms:( -frac{2pi}{3} + 2kpi < 2x < frac{pi}{3} + 2kpi )Divide all parts by 2:( -frac{pi}{3} + kpi < x < frac{pi}{6} + kpi )So, the function is increasing on the intervals ( left( kpi - frac{pi}{3}, kpi + frac{pi}{6} right) ) for any integer ( k ).Now, for the axis of symmetry. The axis of symmetry for a sine function occurs at the midpoints between its peaks and troughs, which are the points where the function reaches its maximum and minimum. Alternatively, it can be found by solving for the vertical lines where the function has symmetry.Another approach is to realize that the axis of symmetry occurs at the points where the function reaches its midline, which is the average of the maximum and minimum values. Since the function oscillates between 2 and -2, the midline is 0. However, the axis of symmetry in terms of x-values occurs at the points where the function crosses the midline going upwards or downwards.But more accurately, the axis of symmetry for a sinusoidal function is the vertical line passing through the maximum or minimum points. Wait, actually, for a sine function, the axis of symmetry would be the vertical lines where the function has reflection symmetry. For the standard sine function, these occur at the midpoints between peaks and troughs.Alternatively, considering the function ( f(x) = 2sin(2x + frac{pi}{6}) ), its graph is symmetric about the vertical lines passing through its peaks and troughs. So, the axis of symmetry can be found by solving for when the argument of the sine function is equal to ( frac{pi}{2} + kpi ), which are the points where the sine function reaches its maximum or minimum.So, set ( 2x + frac{pi}{6} = frac{pi}{2} + kpi ):( 2x = frac{pi}{2} - frac{pi}{6} + kpi = frac{3pi}{6} - frac{pi}{6} + kpi = frac{2pi}{6} + kpi = frac{pi}{3} + kpi )Divide by 2:( x = frac{pi}{6} + frac{kpi}{2} )So, the equation of the axis of symmetry is ( x = frac{pi}{6} + frac{kpi}{2} ) for any integer ( k ).Wait, let me verify that. If I plug in ( k = 0 ), I get ( x = frac{pi}{6} ), which is the x-coordinate of the highest point. Similarly, for ( k = 1 ), ( x = frac{pi}{6} + frac{pi}{2} = frac{2pi}{3} ), which is the x-coordinate of the lowest point. So, these are indeed the points where the function reaches its maximum and minimum, and the axis of symmetry would be these vertical lines.But actually, the axis of symmetry should be the vertical lines equidistant between the peaks and troughs. Wait, no, for a sine function, the axis of symmetry is not necessarily between the peaks and troughs but rather at the points where the function is symmetric. Since the sine function is symmetric about its peaks and troughs, the vertical lines through these points are axes of symmetry.So, yes, the axes of symmetry are at ( x = frac{pi}{6} + frac{kpi}{2} ) for any integer ( k ).Let me just recap:1. Found the amplitude ( A = 2 ) from the maximum value.2. Calculated the period from the distance between a peak and a trough, which gave ( w = 2 ).3. Determined the phase shift ( varphi = frac{pi}{6} ) using the highest point.4. For the intervals of increase, found where the derivative is positive, leading to intervals ( left( kpi - frac{pi}{3}, kpi + frac{pi}{6} right) ).5. For the axis of symmetry, solved for when the function reaches its maxima and minima, resulting in ( x = frac{pi}{6} + frac{kpi}{2} ).I think that covers everything. Let me just make sure I didn't make any calculation mistakes.Checking the period calculation:Distance between ( frac{pi}{6} ) and ( frac{2pi}{3} ) is ( frac{pi}{2} ), which is half the period, so full period is ( pi ). Then ( w = frac{2pi}{T} = frac{2pi}{pi} = 2 ). Correct.For the phase shift, using ( x = frac{pi}{6} ):( 2sinleft(2 cdot frac{pi}{6} + varphiright) = 2 )So, ( sinleft(frac{pi}{3} + varphiright) = 1 ), which gives ( frac{pi}{3} + varphi = frac{pi}{2} + 2kpi ), so ( varphi = frac{pi}{6} + 2kpi ). Since ( |varphi| < frac{pi}{2} ), ( k = 0 ), so ( varphi = frac{pi}{6} ). Correct.For the derivative:( f'(x) = 4cos(2x + frac{pi}{6}) ). Setting this greater than 0 gives ( cos(2x + frac{pi}{6}) > 0 ). The solution is ( 2x + frac{pi}{6} ) in ( (-frac{pi}{2} + 2kpi, frac{pi}{2} + 2kpi) ), leading to ( x ) in ( (kpi - frac{pi}{3}, kpi + frac{pi}{6}) ). Correct.For the axis of symmetry:Setting ( 2x + frac{pi}{6} = frac{pi}{2} + kpi ), solving for ( x ) gives ( x = frac{pi}{6} + frac{kpi}{2} ). Correct.Everything seems to check out. I think I'm confident with these answers.**Final Answer**(1) The analytical expression of the function is (boxed{y = 2sinleft(2x + frac{pi}{6}right)}).(2) The intervals of monotonic increase are (boxed{left[kpi - frac{pi}{3}, kpi + frac{pi}{6}right]}) for any integer (k), and the equation of the axis of symmetry is (boxed{x = frac{kpi}{2} + frac{pi}{6}}) for any integer (k).

💡Okay, so I have this problem: I need to find the coefficient of the x term in the expansion of (x² - 3x + 3)³. Hmm, that sounds a bit tricky, but let me see if I can figure it out step by step.First, I remember that when you have an expression raised to a power like this, you can use the binomial theorem or maybe even the multinomial theorem to expand it. But since there are three terms here (x², -3x, and 3), the binomial theorem might not be directly applicable. Maybe the multinomial theorem is the way to go.Wait, actually, maybe I can think of it as a binomial with two parts: (x² - 3x) and 3. So, it's like (A + B)³ where A is (x² - 3x) and B is 3. That might make it easier to handle. Let me try that approach.So, using the binomial theorem, (A + B)³ = A³ + 3A²B + 3AB² + B³. That should expand correctly. So, substituting A and B:(A + B)³ = (x² - 3x)³ + 3(x² - 3x)² * 3 + 3(x² - 3x) * 3² + 3³.Now, let me compute each term separately.First term: (x² - 3x)³. Hmm, that's another binomial expansion. Let me expand that.Using the binomial theorem again, (x² - 3x)³ = (x²)³ + 3(x²)²(-3x) + 3(x²)(-3x)² + (-3x)³.Calculating each part:(x²)³ = x⁶.3(x²)²(-3x) = 3x⁴ * (-3x) = -9x⁵.3(x²)(-3x)² = 3x² * 9x² = 27x⁴.(-3x)³ = -27x³.So, putting it all together, (x² - 3x)³ = x⁶ - 9x⁵ + 27x⁴ - 27x³.Okay, that's the first term.Second term: 3(x² - 3x)² * 3. Let me compute (x² - 3x)² first.(x² - 3x)² = (x²)² + 2(x²)(-3x) + (-3x)² = x⁴ - 6x³ + 9x².Now, multiply this by 3 and then by 3 again? Wait, no, the term is 3*(x² - 3x)²*3, which is 3*3*(x² - 3x)² = 9*(x² - 3x)².So, 9*(x⁴ - 6x³ + 9x²) = 9x⁴ - 54x³ + 81x².Third term: 3(x² - 3x)*3². Let's compute that.3² is 9, so it's 3*9*(x² - 3x) = 27*(x² - 3x) = 27x² - 81x.Fourth term: 3³ = 27.So, now, putting all four terms together:First term: x⁶ - 9x⁵ + 27x⁴ - 27x³Second term: + 9x⁴ - 54x³ + 81x²Third term: + 27x² - 81xFourth term: + 27Now, let's combine like terms.Starting from the highest power:x⁶: 1x⁶x⁵: -9x⁵x⁴: 27x⁴ + 9x⁴ = 36x⁴x³: -27x³ -54x³ = -81x³x²: 81x² + 27x² = 108x²x: -81xConstants: +27So, putting it all together, the expansion is:x⁶ - 9x⁵ + 36x⁴ - 81x³ + 108x² - 81x + 27.Now, the question is asking for the coefficient of the x term. Looking at the expansion, the x term is -81x, so the coefficient is -81.Wait, let me double-check to make sure I didn't make any mistakes in my calculations.First, when I expanded (x² - 3x)³, I got x⁶ - 9x⁵ + 27x⁴ - 27x³. Let me verify that:(x² - 3x)³ = (x²)³ + 3(x²)²(-3x) + 3(x²)(-3x)² + (-3x)³= x⁶ + 3x⁴*(-3x) + 3x²*(9x²) + (-27x³)= x⁶ - 9x⁵ + 27x⁴ - 27x³. Yep, that looks correct.Next, (x² - 3x)² = x⁴ - 6x³ + 9x². Correct.Then, multiplying by 9: 9x⁴ - 54x³ + 81x². Correct.Third term: 27x² - 81x. Correct.Fourth term: 27. Correct.Combining all terms:x⁶ -9x⁵ + (27x⁴ +9x⁴)=36x⁴; (-27x³ -54x³)= -81x³; (81x² +27x²)=108x²; -81x; +27.Yes, that seems right. So, the coefficient of x is indeed -81.I think that's the correct answer. I don't see any mistakes in my steps. So, I'm confident that the coefficient is -81.**Final Answer**The coefficient of the ( x ) term is boxed{-81}.