answer：First, I need to evaluate the inner absolute value expression ( |pi + 7| ). Since both ( pi ) and 7 are positive numbers, their sum is also positive. Therefore, ( |pi + 7| = pi + 7 ).Next, I substitute this result back into the original expression:[3pi - |pi + 7| = 3pi - (pi + 7) = 2pi - 7]Now, I need to evaluate the absolute value of ( 2pi - 7 ). Knowing that ( pi ) is approximately 3.14, ( 2pi ) is approximately 6.28. Since 6.28 is less than 7, ( 2pi - 7 ) is a negative number. The absolute value of a negative number is its positive counterpart, so:[|2pi - 7| = -(2pi - 7) = 7 - 2pi]Therefore, the exact value of the expression is ( 7 - 2pi ).

answer：First, I need to determine how much it rained on Monday based on the information provided about Tuesday's rainfall.I know that it rained 0.2 inches on Tuesday, and this amount was 0.7 inches less than what it rained on Monday.To find Monday's rainfall, I can set up an equation where Monday's rainfall minus 0.7 inches equals Tuesday's rainfall.Let ( M ) represent the rainfall on Monday. Then, the equation becomes:( M - 0.7 = 0.2 )To solve for ( M ), I add 0.7 inches to both sides of the equation:( M = 0.2 + 0.7 )Calculating the sum gives:( M = 0.9 )Therefore, it rained 0.9 inches on Monday.

answer：Alright, so I've got this problem about buying disinfectant for my school. It's divided into three parts, and I need to solve each one step by step. Let me start by understanding what each part is asking.**Part 1: Finding the price per bottle of each brand**Okay, so we have two brands of disinfectant. The first brand has a volume of 200ml per bottle, and the second brand has 500ml per bottle. The problem gives me two different purchase scenarios with their total costs:1. Buying 3 bottles of the first brand and 2 bottles of the second brand costs 80 yuan.2. Buying 1 bottle of the first brand and 4 bottles of the second brand costs 110 yuan.I need to find out the price per bottle for each brand. This sounds like a system of equations problem. Let me denote:- Let x be the price per bottle of the first brand.- Let y be the price per bottle of the second brand.From the first scenario: 3x + 2y = 80From the second scenario: x + 4y = 110So, I have two equations:1. 3x + 2y = 802. x + 4y = 110I need to solve this system of equations to find x and y. I can use either substitution or elimination. Maybe elimination is easier here. If I multiply the second equation by 3, I'll have:3*(x + 4y) = 3*110 => 3x + 12y = 330Now, subtract the first equation from this new equation:(3x + 12y) - (3x + 2y) = 330 - 80Simplify:3x + 12y - 3x - 2y = 25010y = 250So, y = 250 / 10 = 25Now that I have y, I can plug it back into one of the original equations to find x. Let's use the second equation:x + 4y = 110x + 4*25 = 110x + 100 = 110x = 110 - 100 = 10So, the price per bottle of the first brand is 10 yuan, and the second brand is 25 yuan.**Part 2: Purchasing plans for 4000ml of disinfectant**Now, the school needs to purchase a total of 4000ml of both brands. Both brands need to be purchased, so we can't have zero bottles of either. I need to find all possible purchasing plans.Let me denote:- m = number of bottles of the first brand (200ml)- n = number of bottles of the second brand (500ml)The total volume is 4000ml, so:200m + 500n = 4000I can simplify this equation by dividing all terms by 100:2m + 5n = 40Now, I need to find all pairs of positive integers (m, n) that satisfy this equation.Let me solve for m:2m = 40 - 5nm = (40 - 5n)/2Since m must be an integer, (40 - 5n) must be even. 5n is always ending with 0 or 5, so 40 - 5n must be even. Let's see:If n is even, 5n is even, so 40 - even is even. If n is odd, 5n is odd, so 40 - odd is odd, which is not divisible by 2. Therefore, n must be even.So, n can be 0, 2, 4, 6, ... but since both brands need to be purchased, n cannot be 0. Also, n must be such that m is positive.Let's find possible values of n:Start with n=2:m = (40 - 5*2)/2 = (40 -10)/2 = 30/2 =15n=4:m=(40 -5*4)/2=(40-20)/2=20/2=10n=6:m=(40 -5*6)/2=(40-30)/2=10/2=5n=8:m=(40 -5*8)/2=(40-40)/2=0/2=0But m=0 is not allowed since both brands need to be purchased. So n cannot be 8.Therefore, possible values of n are 2,4,6.Corresponding m values are 15,10,5.So, the purchasing plans are:- 15 bottles of first brand and 2 bottles of second brand- 10 bottles of first brand and 4 bottles of second brand- 5 bottles of first brand and 6 bottles of second brand**Part 3: Duration of disinfectant supply**The school spends 2500 yuan on purchasing both brands. There are 1000 students and teachers, each needing to use 10ml per day on average. I need to find how many days this batch can last.First, let's find out how much disinfectant is purchased in total.Let p = number of bottles of first brandq = number of bottles of second brandTotal cost: 10p +25q =2500Total volume: 200p +500q = ?We need to find the total volume, then divide by the daily usage to find the number of days.But first, let's find p and q.We have:10p +25q =2500Simplify this equation by dividing all terms by 5:2p +5q =500Now, we need another equation to solve for p and q. But we only have one equation. However, since we need to find the total volume, maybe we can express total volume in terms of one variable.Let me solve for p:2p =500 -5qp=(500 -5q)/2Now, total volume:200p +500q =200*(500 -5q)/2 +500qSimplify:200*(500 -5q)/2 =100*(500 -5q)=50000 -500qSo total volume=50000 -500q +500q=50000mlWait, that's interesting. The q terms cancel out. So total volume is 50000ml regardless of q.So, total volume is 50000ml.Now, total daily usage:1000 people *10ml=10,000ml per daySo, number of days=50000 /10000=5 daysSo, the disinfectant will last for 5 days.**Final Answer**1. The price per bottle of the first brand is boxed{10} yuan, and the second brand is boxed{25} yuan.2. The possible purchasing plans are: - boxed{15} bottles of the first brand and boxed{2} bottles of the second brand - boxed{10} bottles of the first brand and boxed{4} bottles of the second brand - boxed{5} bottles of the first brand and boxed{6} bottles of the second brand3. The disinfectant will last for boxed{5} days.

answer：First, I need to find the prime factors of 2210. I'll start by dividing 2210 by the smallest prime numbers.2210 divided by 2 is 1105.1105 divided by 5 is 221.221 divided by 13 is 17.So, the prime factors of 2210 are 2, 5, 13, and 17.Next, I'll look for ways to combine these prime factors to form a two-digit number and a three-digit number.Combining 2 and 5 gives me 10, which is a two-digit number. The remaining factors, 13 and 17, multiply to 221, which is a three-digit number.I'll check if there are any other combinations, but it seems that this is the only valid way to factor 2210 into a two-digit and a three-digit number without repeating the same pair in reverse.Therefore, there is only one unique way to factor 2210 as a product of a two-digit number and a three-digit number.

answer：Okay, so I have this problem where there are 15 boys and 20 girls sitting around a round table. The problem says that the number of pairs of boys sitting next to each other is one and a half times less than the number of pairs of girls sitting next to each other. I need to find the number of boy-girl pairs sitting next to each other.First, let me try to understand what the problem is saying. We have a circular table with 15 boys and 20 girls. So, the total number of children around the table is 15 + 20 = 35. Since it's a round table, the arrangement is circular, meaning the first and last positions are also adjacent.Now, the problem mentions pairs of boys sitting next to each other and pairs of girls sitting next to each other. It also says that the number of boy pairs is one and a half times less than the number of girl pairs. I need to translate this into a mathematical relationship.Let me denote:- B as the number of pairs of boys sitting next to each other.- G as the number of pairs of girls sitting next to each other.- M as the number of boy-girl pairs sitting next to each other.Since the table is circular, the total number of adjacent pairs is equal to the number of children, which is 35. So, we have:B + G + M = 35But wait, actually, in a circular arrangement, each child has two neighbors, so the total number of adjacent pairs is equal to the number of children, which is 35. So, each pair is counted once, so yes, B + G + M = 35.Now, the problem states that B is one and a half times less than G. Hmm, "one and a half times less" is a bit ambiguous. It could mean that B is half of G, or it could mean that B is G minus one and a half times something. I think in this context, it probably means that B is half of G because "one and a half times less" is often interpreted as dividing by one and a half, which is the same as multiplying by two-thirds. Wait, no, actually, if something is one and a half times less, it's like reducing it by one and a half times, which would be subtracting one and a half times the original. But that might not make sense here because B and G are counts of pairs.Alternatively, maybe it's better to interpret it as B = G / 1.5, which is the same as B = (2/3)G. So, B is two-thirds of G. That seems more reasonable because it's a proportional relationship.So, let's go with that interpretation:B = (2/3)GNow, we have two equations:1. B + G + M = 352. B = (2/3)GWe can substitute equation 2 into equation 1:(2/3)G + G + M = 35Combining like terms:(2/3)G + G = (2/3 + 3/3)G = (5/3)GSo, (5/3)G + M = 35But we have two variables here, G and M, and only one equation. So, we need another relationship to solve for M.Wait, maybe I can express M in terms of B and G. Since M is the number of boy-girl pairs, and in a circular arrangement, the number of boy-girl pairs is related to the number of groups of boys and girls.Let me think about this. If we have groups of boys and girls sitting alternately around the table, the number of boy-girl pairs would be equal to twice the number of groups (since each group is surrounded by two opposite-gender groups). But wait, in a circular arrangement, the number of groups of boys and girls must be equal because they alternate. So, if there are X groups of boys, there are also X groups of girls.Each group of boys contributes (number of boys in the group - 1) pairs of boys sitting next to each other, and similarly for girls. So, the total number of boy pairs B is the sum over all boy groups of (size of group - 1). Similarly, G is the sum over all girl groups of (size of group - 1).Let me denote:- X as the number of groups of boys (which is equal to the number of groups of girls, since they alternate)- For each boy group, let the size be b_i, so the number of boy pairs in that group is b_i - 1- Similarly, for each girl group, let the size be g_i, so the number of girl pairs in that group is g_i - 1So, total boy pairs:B = sum_{i=1 to X} (b_i - 1) = sum b_i - X = 15 - XSimilarly, total girl pairs:G = sum_{i=1 to X} (g_i - 1) = sum g_i - X = 20 - XSo, we have:B = 15 - XG = 20 - XFrom earlier, we have B = (2/3)GSo, substituting:15 - X = (2/3)(20 - X)Let's solve for X:15 - X = (2/3)(20 - X)Multiply both sides by 3 to eliminate the fraction:3*(15 - X) = 2*(20 - X)45 - 3X = 40 - 2X45 - 40 = 3X - 2X5 = XSo, X = 5That means there are 5 groups of boys and 5 groups of girls.Now, let's find B and G:B = 15 - 5 = 10G = 20 - 5 = 15Now, going back to the total number of pairs:B + G + M = 3510 + 15 + M = 3525 + M = 35M = 10So, the number of boy-girl pairs sitting next to each other is 10.Wait, let me double-check this. If there are 5 groups of boys and 5 groups of girls, each group of boys contributes (b_i - 1) pairs, and similarly for girls. The total boy pairs are 10, and girl pairs are 15. The mixed pairs would be 2*X, since each group of boys is adjacent to two groups of girls, but since it's circular, each mixed pair is counted once. Wait, actually, in a circular arrangement with X groups, the number of mixed pairs is 2X, but since each mixed pair is between a boy group and a girl group, and there are X boy groups and X girl groups, the number of mixed pairs is 2X. But in our case, X=5, so mixed pairs would be 10, which matches our earlier result.Yes, that makes sense. So, the number of boy-girl pairs is 10.

answer：Alright, so I have this triangle problem here, and I need to find sin(C/2). Let me read it again to make sure I understand all the given information.In triangle ABC, the sides opposite angles A, B, and C are a, b, and c respectively. It's given that a + sqrt(2)c = 2b, and sin B = sqrt(2) sin C. I need to find sin(C/2).Hmm, okay. So, let's break this down. First, I remember that in any triangle, the Law of Sines relates the sides and angles: a/sin A = b/sin B = c/sin C. Maybe that can help here since we have some sine relationships.Given that sin B = sqrt(2) sin C, I can use the Law of Sines to relate the sides. Let me write that down:From the Law of Sines, we have:a / sin A = b / sin B = c / sin C.Given sin B = sqrt(2) sin C, so sin B / sin C = sqrt(2). Therefore, b / c = sqrt(2), which means b = sqrt(2) c.Wait, is that right? Because if sin B / sin C = sqrt(2), then b / c = sqrt(2), so b = sqrt(2) c. Yeah, that seems correct.So, b = sqrt(2) c. Got that. Now, the other equation given is a + sqrt(2)c = 2b. Let me substitute b from the previous equation into this.So, a + sqrt(2)c = 2b = 2 * sqrt(2) c.So, a + sqrt(2)c = 2 sqrt(2) c.Let me solve for a. Subtract sqrt(2)c from both sides:a = 2 sqrt(2) c - sqrt(2)c = sqrt(2)c.So, a = sqrt(2)c.Wait, so a = sqrt(2)c and b = sqrt(2)c. That means a = b.So, sides a and b are equal. Therefore, triangle ABC is isoceles with sides a = b, so angles A and B are equal.Wait, is that necessarily true? Because in a triangle, if two sides are equal, then the angles opposite them are equal. So, since a = b, then angles A and B are equal.But wait, hold on. The given information is sin B = sqrt(2) sin C. If angles A and B are equal, then sin A = sin B.But from sin B = sqrt(2) sin C, that would mean sin A = sqrt(2) sin C.But in a triangle, the sum of angles is 180 degrees, so A + B + C = 180. Since A = B, that would be 2A + C = 180, so C = 180 - 2A.Hmm, maybe I can use that to relate the sines.Alternatively, maybe I can use the Law of Cosines since I have relationships between the sides.Given that a = sqrt(2)c and b = sqrt(2)c, so sides a and b are both sqrt(2)c, and side c is c.So, sides are a = sqrt(2)c, b = sqrt(2)c, c = c.So, triangle sides are sqrt(2)c, sqrt(2)c, c. So, it's an isoceles triangle with two sides equal to sqrt(2)c and the base c.Let me try to find angle C. Since angle C is opposite side c, which is the smallest side, so angle C is the smallest angle.Alternatively, maybe I can use the Law of Cosines to find angle C.Law of Cosines says that c^2 = a^2 + b^2 - 2ab cos C.Wait, let me write that down:c^2 = a^2 + b^2 - 2ab cos C.Given that a = sqrt(2)c and b = sqrt(2)c, let's substitute those in:c^2 = (sqrt(2)c)^2 + (sqrt(2)c)^2 - 2*(sqrt(2)c)*(sqrt(2)c)*cos C.Calculating each term:(sqrt(2)c)^2 = 2c^2.So, c^2 = 2c^2 + 2c^2 - 2*(sqrt(2)c)*(sqrt(2)c)*cos C.Simplify:c^2 = 4c^2 - 2*(2c^2)*cos C.Wait, because (sqrt(2)c)*(sqrt(2)c) = 2c^2.So, 2*(sqrt(2)c)*(sqrt(2)c) = 2*(2c^2) = 4c^2.Wait, no, hold on. Let me recast the equation:c^2 = (2c^2) + (2c^2) - 2*(sqrt(2)c)*(sqrt(2)c)*cos C.So, that's c^2 = 4c^2 - 4c^2 cos C.Let me rearrange this:c^2 - 4c^2 = -4c^2 cos C.-3c^2 = -4c^2 cos C.Divide both sides by -c^2 (assuming c ≠ 0, which it isn't in a triangle):-3 = -4 cos C.So, -3 = -4 cos C.Divide both sides by -4:3/4 = cos C.So, cos C = 3/4.Okay, so angle C has cosine 3/4. Therefore, sin C can be found using sin^2 C + cos^2 C = 1.So, sin C = sqrt(1 - (3/4)^2) = sqrt(1 - 9/16) = sqrt(7/16) = sqrt7 / 4.But wait, in the given problem, sin B = sqrt(2) sin C. So, sin B = sqrt(2)*(sqrt7 / 4) = sqrt14 / 4.But from the Law of Sines, sin B / sin C = b / c = sqrt(2). Let me check if that's consistent.sin B / sin C = (sqrt14 / 4) / (sqrt7 / 4) = sqrt14 / sqrt7 = sqrt(2). Yes, that checks out.So, that's consistent. So, angle C has cos C = 3/4, sin C = sqrt7 / 4.Now, I need to find sin(C/2). There's a half-angle identity for sine:sin(C/2) = sqrt[(1 - cos C)/2].So, plugging in cos C = 3/4:sin(C/2) = sqrt[(1 - 3/4)/2] = sqrt[(1/4)/2] = sqrt(1/8) = (sqrt2)/4.Wait, sqrt(1/8) is equal to 1/(2 sqrt2), which rationalizes to sqrt2 / 4. So, sin(C/2) = sqrt2 / 4.So, that's the answer.But let me double-check my steps to make sure I didn't make any mistakes.1. From the given, sin B = sqrt(2) sin C. Using Law of Sines, b = sqrt(2) c.2. Given a + sqrt(2)c = 2b, substitute b = sqrt(2)c, so a + sqrt(2)c = 2 sqrt(2)c, leading to a = sqrt(2)c.3. So, sides a = sqrt(2)c, b = sqrt(2)c, c = c. So, triangle is isoceles with a = b.4. Using Law of Cosines to find angle C: c^2 = a^2 + b^2 - 2ab cos C.5. Substituted a and b, simplified to find cos C = 3/4.6. Then, used half-angle identity to find sin(C/2) = sqrt[(1 - cos C)/2] = sqrt[(1 - 3/4)/2] = sqrt(1/8) = sqrt2 / 4.Everything seems to check out. So, I think that's correct.**Final Answer**The value of sin frac{C}{2} is boxed{dfrac{sqrt{2}}{4}}.