answer：Okay, so I need to convert the cylindrical coordinates (10, π/6, 2) to rectangular coordinates. Hmm, I remember that cylindrical coordinates have three components: r, θ, and z. Rectangular coordinates are x, y, z. So, I think the z-coordinate stays the same, right? That part is straightforward.Now, for the x and y coordinates, I think there are some trigonometric functions involved. Let me recall the formulas. I believe x is equal to r multiplied by the cosine of θ, and y is equal to r multiplied by the sine of θ. So, x = r cos θ and y = r sin θ. That makes sense because in polar coordinates, which are similar, you use cosine for the x-component and sine for the y-component.Alright, so let's plug in the values. The r value is 10, θ is π/6, and z is 2. So, for x, I need to calculate 10 times the cosine of π/6. What's the cosine of π/6? I think π/6 is 30 degrees, right? And the cosine of 30 degrees is √3/2. So, x should be 10 times √3/2. Let me compute that: 10 times √3 is 10√3, and dividing by 2 gives me 5√3. Okay, so x is 5√3.Now, for y, it's 10 times the sine of π/6. What's the sine of π/6? That's 30 degrees as well, and the sine of 30 degrees is 1/2. So, y is 10 times 1/2, which is 5. Got it, so y is 5.And as I thought earlier, the z-coordinate remains the same, so z is 2.Let me just double-check my calculations to make sure I didn't make any mistakes. Cosine of π/6 is indeed √3/2, and multiplying that by 10 gives 5√3. Sine of π/6 is 1/2, and multiplying that by 10 gives 5. Yep, that all checks out.So, putting it all together, the rectangular coordinates should be (5√3, 5, 2). I think that's correct. I don't see any errors in my reasoning or calculations. It all follows from the standard conversion formulas between cylindrical and rectangular coordinates.Just to visualize it a bit, in cylindrical coordinates, r is the radial distance from the z-axis, θ is the angle made with the positive x-axis, and z is the height. So, with r = 10, θ = π/6, and z = 2, the point is 10 units away from the z-axis, at a 30-degree angle from the x-axis, and 2 units up along the z-axis. Converting that to rectangular coordinates gives me the specific x and y positions in the plane, which are 5√3 and 5, respectively.I think I've covered all the steps and verified each part. I don't see any reason to doubt this result. It seems solid.

answer：Okay, so I have this sequence problem here. It defines a sequence {b_n} where b_1 is 3, and each subsequent term is given by b_{n+1} = b_n + 3n + 1 for n ≥ 1. I need to find b_{50}. Hmm, let me think about how to approach this.First, let me write down what I know. The first term is b_1 = 3. Then, to get the next term, I add 3n + 1, where n is the current term's index. So, for example, b_2 would be b_1 + 3*1 + 1, which is 3 + 3 + 1 = 7. Similarly, b_3 would be b_2 + 3*2 + 1 = 7 + 6 + 1 = 14, and so on.Wait, so each time, the amount I'm adding increases by 3 each step? Let me check: from b_1 to b_2, I add 4 (which is 3*1 + 1), from b_2 to b_3, I add 7 (which is 3*2 + 1), from b_3 to b_4, I add 10 (which is 3*3 + 1), and so on. So the difference between consecutive terms is increasing by 3 each time. That seems like an arithmetic sequence for the differences.So, if I think about it, the sequence {b_n} is built by adding terms of an arithmetic sequence. The differences themselves form an arithmetic sequence with the first term 4 and a common difference of 3. That might help me find a formula for b_n.I remember that for sequences where the differences form an arithmetic sequence, the original sequence is a quadratic sequence. That is, the nth term can be expressed as a quadratic function of n: b_n = an^2 + bn + c. So, maybe I can find such a formula.Let me try to find the coefficients a, b, and c. I know the first few terms:- b_1 = 3- b_2 = 7- b_3 = 14- b_4 = 24Wait, let me calculate b_4 to make sure. b_4 = b_3 + 3*3 + 1 = 14 + 9 + 1 = 24. Okay, that seems right.So, plugging these into the quadratic formula:For n = 1: a(1)^2 + b(1) + c = a + b + c = 3For n = 2: a(2)^2 + b(2) + c = 4a + 2b + c = 7For n = 3: a(3)^2 + b(3) + c = 9a + 3b + c = 14Now, I have a system of three equations:1) a + b + c = 32) 4a + 2b + c = 73) 9a + 3b + c = 14I can solve this system step by step. Let me subtract equation 1 from equation 2:(4a + 2b + c) - (a + b + c) = 7 - 3Which simplifies to:3a + b = 4Similarly, subtract equation 2 from equation 3:(9a + 3b + c) - (4a + 2b + c) = 14 - 7Which simplifies to:5a + b = 7Now, I have two equations:4) 3a + b = 45) 5a + b = 7Subtract equation 4 from equation 5:(5a + b) - (3a + b) = 7 - 4Which gives:2a = 3So, a = 3/2 or 1.5Now, plug a back into equation 4:3*(1.5) + b = 44.5 + b = 4So, b = 4 - 4.5 = -0.5Now, plug a and b back into equation 1:1.5 + (-0.5) + c = 31.5 - 0.5 + c = 31 + c = 3So, c = 2Therefore, the formula for b_n is:b_n = 1.5n^2 - 0.5n + 2Let me check this formula with the known terms to make sure.For n = 1: 1.5*(1)^2 - 0.5*(1) + 2 = 1.5 - 0.5 + 2 = 3. Correct.For n = 2: 1.5*(4) - 0.5*(2) + 2 = 6 - 1 + 2 = 7. Correct.For n = 3: 1.5*(9) - 0.5*(3) + 2 = 13.5 - 1.5 + 2 = 14. Correct.For n = 4: 1.5*(16) - 0.5*(4) + 2 = 24 - 2 + 2 = 24. Correct.Okay, so the formula seems to work for the first few terms. Good.Now, I need to find b_{50}. Let's plug n = 50 into the formula:b_{50} = 1.5*(50)^2 - 0.5*(50) + 2First, calculate 50 squared: 50*50 = 2500Then, 1.5*2500 = 3750Next, 0.5*50 = 25So, putting it all together:b_{50} = 3750 - 25 + 2 = 3750 - 25 is 3725, plus 2 is 3727.So, b_{50} is 3727.Wait, let me double-check my calculations to be sure.50 squared is 2500, correct.1.5 times 2500: 1 times 2500 is 2500, 0.5 times 2500 is 1250, so total is 2500 + 1250 = 3750. Correct.0.5 times 50 is 25. Correct.So, 3750 - 25 is 3725, plus 2 is 3727. Yep, that seems right.Alternatively, I could think of this as a recursive sequence and try to find the sum of the differences up to n=50.Since each term is built by adding 3n + 1, starting from n=1 up to n=49 (since b_{50} = b_{49} + 3*49 + 1). So, the total addition from b_1 to b_{50} would be the sum from k=1 to k=49 of (3k + 1).Let me compute that sum:Sum = Σ_{k=1}^{49} (3k + 1) = 3*Σk + Σ1 = 3*(49*50)/2 + 49*1Compute Σk from 1 to 49: (49*50)/2 = 1225So, 3*1225 = 3675Σ1 from 1 to 49 is 49So, total sum is 3675 + 49 = 3724Then, b_{50} = b_1 + 3724 = 3 + 3724 = 3727Wait, that's the same answer as before. So, that confirms it.Alternatively, I could have used the formula for the sum of an arithmetic series since the differences form an arithmetic sequence.The differences are 4, 7, 10, ..., up to the 49th term.First term, a_1 = 4, common difference d = 3, number of terms N = 49.Sum = N/2 * [2a_1 + (N - 1)d] = 49/2 * [8 + 48*3] = 49/2 * [8 + 144] = 49/2 * 152Compute 49/2 * 152: 49 * 76 = let's compute 50*76 = 3800, minus 1*76 = 76, so 3800 - 76 = 3724Then, b_{50} = b_1 + 3724 = 3 + 3724 = 3727Same result again. So, I'm confident that 3727 is the correct answer.Therefore, the answer is C) 3727.

answer：First, I need to determine the total volume of the wheat seed batch, which is 2 liters. Since 1 liter is equal to 1000 milliliters, the total volume in milliliters is:2 liters × 1000 milliliters/liter = 2000 millilitersNext, I need to find out what fraction of the total volume is being sampled. The sample size is 10 milliliters. Therefore, the probability of selecting the infected seed is the ratio of the sample volume to the total volume:Probability = Sample Volume / Total Volume = 10 milliliters / 2000 milliliters = 1/200Thus, the probability of including the seed with powdery mildew in the sample is 1/200.

answer：Okay, so I have this geometry problem here. Let me try to understand it step by step. The problem is about triangle ABC where AB is 52 units, BC is 34 units, and CA is 50 units. We need to split BC into n equal segments by placing n-1 new points. Among these points, there should be the feet of the altitude, median, and angle bisector from point A. The question is asking for the smallest possible value of n.Alright, so first, let me visualize triangle ABC. Let me note down the sides: AB = 52, BC = 34, and CA = 50. So, triangle ABC has sides of lengths 52, 34, and 50. I need to find the feet of the altitude, median, and angle bisector from A onto BC. These three feet should lie on the n equal segments of BC. So, BC is divided into n equal parts, and each segment has length 34/n. The points where the altitude, median, and angle bisector meet BC should coincide with these division points.So, to find the minimal n, I need to find the least common multiple or something related to the positions of these feet along BC. Maybe I should find the exact positions of these feet and then see what n would make all these positions fall on the division points.Let me start by finding the midpoint of BC because that's the foot of the median from A. Since BC is 34, the midpoint M divides BC into two equal parts, each of length 17. So, BM = MC = 17.Next, I need to find the foot of the altitude from A to BC. Let's call this point D. To find D, I might need to use some coordinate geometry or maybe the Pythagorean theorem. Alternatively, I can use the formula for the foot of the altitude in terms of the sides of the triangle.Wait, maybe I can use the formula for the length of the altitude. The area of triangle ABC can be found using Heron's formula, and then I can find the altitude from A.Let me compute the area first. The semi-perimeter (s) of triangle ABC is (52 + 34 + 50)/2 = (136)/2 = 68.Using Heron's formula, the area is sqrt[s(s - AB)(s - BC)(s - CA)] = sqrt[68*(68 - 52)*(68 - 34)*(68 - 50)].Calculating each term:68 - 52 = 1668 - 34 = 3468 - 50 = 18So, area = sqrt[68 * 16 * 34 * 18]Let me compute this step by step.First, 68 * 16 = 1088Then, 34 * 18 = 612So, area = sqrt[1088 * 612]Hmm, that's a big number. Let me see if I can factor this to simplify.Alternatively, maybe I can compute it numerically.Wait, 1088 * 612. Let me compute 1088 * 600 = 652,800 and 1088 * 12 = 13,056. So total is 652,800 + 13,056 = 665,856.So, area = sqrt[665,856]. Let me compute this square root.I know that 816^2 = 665,856 because 800^2 = 640,000 and 16^2 = 256, and cross terms would give 2*800*16 = 25,600, so (800 + 16)^2 = 800^2 + 2*800*16 + 16^2 = 640,000 + 25,600 + 256 = 665,856.So, the area is 816.Now, the area can also be expressed as (1/2)*BC*AD, where AD is the altitude from A to BC.So, 816 = (1/2)*34*ADTherefore, 816 = 17*ADSo, AD = 816 / 17 = 48.So, the altitude from A is 48 units long.Now, to find the foot D of the altitude on BC, I can use the Pythagorean theorem in triangle ADC.In triangle ADC, AC = 50, AD = 48, so DC = sqrt(AC^2 - AD^2) = sqrt(50^2 - 48^2) = sqrt(2500 - 2304) = sqrt(196) = 14.So, DC = 14, which means BD = BC - DC = 34 - 14 = 20.So, the foot of the altitude D is 20 units from B and 14 units from C.Alright, so now I have the median at 17 units from B, the altitude at 20 units from B, and the angle bisector somewhere on BC.Wait, I need to find the foot of the angle bisector from A onto BC. Let me denote this point as E.To find E, I can use the Angle Bisector Theorem, which states that the angle bisector divides the opposite side in the ratio of the adjacent sides.So, in triangle ABC, the angle bisector from A will divide BC into segments proportional to AB and AC.So, BE / EC = AB / AC = 52 / 50 = 26 / 25.So, BE = (26/51)*BC and EC = (25/51)*BC.Since BC is 34, BE = (26/51)*34 = (26*34)/51.Simplify 34/51 = 2/3, so BE = 26*(2/3) = 52/3 ≈ 17.333...Similarly, EC = 25/51*34 = (25*34)/51 = (25*2)/3 = 50/3 ≈ 16.666...So, the foot of the angle bisector E is 52/3 units from B and 50/3 units from C.So, summarizing:- Median from A meets BC at M, which is 17 units from B.- Altitude from A meets BC at D, which is 20 units from B.- Angle bisector from A meets BC at E, which is 52/3 ≈ 17.333 units from B.So, now, I need to place n-1 points on BC, dividing it into n equal segments, such that the points M, D, and E are among these division points.So, the positions of M, D, and E on BC are at 17, 20, and 52/3 units from B, respectively.So, BC is 34 units long, so each segment will be 34/n units long.We need 17, 20, and 52/3 to be integer multiples of 34/n.In other words, 17 = k*(34/n), 20 = m*(34/n), and 52/3 = p*(34/n), where k, m, p are integers.Simplifying each equation:17 = k*(34/n) => k = (17n)/34 = n/220 = m*(34/n) => m = (20n)/34 = (10n)/1752/3 = p*(34/n) => p = (52/3)*(n/34) = (52n)/(3*34) = (26n)/(51) = (26n)/51So, for k, m, p to be integers, n must be such that:n/2 is integer => n must be even.(10n)/17 is integer => n must be a multiple of 17.(26n)/51 is integer => 26n must be divisible by 51. Since 26 and 51 are coprime (their GCD is 13 and 51 is 3*17, so no common factors), n must be a multiple of 51.Wait, 26 and 51: 26 is 2*13, 51 is 3*17, so yes, they are coprime. So, n must be a multiple of 51 for (26n)/51 to be integer.So, n must satisfy:- n is even,- n is a multiple of 17,- n is a multiple of 51.So, the least common multiple (LCM) of 2, 17, and 51.But 51 is already a multiple of 17 (51 = 3*17). So, LCM of 2 and 51 is 102.Therefore, the smallest n is 102.Wait, let me verify this.If n = 102, then each segment is 34/102 = 1/3 units.So, the points are at 1/3, 2/3, 3/3, ..., 102/3 = 34 units.So, 17 units is 51*(1/3), which is the 51st point.20 units is 60*(1/3), which is the 60th point.52/3 units is approximately 17.333, which is 52/3 = 17.333... So, 52/3 divided by 1/3 is 52, so the 52nd point.So, yes, all three points M, D, E fall on the division points when n = 102.Is there a smaller n? Let's see.We concluded that n must be a multiple of 102. But let me think again.Wait, n must be a multiple of 2, 17, and 51. Since 51 is 3*17, and 2 is prime, the LCM is 2*3*17 = 102.So, 102 is indeed the minimal n.Therefore, the smallest possible value of n is 102.

answer：Okay, so I have this problem where point A is on the z-axis, and point B is at (2, 1, -3). The distance between A and B is given as 3 units. I need to find the coordinates of point A.First, since point A is on the z-axis, its x and y coordinates must be zero. So, point A can be represented as (0, 0, z), where z is some value I need to find.Now, the distance between two points in three-dimensional space can be calculated using the distance formula. The formula is:Distance = sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2]In this case, point A is (0, 0, z) and point B is (2, 1, -3). Plugging these into the distance formula, I get:Distance = sqrt[(2 - 0)^2 + (1 - 0)^2 + (-3 - z)^2]Simplifying inside the square root:= sqrt[(2)^2 + (1)^2 + (-3 - z)^2]= sqrt[4 + 1 + ( -3 - z)^2]= sqrt[5 + ( -3 - z)^2]We know this distance is equal to 3, so:sqrt[5 + (-3 - z)^2] = 3To solve for z, I'll square both sides to eliminate the square root:[sqrt[5 + (-3 - z)^2]]^2 = 3^25 + (-3 - z)^2 = 9Now, subtract 5 from both sides:(-3 - z)^2 = 4Taking the square root of both sides:-3 - z = ±2This gives two equations:1) -3 - z = 22) -3 - z = -2Solving the first equation:-3 - z = 2Subtract -3 from both sides:-z = 2 + 3-z = 5Multiply both sides by -1:z = -5Solving the second equation:-3 - z = -2Subtract -3 from both sides:-z = -2 + 3-z = 1Multiply both sides by -1:z = -1So, the possible z-values for point A are -5 and -1. Therefore, the coordinates of point A are either (0, 0, -5) or (0, 0, -1).I should double-check my calculations to make sure I didn't make any mistakes. Let's verify both solutions.First, for z = -5:Distance = sqrt[(2 - 0)^2 + (1 - 0)^2 + (-3 - (-5))^2]= sqrt[4 + 1 + (2)^2]= sqrt[4 + 1 + 4]= sqrt[9]= 3That's correct.Now, for z = -1:Distance = sqrt[(2 - 0)^2 + (1 - 0)^2 + (-3 - (-1))^2]= sqrt[4 + 1 + (-2)^2]= sqrt[4 + 1 + 4]= sqrt[9]= 3That also checks out. So both solutions are valid.I think I've considered all possible cases and my calculations seem correct. Therefore, the coordinates of point A are (0, 0, -5) and (0, 0, -1).

answer：Okay, so I have this problem here where I need to find the maximum value of ( xy ) given that ( x > 0 ), ( y > 0 ), and ( 2x + 3y = 6 ). Hmm, let me think about how to approach this.First, I remember that when dealing with optimization problems like finding maximums or minimums, calculus is often useful. Maybe I can express ( y ) in terms of ( x ) using the equation ( 2x + 3y = 6 ) and then substitute that into ( xy ) to make it a function of a single variable. That way, I can take the derivative and find the maximum.Let me try that. Starting with the equation ( 2x + 3y = 6 ), I can solve for ( y ):( 3y = 6 - 2x )Divide both sides by 3:( y = frac{6 - 2x}{3} )Simplify that:( y = 2 - frac{2}{3}x )Okay, so now I can substitute this expression for ( y ) into ( xy ):( xy = x left(2 - frac{2}{3}xright) )Let me expand that:( xy = 2x - frac{2}{3}x^2 )So, now I have ( xy ) as a function of ( x ): ( f(x) = 2x - frac{2}{3}x^2 ). To find the maximum value, I need to find the critical points by taking the derivative of ( f(x) ) with respect to ( x ) and setting it equal to zero.Calculating the derivative:( f'(x) = frac{d}{dx}left(2x - frac{2}{3}x^2right) = 2 - frac{4}{3}x )Set the derivative equal to zero to find critical points:( 2 - frac{4}{3}x = 0 )Solving for ( x ):( frac{4}{3}x = 2 )Multiply both sides by ( frac{3}{4} ):( x = 2 times frac{3}{4} = frac{6}{4} = frac{3}{2} )So, ( x = frac{3}{2} ). Now, I need to find the corresponding ( y ) value using the equation ( y = 2 - frac{2}{3}x ):( y = 2 - frac{2}{3} times frac{3}{2} )Simplify that:( y = 2 - 1 = 1 )So, ( y = 1 ). Therefore, the critical point is at ( x = frac{3}{2} ) and ( y = 1 ).Now, I should check whether this critical point is indeed a maximum. Since the function ( f(x) = 2x - frac{2}{3}x^2 ) is a quadratic function opening downwards (because the coefficient of ( x^2 ) is negative), the critical point we found must be a maximum.Alternatively, I can use the second derivative test. Let's compute the second derivative:( f''(x) = frac{d}{dx}left(2 - frac{4}{3}xright) = -frac{4}{3} )Since ( f''(x) = -frac{4}{3} < 0 ), the function is concave down at this point, confirming that it's a maximum.Therefore, the maximum value of ( xy ) occurs at ( x = frac{3}{2} ) and ( y = 1 ). Plugging these back into ( xy ):( xy = frac{3}{2} times 1 = frac{3}{2} )So, the maximum value of ( xy ) is ( frac{3}{2} ).Wait, let me double-check my steps to make sure I didn't make any errors. Starting from the beginning, I solved for ( y ) correctly, substituted into ( xy ), took the derivative properly, solved for ( x ), found ( y ), and confirmed it's a maximum. Everything seems to check out.Alternatively, I remember that sometimes using the AM-GM inequality can help in such optimization problems, especially when dealing with products and sums. Let me try that approach as a verification.The AM-GM inequality states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. In this case, we have ( 2x + 3y = 6 ). Let me consider the terms ( 2x ) and ( 3y ). Applying AM-GM:( frac{2x + 3y}{2} geq sqrt{2x times 3y} )Plugging in ( 2x + 3y = 6 ):( frac{6}{2} geq sqrt{6xy} )Simplify:( 3 geq sqrt{6xy} )Square both sides:( 9 geq 6xy )Divide both sides by 6:( frac{9}{6} geq xy )Simplify:( frac{3}{2} geq xy )So, the maximum value of ( xy ) is ( frac{3}{2} ), which matches the result I got using calculus. This gives me more confidence that the answer is correct.Equality in AM-GM holds when all the terms are equal. So, in this case, ( 2x = 3y ). Let's solve for ( x ) and ( y ):From ( 2x = 3y ), we get ( y = frac{2}{3}x ). Plugging this into the original equation ( 2x + 3y = 6 ):( 2x + 3 times frac{2}{3}x = 6 )Simplify:( 2x + 2x = 6 )( 4x = 6 )( x = frac{6}{4} = frac{3}{2} )Then, ( y = frac{2}{3} times frac{3}{2} = 1 )So again, we get ( x = frac{3}{2} ) and ( y = 1 ), confirming the maximum occurs at these values.Just to make sure, let me plug these values back into the original equation to verify:( 2x + 3y = 2 times frac{3}{2} + 3 times 1 = 3 + 3 = 6 )Yes, that's correct. So, everything checks out.Another way to think about this is using substitution or maybe even completing the square, but I think the methods I've used are sufficient and have confirmed the result.In summary, whether I use calculus by expressing ( y ) in terms of ( x ), finding the derivative, and determining the critical point, or use the AM-GM inequality, I arrive at the same conclusion that the maximum value of ( xy ) is ( frac{3}{2} ) when ( x = frac{3}{2} ) and ( y = 1 ).I don't see any mistakes in my reasoning, and both methods corroborate each other, so I feel confident about the answer.**Final Answer**The maximum value of ( xy ) is boxed{dfrac{3}{2}}.