💡Okay, so I need to find the value of x such that it is the mean, median, and mode of the given 9 data values: -10, -5, x, x, 0, 15, 20, 25, 30. Hmm, let me break this down step by step.First, let's recall what mean, median, and mode mean. The mean is the average of all the numbers. The median is the middle number when the data is ordered from least to greatest. The mode is the number that appears most frequently.Given that x is the mode, that means x must appear more frequently than any other number in the dataset. Looking at the data, I see that x appears twice, while all the other numbers appear only once. So, x is already the mode because it appears more frequently. That part seems straightforward.Next, let's tackle the median. Since there are 9 data points, the median will be the 5th number when the data is ordered from smallest to largest. Let me write out the data in order, keeping in mind that x could be anywhere in the sequence depending on its value.The current data points are: -10, -5, x, x, 0, 15, 20, 25, 30. Wait, actually, that's not in order. Let me reorder them properly.If I arrange them in ascending order, it would be: -10, -5, 0, x, x, 15, 20, 25, 30. But I need to make sure where x fits. If x is less than 0, it would be before 0, and if x is greater than 15, it would be after 15. But since x is appearing twice, it's likely that x is somewhere in the middle.Wait, actually, the original list is: -10, -5, x, x, 0, 15, 20, 25, 30. So, in order, it's: -10, -5, 0, x, x, 15, 20, 25, 30. Hmm, but that's assuming x is between 0 and 15. If x is less than 0, the order would be different. Similarly, if x is greater than 15, the order would also change.But since x is supposed to be the median, which is the 5th term in the ordered list, let's see. If x is between 0 and 15, then the order would be: -10, -5, 0, x, x, 15, 20, 25, 30. So, the 5th term is x, which is what we want because the median should be x. So, as long as x is between 0 and 15, it will be the median. If x is less than 0, the order would be: -10, -5, x, x, 0, 15, 20, 25, 30, making the 5th term 0, which is not x. Similarly, if x is greater than 15, the order would be: -10, -5, 0, 15, x, x, 20, 25, 30, making the 5th term x. So, x can be either between 0 and 15 or greater than 15 to be the median. But since x is also the mode, which is already satisfied because it appears twice, we just need to make sure it's the mean as well.Now, let's calculate the mean. The mean is the sum of all the numbers divided by the number of numbers. There are 9 numbers, so the mean is:Mean = (-10 + (-5) + x + x + 0 + 15 + 20 + 25 + 30) / 9Let's compute the sum:-10 -5 + 0 + 15 + 20 + 25 + 30 = (-15) + (15 + 20 + 25 + 30) = (-15) + 90 = 75So, the sum is 75 + 2x (since there are two x's). Therefore, the mean is (75 + 2x)/9.We are told that the mean is x, so:x = (75 + 2x)/9Let's solve for x:Multiply both sides by 9:9x = 75 + 2xSubtract 2x from both sides:7x = 75Divide both sides by 7:x = 75/7Hmm, 75 divided by 7 is approximately 10.71. Let me check if this value of x satisfies the conditions for median and mode.First, mode: x appears twice, which is more than any other number, so that's good.Median: Let's order the data with x = 75/7 ≈ 10.71.The data points are: -10, -5, 0, 10.71, 10.71, 15, 20, 25, 30.So, the ordered list is: -10, -5, 0, 10.71, 10.71, 15, 20, 25, 30.The 5th term is 10.71, which is x, so that works.Therefore, x = 75/7 satisfies all three conditions: mean, median, and mode.Wait, let me double-check the calculations to make sure I didn't make a mistake.Sum of the known numbers: -10 -5 + 0 + 15 + 20 + 25 + 30 = (-15) + 90 = 75.Adding 2x: 75 + 2x.Mean: (75 + 2x)/9 = x.Multiply both sides by 9: 75 + 2x = 9x.Subtract 2x: 75 = 7x.Divide by 7: x = 75/7.Yes, that seems correct.So, the value of x is 75/7.

💡First, determine the number of 10 bills by dividing the total amount in 10 bills by the value of each 10 bill:Number of 10 bills = 50 / 10 = 5Next, calculate the total amount in 20 bills by multiplying the number of 20 bills by the value of each 20 bill:Total amount in 20 bills = 4 * 20 = 80Subtract the amounts in 10 and 20 bills from the total amount to find the amount in 5 bills:Amount in 5 bills = 150 - (50 + 80) = 20Determine the number of 5 bills by dividing the total amount in 5 bills by the value of each 5 bill:Number of 5 bills = 20 / 5 = 4Finally, add up the number of bills in each denomination to find the total number of bills:Total number of bills = 4 + 5 + 4 = 13

💡Alright, so I've got this problem about a company trying to reduce carbon emissions by converting CO2 into a chemical product. They've given me a function that relates the monthly processing cost to the volume of CO2 processed. The function is f(x) = (1/2)x² - 300x + 64800, where x is the processing volume in tons, and f(x) is the cost in yuan. The company can process between 30 tons and 400 tons per month.There are two parts to the problem. The first part asks for the monthly processing volume that results in the lowest monthly processing cost and what that lowest cost is. The second part asks for the processing volume that gives the lowest average processing cost per ton and what that cost is.Starting with part (1), I need to find the minimum of the function f(x). Since it's a quadratic function, I know it has a parabola shape. The coefficient of x² is positive (1/2), which means the parabola opens upwards, so the vertex will be the minimum point.To find the vertex of a quadratic function ax² + bx + c, the x-coordinate is given by -b/(2a). In this case, a is 1/2 and b is -300. Plugging these into the formula: x = -(-300)/(2*(1/2)) = 300/1 = 300. So, the minimum cost occurs at x = 300 tons.Now, to find the minimum cost, I plug x = 300 back into the function: f(300) = (1/2)*(300)² - 300*(300) + 64800. Calculating each term: (1/2)*90000 = 45000, 300*300 = 90000, so f(300) = 45000 - 90000 + 64800 = 19800 yuan.So, for part (1), the company achieves the lowest monthly processing cost at 300 tons, and the cost is 19800 yuan.Moving on to part (2), I need to find the processing volume that gives the lowest average processing cost per ton. The average cost per ton is f(x)/x. So, I need to minimize the function g(x) = f(x)/x = [(1/2)x² - 300x + 64800]/x.Simplifying g(x): g(x) = (1/2)x - 300 + 64800/x.To find the minimum of this function, I can take its derivative and set it equal to zero. The derivative of g(x) with respect to x is g'(x) = 1/2 - 64800/x².Setting g'(x) = 0: 1/2 - 64800/x² = 0. Solving for x: 1/2 = 64800/x² => x² = 64800*2 = 129600 => x = sqrt(129600) = 360.So, the average cost per ton is minimized at x = 360 tons.Now, to find the minimum average cost, plug x = 360 into g(x): g(360) = (1/2)*360 - 300 + 64800/360.Calculating each term: (1/2)*360 = 180, 64800/360 = 180. So, g(360) = 180 - 300 + 180 = 60 yuan per ton.Therefore, for part (2), the company achieves the lowest average processing cost per ton at 360 tons, and the cost is 60 yuan per ton.Wait, let me double-check my calculations to make sure I didn't make any mistakes.For part (1), f(300) = (1/2)*(300)^2 - 300*300 + 64800. So, (1/2)*90000 = 45000, 300*300 = 90000, so 45000 - 90000 = -45000, then -45000 + 64800 = 19800. That seems correct.For part (2), g(x) = (1/2)x - 300 + 64800/x. Taking derivative: 1/2 - 64800/x². Setting to zero: 1/2 = 64800/x² => x² = 129600 => x = 360. Plugging back in: (1/2)*360 = 180, 64800/360 = 180, so 180 - 300 + 180 = 60. That also seems correct.I think I did everything right. So, the answers are:1) 300 tons, 19800 yuan.2) 360 tons, 60 yuan per ton.

💡First, I need to add the two base 6 numbers, (35_6) and (14_6). I'll start by adding the units digits: (5_6 + 4_6 = 9_{10}). Since (9_{10}) is equal to (13_6) in base 6, I'll write down (3_6) in the units place and carry over (1_6) to the next higher digit.Next, I'll add the carry-over to the higher digits: (3_6 + 1_6 + 1_6 = 5_6). There's no need for another carry-over here. So, the sum of (35_6 + 14_6) is (53_6).Now, I need to subtract (20_6) from (53_6). Starting with the units digit: (3_6 - 0_6 = 3_6). Then, subtracting the higher digits: (5_6 - 2_6 = 3_6). Therefore, the result of (53_6 - 20_6) is (33_6).The final result of (35_6 + 14_6 - 20_6) in base 6 is (33_6).

💡First, I need to determine the total number of ways to select three plates from the total of 11 plates. This can be calculated using the combination formula ( binom{11}{3} ), which equals 165.Next, I'll calculate the number of ways to select three plates of the same color. There are two possibilities: all red plates or all blue plates.For the red plates, there are 6 available, so the number of ways to choose three red plates is ( binom{6}{3} = 20 ).For the blue plates, there are 5 available, so the number of ways to choose three blue plates is ( binom{5}{3} = 10 ).Adding these together, the total number of favorable outcomes is ( 20 + 10 = 30 ).Finally, the probability that all three selected plates are the same color is the ratio of favorable outcomes to the total number of possible outcomes, which is ( frac{30}{165} = frac{2}{11} ).

💡Okay, so I have this problem about an ellipse, and I need to solve two parts. Let me try to understand what's given and what I need to find.First, the ellipse is given by the equation (frac{x^2}{a^2} + frac{y^2}{b^2} = 1), where (a > b > 0). The left and right vertices are A and B, respectively, and the distance between them, |AB|, is 4. The eccentricity of the ellipse is (frac{1}{2}).**Part I: Find the equation of ellipse C.**Alright, so I know that for an ellipse, the distance between the two vertices is (2a). Since |AB| = 4, that means (2a = 4), so (a = 2). Got that.Next, the eccentricity (e) is given by (e = frac{c}{a}), where (c) is the distance from the center to each focus. They say the eccentricity is (frac{1}{2}), so:[frac{c}{a} = frac{1}{2} implies c = frac{a}{2} = frac{2}{2} = 1]So, (c = 1). Now, for an ellipse, the relationship between (a), (b), and (c) is (c^2 = a^2 - b^2). Plugging in the known values:[1^2 = 2^2 - b^2 implies 1 = 4 - b^2 implies b^2 = 4 - 1 = 3]Therefore, (b = sqrt{3}). So, the equation of the ellipse is:[frac{x^2}{2^2} + frac{y^2}{(sqrt{3})^2} = 1 implies frac{x^2}{4} + frac{y^2}{3} = 1]That should be the equation for part I. Let me just double-check:- Vertices at ((pm 2, 0)), so |AB| = 4, which matches.- Eccentricity (e = frac{c}{a} = frac{1}{2}), which is correct.- (b^2 = 3), so the equation looks good.**Part II: Determine if there's a point P such that quadrilateral APQM is a trapezoid.**Hmm, okay. Let me parse this.Point Q is given as (4, 0). Point P lies on the line (x = 4), so its coordinates are (4, t) for some t. Line BP intersects the ellipse at another point M. We need to check if there exists such a P so that APQM is a trapezoid.First, let me recall that a trapezoid is a quadrilateral with at least one pair of parallel sides. So, in quadrilateral APQM, we need at least one pair of sides to be parallel.Let me visualize this. Points A and B are the vertices of the ellipse, so A is (-2, 0) and B is (2, 0). Point Q is (4, 0), which is to the right of B. Point P is somewhere on the vertical line (x = 4), so it's somewhere above or below Q.Quadrilateral APQM has vertices at A, P, Q, M. So, the sides are AP, PQ, QM, and MA.I need to see if any two sides are parallel. Let's consider the possibilities:1. AP parallel to QM2. AQ parallel to PM3. PQ parallel to AMBut since AP, AQ, and PQ are all connected to P or Q, which are on the line (x = 4), maybe it's more likely that AP is parallel to QM or something like that.Wait, but let me think step by step.First, let's note the coordinates:- A is (-2, 0)- B is (2, 0)- Q is (4, 0)- P is (4, t), where t is unknown- M is another intersection point of line BP with the ellipse.So, line BP connects B (2, 0) to P (4, t). Let me find the parametric equation of BP.Parametric equations for BP can be written as:[x = 2 + 2s][y = 0 + t s]where (s) ranges from 0 to 1. When (s = 0), we are at B (2, 0), and when (s = 1), we are at P (4, t).But since M is another intersection point with the ellipse, we can plug these parametric equations into the ellipse equation and solve for (s).So, substituting into (frac{x^2}{4} + frac{y^2}{3} = 1):[frac{(2 + 2s)^2}{4} + frac{(ts)^2}{3} = 1]Simplify:[frac{4(1 + 2s + s^2)}{4} + frac{t^2 s^2}{3} = 1][(1 + 2s + s^2) + frac{t^2 s^2}{3} = 1][1 + 2s + s^2 + frac{t^2 s^2}{3} = 1]Subtract 1 from both sides:[2s + s^2 + frac{t^2 s^2}{3} = 0]Factor out s:[s left(2 + s + frac{t^2 s}{3}right) = 0]So, solutions are (s = 0) (which is point B) and:[2 + s + frac{t^2 s}{3} = 0][s left(1 + frac{t^2}{3}right) + 2 = 0][s = frac{-2}{1 + frac{t^2}{3}} = frac{-6}{3 + t^2}]So, the parameter (s) for point M is (-6/(3 + t^2)). Therefore, the coordinates of M are:[x = 2 + 2s = 2 + 2 left(frac{-6}{3 + t^2}right) = 2 - frac{12}{3 + t^2}][y = t s = t left(frac{-6}{3 + t^2}right) = frac{-6t}{3 + t^2}]So, M is (left(2 - frac{12}{3 + t^2}, frac{-6t}{3 + t^2}right)).Now, quadrilateral APQM has coordinates:- A: (-2, 0)- P: (4, t)- Q: (4, 0)- M: (left(2 - frac{12}{3 + t^2}, frac{-6t}{3 + t^2}right))We need to check if any sides are parallel. Let's compute the slopes of the sides:1. Slope of AP: from A(-2, 0) to P(4, t) [ m_{AP} = frac{t - 0}{4 - (-2)} = frac{t}{6} ]2. Slope of PQ: from P(4, t) to Q(4, 0) [ m_{PQ} = frac{0 - t}{4 - 4} = text{undefined (vertical line)} ]3. Slope of QM: from Q(4, 0) to M(left(2 - frac{12}{3 + t^2}, frac{-6t}{3 + t^2}right)) [ m_{QM} = frac{frac{-6t}{3 + t^2} - 0}{2 - frac{12}{3 + t^2} - 4} = frac{frac{-6t}{3 + t^2}}{-2 - frac{12}{3 + t^2}} = frac{-6t}{-2(3 + t^2) - 12} cdot frac{1}{3 + t^2} ] Wait, that seems complicated. Let me compute the denominator first: Denominator: (2 - frac{12}{3 + t^2} - 4 = (-2) - frac{12}{3 + t^2}) So, [ m_{QM} = frac{frac{-6t}{3 + t^2}}{-2 - frac{12}{3 + t^2}} = frac{-6t}{-2(3 + t^2) - 12} cdot frac{1}{3 + t^2} ] Wait, maybe I should factor out the denominator: Let me write denominator as: [ -2 - frac{12}{3 + t^2} = -left(2 + frac{12}{3 + t^2}right) = -left(frac{2(3 + t^2) + 12}{3 + t^2}right) = -left(frac{6 + 2t^2 + 12}{3 + t^2}right) = -left(frac{18 + 2t^2}{3 + t^2}right) = -frac{2(9 + t^2)}{3 + t^2} ] So, numerator is (-6t/(3 + t^2)), denominator is (-2(9 + t^2)/(3 + t^2)). Therefore, [ m_{QM} = frac{-6t/(3 + t^2)}{-2(9 + t^2)/(3 + t^2)} = frac{-6t}{-2(9 + t^2)} = frac{6t}{2(9 + t^2)} = frac{3t}{9 + t^2} ]4. Slope of MA: from M to A(-2, 0) [ m_{MA} = frac{0 - left(frac{-6t}{3 + t^2}right)}{-2 - left(2 - frac{12}{3 + t^2}right)} = frac{frac{6t}{3 + t^2}}{-4 + frac{12}{3 + t^2}} ] Let me compute the denominator: [ -4 + frac{12}{3 + t^2} = -frac{4(3 + t^2)}{3 + t^2} + frac{12}{3 + t^2} = frac{-12 - 4t^2 + 12}{3 + t^2} = frac{-4t^2}{3 + t^2} ] So, [ m_{MA} = frac{frac{6t}{3 + t^2}}{frac{-4t^2}{3 + t^2}} = frac{6t}{-4t^2} = -frac{3}{2t} ]So, now we have the slopes:- (m_{AP} = frac{t}{6})- (m_{PQ}) is vertical (undefined)- (m_{QM} = frac{3t}{9 + t^2})- (m_{MA} = -frac{3}{2t})Now, for quadrilateral APQM to be a trapezoid, at least one pair of sides must be parallel. So, we can check the possible pairs:1. AP and QM: Check if (m_{AP} = m_{QM})2. PQ and MA: PQ is vertical, MA has slope (-3/(2t)). For PQ and MA to be parallel, MA would also have to be vertical, which would require the slope to be undefined, but (-3/(2t)) is defined unless t is infinity, which isn't the case here. So, PQ and MA can't be parallel.3. AP and MA: Check if (m_{AP} = m_{MA})4. QM and MA: Check if (m_{QM} = m_{MA})Let's check each possibility.**1. AP parallel to QM:**Set (m_{AP} = m_{QM}):[frac{t}{6} = frac{3t}{9 + t^2}]Multiply both sides by 6(9 + t^2):[t(9 + t^2) = 18t]Assuming (t neq 0) (since if t=0, P would coincide with Q, which might not form a quadrilateral), we can divide both sides by t:[9 + t^2 = 18][t^2 = 9][t = pm 3]So, if t = 3 or t = -3, then AP is parallel to QM, making APQM a trapezoid.Wait, but let me check if these values are valid. If t = 3, then P is (4, 3), and M would be:[x = 2 - frac{12}{3 + 9} = 2 - frac{12}{12} = 2 - 1 = 1][y = frac{-6*3}{12} = frac{-18}{12} = -1.5]So, M is (1, -1.5). Similarly, for t = -3, M would be (1, 1.5).So, these points are valid on the ellipse.**2. AP parallel to MA:**Set (m_{AP} = m_{MA}):[frac{t}{6} = -frac{3}{2t}]Multiply both sides by 6*2t:[2t^2 = -18][t^2 = -9]Which has no real solutions. So, this case is impossible.**3. QM parallel to MA:**Set (m_{QM} = m_{MA}):[frac{3t}{9 + t^2} = -frac{3}{2t}]Multiply both sides by (2t(9 + t^2)):[6t^2 = -3(9 + t^2)][6t^2 = -27 - 3t^2][6t^2 + 3t^2 = -27][9t^2 = -27][t^2 = -3]Again, no real solutions. So, this case is impossible.Therefore, the only possibility is when AP is parallel to QM, which happens when t = ±3. Therefore, such points P exist at (4, 3) and (4, -3).Wait, but let me just visualize this. If P is at (4, 3), then M is at (1, -1.5). So, quadrilateral APQM would have points A(-2,0), P(4,3), Q(4,0), M(1,-1.5). Let me check if AP is indeed parallel to QM.Slope of AP: (3 - 0)/(4 - (-2)) = 3/6 = 0.5Slope of QM: (-1.5 - 0)/(1 - 4) = (-1.5)/(-3) = 0.5Yes, they are equal. Similarly, for P(4, -3), M would be (1, 1.5), and slope of QM would be (1.5 - 0)/(1 - 4) = 1.5/(-3) = -0.5, and slope of AP would be (-3)/6 = -0.5. So, they are equal as well.Therefore, such points P exist at (4, 3) and (4, -3).I think that's it. So, the answer is yes, and the coordinates of P are (4, 3) and (4, -3).