💡Alright, let's tackle this problem step by step. So, we have 19 female students, all with different heights, and they need to select the 10 tallest students for a volleyball competition. Xiaohong wants to know if she'll be selected. The question is asking what statistical measure she needs to know to determine her chances.First, I need to understand the selection process. Since they're selecting the 10 tallest students out of 19, it means that the 10th tallest student will be the cutoff for selection. If Xiaohong's height is greater than or equal to the height of the 10th tallest student, she'll be selected; otherwise, she won't.Now, let's think about the statistical measures given in the options: median, mean, minimum value, and variance. I need to figure out which one of these is relevant for Xiaohong to determine her selection.Starting with the median. The median is the middle value in an ordered list of numbers. Since there are 19 students, the median would be the 10th value when the heights are arranged in ascending order. That means the median is exactly the cutoff point for selection because the 10th tallest student is the median. So, if Xiaohong knows the median height, she can compare her height to it and determine if she's among the tallest 10.Next, the mean. The mean is the average height of all the students. While knowing the average might give some general idea about the group's height, it doesn't directly tell Xiaohong where she stands in terms of selection. The mean could be skewed by extremely tall or short students, so it's not the most reliable measure for this specific purpose.The minimum value is the shortest height in the group. Knowing the minimum value doesn't help Xiaohong because she's concerned about being among the tallest, not the shortest. This measure is irrelevant for determining her selection.Lastly, variance. Variance measures how spread out the heights are from the mean. While variance gives information about the distribution of heights, it doesn't provide a specific cutoff point for selection. It's more about the variability rather than the actual selection criteria.So, after evaluating all the options, the median seems to be the most relevant statistical measure for Xiaohong to determine her selection. It directly corresponds to the cutoff point for being among the tallest 10 students.To summarize, Xiaohong needs to know the median height of the group. If her height is equal to or greater than the median, she'll be selected for the volleyball team. This makes the median the correct answer.

💡First, I need to compare the fractions (frac{21}{17}), (frac{22}{19}), (frac{18}{15}), and (frac{20}{16}) to determine their order from least to greatest.I'll start by comparing (frac{21}{17}) and (frac{22}{19}) using cross-multiplication:[21 times 19 = 399 quad text{and} quad 22 times 17 = 374]Since (399 > 374), it means (frac{21}{17} > frac{22}{19}).Next, I'll compare (frac{21}{17}) with (frac{18}{15}):[21 times 15 = 315 quad text{and} quad 18 times 17 = 306]Here, (315 > 306), so (frac{21}{17} > frac{18}{15}).Then, I'll compare (frac{21}{17}) with (frac{20}{16}):[21 times 16 = 336 quad text{and} quad 20 times 17 = 340]Since (336 < 340), it follows that (frac{21}{17} < frac{20}{16}).Moving on, I'll compare (frac{22}{19}) with (frac{18}{15}):[22 times 15 = 330 quad text{and} quad 18 times 19 = 342]Here, (330 < 342), so (frac{22}{19} < frac{18}{15}).Next, I'll compare (frac{22}{19}) with (frac{20}{16}):[22 times 16 = 352 quad text{and} quad 20 times 19 = 380]Since (352 < 380), it means (frac{22}{19} < frac{20}{16}).Finally, I'll compare (frac{18}{15}) with (frac{20}{16}):[18 times 16 = 288 quad text{and} quad 20 times 15 = 300]Here, (288 < 300), so (frac{18}{15} < frac{20}{16}).Putting all these comparisons together, the order from least to greatest is:[frac{22}{19} < frac{18}{15} < frac{21}{17} < frac{20}{16}]

💡Okay, so I have this problem here: I need to find the length of the chord intercepted by the line ( x + y = 3 ) on the curve ( x^2 + y^2 - 2y - 3 = 0 ). Hmm, let me try to figure this out step by step.First, I remember that when a line intersects a circle, it can create a chord. The length of this chord can be found if I know the radius of the circle and the distance from the center of the circle to the line. I think the formula for the chord length is something like ( 2sqrt{r^2 - d^2} ), where ( r ) is the radius and ( d ) is the distance from the center to the line. Let me make sure that's correct. Yeah, I think that's right because it's derived from the Pythagorean theorem, considering the right triangle formed by the radius, the distance, and half the chord.So, to apply this formula, I need two things: the radius of the circle and the distance from the center of the circle to the given line. Let me start by rewriting the given curve equation in a standard form so I can identify the center and radius.The given curve is ( x^2 + y^2 - 2y - 3 = 0 ). I notice that this looks like the equation of a circle, but it's not in the standard form. The standard form of a circle is ( (x - h)^2 + (y - k)^2 = r^2 ), where ( (h, k) ) is the center and ( r ) is the radius.To convert ( x^2 + y^2 - 2y - 3 = 0 ) into standard form, I need to complete the square for the y-terms. Let's see:Starting with the given equation:( x^2 + y^2 - 2y - 3 = 0 )I can rearrange the terms:( x^2 + (y^2 - 2y) = 3 )Now, to complete the square for ( y^2 - 2y ), I take the coefficient of y, which is -2, divide it by 2 to get -1, and then square it to get 1. So, I add and subtract 1 on the left side to maintain equality:( x^2 + (y^2 - 2y + 1 - 1) = 3 )This simplifies to:( x^2 + (y - 1)^2 - 1 = 3 )Then, adding 1 to both sides:( x^2 + (y - 1)^2 = 4 )Okay, so now it's in standard form. The center of the circle is at ( (0, 1) ) and the radius ( r ) is ( sqrt{4} = 2 ). Got that down.Next, I need to find the distance from the center of the circle to the line ( x + y = 3 ). The formula for the distance ( d ) from a point ( (h, k) ) to the line ( ax + by + c = 0 ) is:( d = frac{|ah + bk + c|}{sqrt{a^2 + b^2}} )First, let me write the line equation in the standard form ( ax + by + c = 0 ). The given line is ( x + y = 3 ). Subtracting 3 from both sides gives:( x + y - 3 = 0 )So, ( a = 1 ), ( b = 1 ), and ( c = -3 ). The center of the circle is ( (0, 1) ), so ( h = 0 ) and ( k = 1 ).Plugging these into the distance formula:( d = frac{|1*0 + 1*1 - 3|}{sqrt{1^2 + 1^2}} = frac{|0 + 1 - 3|}{sqrt{2}} = frac{|-2|}{sqrt{2}} = frac{2}{sqrt{2}} )Simplifying ( frac{2}{sqrt{2}} ), I can rationalize the denominator:( frac{2}{sqrt{2}} = frac{2sqrt{2}}{2} = sqrt{2} )So, the distance from the center to the line is ( sqrt{2} ).Now, using the chord length formula ( 2sqrt{r^2 - d^2} ):( 2sqrt{2^2 - (sqrt{2})^2} = 2sqrt{4 - 2} = 2sqrt{2} )Wait, let me double-check that. The radius squared is 4, and the distance squared is ( (sqrt{2})^2 = 2 ). So, ( 4 - 2 = 2 ), and the square root of 2 is ( sqrt{2} ). Multiplying by 2 gives ( 2sqrt{2} ). That seems correct.Just to make sure I didn't make a mistake, let me recap:1. Converted the given circle equation to standard form, found center (0,1) and radius 2.2. Used the distance formula from center to line ( x + y = 3 ), got ( sqrt{2} ).3. Applied chord length formula, got ( 2sqrt{2} ).Hmm, seems solid. I don't see any errors in my calculations. Maybe I can also solve it another way to confirm. For example, solving the system of equations to find the points of intersection and then computing the distance between them.Let me try that approach.We have the line ( x + y = 3 ) and the circle ( x^2 + y^2 - 2y - 3 = 0 ). Let's solve for one variable in the line equation and substitute into the circle equation.From ( x + y = 3 ), we can express ( y = 3 - x ).Substitute ( y = 3 - x ) into the circle equation:( x^2 + (3 - x)^2 - 2(3 - x) - 3 = 0 )Let me expand this step by step.First, expand ( (3 - x)^2 ):( (3 - x)^2 = 9 - 6x + x^2 )So, substituting back:( x^2 + (9 - 6x + x^2) - 2(3 - x) - 3 = 0 )Simplify term by term:1. ( x^2 )2. ( + 9 - 6x + x^2 )3. ( - 6 + 2x ) (because ( -2*3 = -6 ) and ( -2*(-x) = +2x ))4. ( - 3 )Now, combine all these:( x^2 + 9 - 6x + x^2 - 6 + 2x - 3 = 0 )Combine like terms:- ( x^2 + x^2 = 2x^2 )- ( -6x + 2x = -4x )- ( 9 - 6 - 3 = 0 )So, the equation simplifies to:( 2x^2 - 4x = 0 )Factor out 2x:( 2x(x - 2) = 0 )So, the solutions are ( x = 0 ) or ( x = 2 ).Now, find the corresponding y-values using ( y = 3 - x ):- If ( x = 0 ), then ( y = 3 - 0 = 3 ). So, one point is (0, 3).- If ( x = 2 ), then ( y = 3 - 2 = 1 ). So, the other point is (2, 1).Now, we have the two points of intersection: (0, 3) and (2, 1). Let's find the distance between these two points.The distance formula is ( sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ).Plugging in the points:( sqrt{(2 - 0)^2 + (1 - 3)^2} = sqrt{(2)^2 + (-2)^2} = sqrt{4 + 4} = sqrt{8} = 2sqrt{2} )So, the length of the chord is ( 2sqrt{2} ), which matches the result I got earlier using the chord length formula. That gives me more confidence that the answer is correct.I think I've covered both methods: using the chord length formula by finding the distance from the center to the line, and solving the system of equations to find the points of intersection and then computing the distance between them. Both approaches led me to the same answer, so I feel pretty certain that ( 2sqrt{2} ) is the correct length of the chord.Just to recap the steps clearly:1. **Identify the circle's center and radius:** - Given equation: ( x^2 + y^2 - 2y - 3 = 0 ) - Completed the square for y: ( x^2 + (y - 1)^2 = 4 ) - Center: (0, 1), Radius: 22. **Find the distance from the center to the line:** - Line equation: ( x + y = 3 ) or ( x + y - 3 = 0 ) - Distance formula: ( d = frac{|1*0 + 1*1 - 3|}{sqrt{1 + 1}} = frac{2}{sqrt{2}} = sqrt{2} )3. **Calculate chord length:** - Formula: ( 2sqrt{r^2 - d^2} = 2sqrt{4 - 2} = 2sqrt{2} )4. **Verification by solving the system:** - Substituted ( y = 3 - x ) into the circle equation - Found intersection points: (0, 3) and (2, 1) - Calculated distance between these points: ( 2sqrt{2} )Everything checks out. I think I'm confident that the chord length is indeed ( 2sqrt{2} ).**Final Answer**The length of the chord is boxed{2sqrt{2}}.

💡Okay, so I have this problem here, and I need to figure out which of the given expressions cannot be calculated using the square difference formula. The options are A, B, C, and D. Let me start by recalling what the square difference formula is. I think it's something like ( a^2 - b^2 = (a + b)(a - b) ). Yeah, that sounds right. So, this formula allows us to factor the difference of two squares into a product of two binomials.Now, looking at the options, each one is a product of two binomials. I need to see if each of these can be rewritten in the form ( (a + b)(a - b) ) so that it fits the square difference formula. If I can do that, then it can be calculated using the formula. If not, then that's the one that can't be calculated that way.Let's go through each option one by one.**Option A: ( (-x - y)(x - y) )**Hmm, okay. Let's see. I can try to factor out a negative sign from the first binomial to make it easier. So, ( (-x - y) ) can be written as ( - (x + y) ). So, the expression becomes ( - (x + y)(x - y) ). Now, this looks similar to the square difference formula. If I let ( a = x ) and ( b = y ), then ( (x + y)(x - y) = x^2 - y^2 ). But here, we have a negative sign in front, so it would be ( - (x^2 - y^2) = -x^2 + y^2 ) or ( y^2 - x^2 ). Either way, this is still a difference of squares, just with the terms reversed. So, I think this can be calculated using the square difference formula.**Option B: ( (-x + y)(-x - y) )**Alright, let's look at this one. Both binomials have negative signs. Maybe I can factor out a negative from one of them. Let's see. If I factor out a negative from the second binomial, ( (-x - y) ), it becomes ( - (x + y) ). So, the expression becomes ( (-x + y) times - (x + y) ). That's ( - (-x + y)(x + y) ). Hmm, that's a bit messy. Maybe another approach. Alternatively, I can rearrange the terms. Let's consider ( (-x + y) ) as ( (y - x) ) and ( (-x - y) ) as ( -(x + y) ). So, the expression is ( (y - x)(- (x + y)) ). That simplifies to ( - (y - x)(x + y) ). Now, ( (y - x)(x + y) ) is similar to ( (a - b)(a + b) ), which is ( a^2 - b^2 ). So, here, ( a = y ) and ( b = x ), so it would be ( y^2 - x^2 ). Then, with the negative sign, it becomes ( - (y^2 - x^2) = -y^2 + x^2 = x^2 - y^2 ). So, again, this is a difference of squares, just with some rearrangement and factoring out negatives. So, this can also be calculated using the square difference formula.**Option C: ( (x + y)(-x + y) )**Okay, moving on to option C. Let's see. ( (x + y) ) is straightforward, and ( (-x + y) ) can be written as ( (y - x) ). So, the expression becomes ( (x + y)(y - x) ). This is similar to ( (a + b)(a - b) ), where ( a = y ) and ( b = x ). So, applying the square difference formula, this would be ( y^2 - x^2 ). That's a straightforward application of the formula. So, this can definitely be calculated using the square difference formula.**Option D: ( (x - y)(-x + y) )**Now, onto option D. Let's see. ( (x - y) ) is one binomial, and ( (-x + y) ) can be written as ( (y - x) ). So, the expression becomes ( (x - y)(y - x) ). Wait a minute, this looks a bit different. Let me think. If I have ( (x - y)(y - x) ), that's essentially ( (x - y)(- (x - y)) ) because ( y - x = - (x - y) ). So, this becomes ( - (x - y)^2 ). Hmm, that's not a difference of squares; it's actually a negative of a square. So, it's ( - (x - y)^2 ), which is not the same as ( a^2 - b^2 ). The square difference formula requires two different terms being squared and subtracted, but here, both terms are the same, and it's just a negative square.Alternatively, if I try to apply the formula directly, ( (x - y)(y - x) ), I can see that both binomials have opposite signs. So, ( x ) is positive in the first and negative in the second, and ( y ) is negative in the first and positive in the second. This means that when I multiply them, I won't get a simple difference of squares. Instead, I'll get a negative square, as I saw earlier.So, this seems like it can't be calculated using the square difference formula because it doesn't fit the form ( (a + b)(a - b) ); instead, it's more like ( (a - b)(- (a - b)) ), which simplifies to a negative square.Wait, let me double-check that. If I expand ( (x - y)(y - x) ), what do I get?First, multiply ( x ) by ( y ) to get ( xy ).Then, ( x ) times ( -x ) is ( -x^2 ).Next, ( -y ) times ( y ) is ( -y^2 ).Finally, ( -y ) times ( -x ) is ( xy ).So, adding all those up: ( xy - x^2 - y^2 + xy ).Combine like terms: ( 2xy - x^2 - y^2 ).Hmm, that's ( -x^2 - y^2 + 2xy ), which can be rewritten as ( -(x^2 + y^2 - 2xy) ).And ( x^2 + y^2 - 2xy ) is ( (x - y)^2 ). So, the entire expression is ( - (x - y)^2 ), which is indeed a negative square, not a difference of squares.Therefore, this expression doesn't fit the square difference formula because it results in a negative square rather than a difference of two distinct squares.So, after going through each option, it seems like option D is the one that cannot be calculated using the square difference formula.

💡First, I identify the given information: the reciprocals of the HCF and LCM are ( frac{1}{13} ) and ( frac{1}{312} ), respectively. This means the HCF is 13 and the LCM is 312. One of the numbers is 24, and I need to find the other number.I recall the relationship between HCF, LCM, and the product of the two numbers: ( text{HCF} times text{LCM} = text{Product of the numbers} ).Using this relationship, I set up the equation:[13 times 312 = 24 times B]Next, I solve for ( B ) by dividing both sides of the equation by 24:[B = frac{13 times 312}{24}]To simplify the calculation, I divide 312 by 24 to get 13:[B = 13 times 13 = 169]Therefore, the other number is 169.

💡Okay, so I have this quadratic equation: 9x² + mx + 36 = 0. The problem says that for two real values of m, this equation has exactly one solution in x. I need to find the positive value of m. Hmm, let me think about how to approach this.First, I remember that a quadratic equation ax² + bx + c = 0 has exactly one real solution when its discriminant is zero. The discriminant is given by b² - 4ac. So, if I calculate the discriminant for this equation and set it equal to zero, I should be able to solve for m.Alright, let's identify the coefficients in the equation. Here, a is 9, b is m, and c is 36. So, plugging these into the discriminant formula, I get:Discriminant = m² - 4 * 9 * 36Let me compute that step by step. First, 4 * 9 is 36, and then 36 * 36 is... wait, that's 1296. So, the discriminant simplifies to m² - 1296.Now, setting the discriminant equal to zero for the equation to have exactly one solution:m² - 1296 = 0To solve for m, I can add 1296 to both sides:m² = 1296Taking the square root of both sides gives me:m = ±√1296I know that √1296 is 36 because 36 * 36 is 1296. So, m can be either 36 or -36.The problem asks for the positive value of m, so that would be 36.Wait, let me just double-check my steps to make sure I didn't make any mistakes. I identified a, b, and c correctly. Calculated the discriminant correctly as m² - 4*9*36, which is m² - 1296. Set that equal to zero, solved for m, and got ±36. Positive value is 36. Yeah, that seems right.I guess another way to think about it is that the quadratic equation will have exactly one solution when the parabola touches the x-axis at exactly one point, meaning it's a perfect square. So, 9x² + mx + 36 should be a perfect square trinomial. Let's see if that's the case when m is 36.If m is 36, the equation becomes 9x² + 36x + 36. Let's factor that. I can factor out a 9 first: 9(x² + 4x + 4). Then, x² + 4x + 4 is a perfect square, (x + 2)². So, the equation becomes 9(x + 2)² = 0, which indeed has exactly one solution at x = -2. That checks out.Similarly, if m is -36, the equation becomes 9x² - 36x + 36. Factoring out 9, we get 9(x² - 4x + 4), which is 9(x - 2)² = 0, also having exactly one solution at x = 2. So, both values of m give exactly one solution, but since we're asked for the positive value, 36 is the answer.I think that's solid. I don't see any errors in my reasoning. The key was remembering the discriminant condition for a single solution and then solving for m accordingly. It's also good to verify by plugging the values back into the equation to ensure it works, which I did. So, yeah, m equals 36 is the positive value that satisfies the condition.