answer:To find out how many siblings the fifth girl had, we first need to calculate the total number of siblings reported by all the girls in the class. We do this by multiplying the mean number of siblings by the total number of girls surveyed. However, we don't know the total number of girls surveyed yet. We know the number of siblings for 8 of the girls (1, 6, 10, 4, 3, 11, 3, and 10), and we need to include the fifth girl in our count. So, the total number of girls surveyed is 8 + 1 = 9. Now, we multiply the mean number of siblings (5.7) by the total number of girls (9) to find the total number of siblings: 5.7 siblings/girl * 9 girls = 51.3 siblings Since the number of siblings must be a whole number, we can round the total to 51 siblings (since we can't have a fraction of a sibling). Next, we add up the number of siblings for the 8 girls we know: 1 + 6 + 10 + 4 + 3 + 11 + 3 + 10 = 48 siblings Now, we subtract the total number of siblings we have from the 8 girls from the total number of siblings for all 9 girls to find out how many siblings the fifth girl has: 51 siblings (total) - 48 siblings (8 girls) = 3 siblings Therefore, the fifth girl has boxed{3} siblings.

answer:1. **Coordinate System Setup:** Place our setup such that the circle ( k ) is centered at the origin ((0,0)) with radius equal to 1. Let the diameter points be ( P(-1, 0) ) and ( Q(1, 0) ). Hence, the equation of the circle ( k ) is: [ x^2 + y^2 = 1 ] 2. **Equation of the Line ( e ):** Let the line ( e ) be vertical having the equation ( x = e ) where ( -1 leq e leq 1 ). 3. **Equation of the Line ( f ):** The line ( f ) is parallel to ( e ) at a fixed distance ( d ). Depending on the position relative to ( e ), its equation will be either ( x = e + d ) or ( x = e - d ). 4. **Finding Intersection ( E_1 ):** The point ( E_1 ) is the intersection of the circle ( k ) and the line ( e ). The coordinates of ( E_1 ) can be calculated as: [ (e, sqrt{1 - e^2}) ] Since we only consider the upper intersection point because of symmetry. 5. **Equation of the Circle Centered at ( P ) through ( E_1 ):** The equation of a circle centered at ( P ) with radius ( PE_1 ) is needed: [ PE_1^2 = left(e + 1 right)^2 + left( sqrt{1 - e^2} right)^2 = (e + 1)^2 + (1 - e^2) = 2 + 2e ] Therefore, the equation of the circle with center ( P(-1, 0) ) and radius ( sqrt{2+2e} ) is: [ (x + 1)^2 + y^2 = 2 + 2e ] 6. **Finding Intersection Points ( F_1 ) and ( F_2 ):** To find the intersection points of this circle with line ( f ), we substitute ( x = e mp d ) into the circle's equation: [ y^2 = 2 + 2e - (e mp d + 1)^2 ] 7. **Simplifying:** Perform the simplification: [ y^2 = 2 + 2e - left( (e mp d + 1)^2 right) = 2 + 2e - (e^2 mp 2ed + d^2 + 2e mp 2d + 1) ] [ y^2 = 2 + 2e - e^2 mp d^2 - 2e mp 2ed + d^2 + 2e mp 2d - 1 = 1 pm 2d - (e mp d)^2 ] [ y^2 = 1 pm 2d - (e mp d)^2 ] 8. **Geometric Analysis:** The points ( F_1 ) and ( F_2 ) lie on circles that are concentric with the original circle ( k ): [ x^2 + y^2 = 1 pm 2d ] - When ( d < 0.5 ), both points lie on the circle: [ x^2 + y^2 = sqrt{1 - 2d} ] - When ( d = 0.5 ), as the circle shrinks to a point: [ x^2 + y^2 = 0 ] the only point is ( O ). - When ( d > 0.5 ), no corresponding ( F ) points on ( k ). 9. **Conclusion:** Hence, the loci of points ( F_1 ) and ( F_2 ) are circles concentric with the original circle ( k ), but their exact appearance depends on ( d ): - For ( d < 0.5 ), they form smaller concentric circles, - For ( d = 0.5 ), they collapse to a single point ( O ), - For ( d > 0.5 ), ( F_1 ) and ( F_2 ) do not exist. [ boxed{} ]

answer:1. **Determine the sum of the original set of numbers:** The mean (average) of the six numbers is given as (10). Since the mean is the sum of the numbers divided by the number of numbers, we can set up the equation: [ frac{text{Sum of 6 numbers}}{6} = 10 ] Multiply both sides by 6 to find the sum: [ text{Sum of 6 numbers} = 10 times 6 = 60 ] 2. **Calculate the sum of the remaining numbers after removing 25:** If the number 25 is removed, we need to subtract 25 from the total sum of the original six numbers: [ text{Sum of remaining 5 numbers} = 60 - 25 = 35 ] 3. **Find the mean of the remaining numbers:** The mean (average) of the remaining five numbers is the sum of these numbers divided by the number of those numbers: [ text{Mean of remaining 5 numbers} = frac{text{Sum of remaining 5 numbers}}{5} = frac{35}{5} = 7 ] # Conclusion: The mean of the remaining five numbers is (boxed{7}).

answer:To find the minimum number of questionnaires that should be mailed in each category to achieve at least 100 responses, we can use the response rates for each category. For Category A with a 60% response rate to get at least 100 responses, we can set up the equation: 0.60 * x = 100 Solving for x (the number of questionnaires to be mailed), we get: x = 100 / 0.60 x = 166.67 Since we can't send a fraction of a questionnaire, we need to round up to the nearest whole number. So, we need to send at least 167 questionnaires to Category A. For Category B with a 75% response rate: 0.75 * y = 100 Solving for y: y = 100 / 0.75 y = 133.33 Rounding up, we need to send at least 134 questionnaires to Category B. For Category C with an 80% response rate: 0.80 * z = 100 Solving for z: z = 100 / 0.80 z = 125 Since 125 is a whole number, we need to send exactly 125 questionnaires to Category C. To find the overall minimum number of questionnaires to be mailed, we add the minimum numbers for each category: Total questionnaires = 167 (Category A) + 134 (Category B) + 125 (Category C) Total questionnaires = 426 Therefore, the minimum number of questionnaires that should be mailed is boxed{167} to Category A, 134 to Category B, 125 to Category C, and 426 overall.