💡Okay, so I have this problem where Adam and Eva are creating decorations using identical white circles. Adam uses four circles arranged in a square, each touching two others, and then adds a red circle in the center that touches all four. Eva uses three circles arranged in a triangle, each touching the others, and adds a green circle in the center that touches all three. I need to find the radii of the red and green circles in terms of the radius of the white circles.First, let's tackle Adam's configuration. He has four white circles arranged in a square. Each white circle touches two others, so the centers of these circles form a square. If the radius of each white circle is r, then the distance between the centers of two adjacent white circles is 2r because they are touching each other.Now, the red circle is placed in the center of this square and touches all four white circles. To find the radius of the red circle, I need to determine the distance from the center of the square to the center of one of the white circles and then subtract the radius r of the white circle.The centers of the four white circles form a square with side length 2r. The diagonal of this square can be found using the Pythagorean theorem. The diagonal d is sqrt[(2r)^2 + (2r)^2] = sqrt[4r^2 + 4r^2] = sqrt[8r^2] = 2*sqrt(2)*r.The distance from the center of the square to the center of one of the white circles is half of this diagonal, so that's (2*sqrt(2)*r)/2 = sqrt(2)*r.Therefore, the radius of the red circle R is this distance minus the radius of the white circle: R = sqrt(2)*r - r = (sqrt(2) - 1)*r.Okay, that seems straightforward. Now, onto Eva's configuration. She has three white circles arranged in an equilateral triangle, each touching the others. The centers of these circles form an equilateral triangle with side length 2r.Eva adds a green circle in the center that touches all three white circles. To find the radius of the green circle, I need to determine the distance from the center of the triangle to the center of one of the white circles and then subtract the radius r of the white circle.In an equilateral triangle, the distance from the centroid to a vertex is (2/3) of the height of the triangle. First, let's find the height h of the equilateral triangle. The height can be found using the Pythagorean theorem: h = sqrt[(2r)^2 - (r)^2] = sqrt[4r^2 - r^2] = sqrt[3r^2] = r*sqrt(3).So, the distance from the centroid to a vertex is (2/3)*h = (2/3)*r*sqrt(3) = (2*sqrt(3)/3)*r.Therefore, the radius of the green circle G is this distance minus the radius of the white circle: G = (2*sqrt(3)/3)*r - r = (2*sqrt(3) - 3)/3 * r.Wait, let me double-check that. So, the distance from the centroid to the vertex is (2*sqrt(3)/3)*r, and then subtracting r gives us the radius of the green circle. Yes, that seems correct.So, summarizing:- The radius of Adam's red circle is (sqrt(2) - 1)*r.- The radius of Eva's green circle is (2*sqrt(3) - 3)/3 * r.I think that makes sense because in Adam's case, the red circle is fitting into a square arrangement, which involves the diagonal, hence the sqrt(2), and in Eva's case, the green circle is fitting into a triangular arrangement, which involves the height of an equilateral triangle, hence the sqrt(3).Just to make sure, let's plug in some numbers. Suppose r = 1.For Adam: R = (sqrt(2) - 1)*1 ≈ (1.4142 - 1) ≈ 0.4142.For Eva: G = (2*sqrt(3) - 3)/3 ≈ (3.4641 - 3)/3 ≈ 0.4641/3 ≈ 0.1547.So, the red circle has a radius of approximately 0.4142, and the green circle has a radius of approximately 0.1547 when r = 1. That seems reasonable, as the red circle is larger than the green one, which makes sense because the square arrangement allows for a larger central circle compared to the triangular arrangement.I think I've got it right. The radii are expressed in terms of r, the radius of the white circles, as (sqrt(2) - 1)r and (2*sqrt(3) - 3)/3 r.

💡Okay, so I have this problem here with two complex numbers, z₁ and z₂. z₁ is given as 1 + bi, and z₂ is -2 + i. The question says that when you divide z₁ by z₂, the real part and the imaginary part of the result are opposites of each other. I need to find the value of b, which is a real number, and choose the correct option from A to D.Alright, let me start by recalling how to divide complex numbers. I remember that to divide two complex numbers, you multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. So, in this case, the conjugate of z₂, which is -2 + i, would be -2 - i.So, let me write down the division:z₁ / z₂ = (1 + bi) / (-2 + i)To simplify this, I'll multiply both the numerator and the denominator by the conjugate of the denominator, which is (-2 - i):= [(1 + bi) * (-2 - i)] / [(-2 + i) * (-2 - i)]Now, I need to carry out the multiplication in both the numerator and the denominator.Starting with the numerator:(1 + bi) * (-2 - i) = 1*(-2) + 1*(-i) + bi*(-2) + bi*(-i)Let me compute each term:1*(-2) = -21*(-i) = -ibi*(-2) = -2bibi*(-i) = -bi²Wait, I remember that i² is equal to -1, so -bi² becomes -b*(-1) = b.So, putting it all together:-2 - i - 2bi + bNow, let me combine like terms. The real parts are -2 and b, and the imaginary parts are -i and -2bi.So, real part: (-2 + b)Imaginary part: (-1 - 2b)iSo, the numerator simplifies to (b - 2) + (-1 - 2b)i.Now, moving on to the denominator:(-2 + i) * (-2 - i)This is a product of a complex number and its conjugate, which should result in a real number. Let me compute it:= (-2)*(-2) + (-2)*(-i) + i*(-2) + i*(-i)Compute each term:(-2)*(-2) = 4(-2)*(-i) = 2ii*(-2) = -2ii*(-i) = -i²Again, i² is -1, so -i² is 1.Now, let me add all these terms:4 + 2i - 2i + 1Simplify:The 2i and -2i cancel each other out, so we have 4 + 1 = 5.So, the denominator is 5.Putting it all together, the division simplifies to:[(b - 2) + (-1 - 2b)i] / 5Which can be written as:(b - 2)/5 + [(-1 - 2b)/5]iSo, the real part is (b - 2)/5, and the imaginary part is (-1 - 2b)/5.According to the problem, the real part and the imaginary part are opposites of each other. That means:Real part = - (Imaginary part)So,(b - 2)/5 = - [(-1 - 2b)/5]Let me write that equation:(b - 2)/5 = - [(-1 - 2b)/5]Simplify the right side:- [(-1 - 2b)/5] = (1 + 2b)/5So, the equation becomes:(b - 2)/5 = (1 + 2b)/5Since both sides have the same denominator, I can multiply both sides by 5 to eliminate the denominators:b - 2 = 1 + 2bNow, let's solve for b.First, subtract b from both sides:b - 2 - b = 1 + 2b - bSimplify:-2 = 1 + bNow, subtract 1 from both sides:-2 - 1 = bWhich simplifies to:-3 = bSo, b is equal to -3.Let me check my steps to make sure I didn't make a mistake.1. Divided z₁ by z₂ by multiplying numerator and denominator by the conjugate of z₂. That seems correct.2. Expanded the numerator: (1 + bi)(-2 - i). Calculated each term, combined like terms. Got (b - 2) + (-1 - 2b)i. That seems correct.3. Expanded the denominator: (-2 + i)(-2 - i). Got 4 + 2i - 2i +1 = 5. Correct.4. So, the division became [(b - 2) + (-1 - 2b)i]/5. Correct.5. Set real part equal to negative of imaginary part: (b - 2)/5 = - [(-1 - 2b)/5]. Simplified to (b - 2)/5 = (1 + 2b)/5. Then multiplied both sides by 5: b - 2 = 1 + 2b. Subtracted b: -2 = 1 + b. Subtracted 1: -3 = b. All steps look correct.So, yeah, b is -3, which is option D.I think that's solid. I don't see any mistakes in my calculations. So, I feel confident that the answer is D: -3.**Final Answer**boxed{D}

💡Okay, so I have this problem here: If the two roots of the function ( f(x) = x^2 - ax - b ) are 2 and 3, then the roots of the function ( g(x) = bx^2 - ax - 1 ) are ______.Hmm, alright, let me try to figure this out step by step. I remember that for a quadratic equation, if we know the roots, we can express the quadratic in its factored form. So, if the roots are 2 and 3, then ( f(x) ) can be written as ( (x - 2)(x - 3) ). Let me expand that to see if it matches the given form.Expanding ( (x - 2)(x - 3) ):( x times x = x^2 ),( x times (-3) = -3x ),( (-2) times x = -2x ),( (-2) times (-3) = 6 ).So, adding those up: ( x^2 - 3x - 2x + 6 = x^2 - 5x + 6 ).Wait, but the given function is ( f(x) = x^2 - ax - b ). Comparing this with my expanded form ( x^2 - 5x + 6 ), I can set up equations for the coefficients.So, the coefficient of x in my expanded form is -5, which should equal -a. Therefore, ( -a = -5 ) implies ( a = 5 ).Similarly, the constant term in my expanded form is +6, which should equal -b. So, ( -b = 6 ) implies ( b = -6 ).Alright, so I found that ( a = 5 ) and ( b = -6 ). Let me write that down:( a = 5 )( b = -6 )Now, the function ( g(x) ) is given as ( bx^2 - ax - 1 ). Let me substitute the values of a and b into this function.Substituting ( b = -6 ) and ( a = 5 ):( g(x) = (-6)x^2 - 5x - 1 )So, ( g(x) = -6x^2 - 5x - 1 )Now, I need to find the roots of this quadratic function. To find the roots, I can set ( g(x) = 0 ) and solve for x.So, the equation is:( -6x^2 - 5x - 1 = 0 )Hmm, solving a quadratic equation. I remember the quadratic formula: ( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where in this case, the coefficients are:( a = -6 ) (coefficient of ( x^2 ))( b = -5 ) (coefficient of x)( c = -1 ) (constant term)Wait, hold on, that might be confusing because the standard quadratic formula uses a, b, c, but in our function, the coefficients are different. Let me clarify:In the quadratic equation ( ax^2 + bx + c = 0 ), the coefficients are:- ( a ) is the coefficient of ( x^2 )- ( b ) is the coefficient of x- ( c ) is the constant termSo, in our case, ( g(x) = -6x^2 -5x -1 ), so:- ( a = -6 )- ( b = -5 )- ( c = -1 )So, plugging into the quadratic formula:( x = frac{-(-5) pm sqrt{(-5)^2 - 4 times (-6) times (-1)}}{2 times (-6)} )Let me compute each part step by step.First, compute the numerator:- ( -(-5) = 5 )Now, compute the discriminant ( D = b^2 - 4ac ):( D = (-5)^2 - 4 times (-6) times (-1) )Let's compute each term:- ( (-5)^2 = 25 )- ( 4 times (-6) times (-1) = 4 times 6 = 24 ) (since negative times negative is positive)So, ( D = 25 - 24 = 1 )So, the discriminant is 1. That's a perfect square, so the roots will be rational.Now, plug back into the formula:( x = frac{5 pm sqrt{1}}{2 times (-6)} )Compute ( sqrt{1} = 1 ), so:( x = frac{5 pm 1}{-12} )So, we have two solutions:1. ( x = frac{5 + 1}{-12} = frac{6}{-12} = -frac{1}{2} )2. ( x = frac{5 - 1}{-12} = frac{4}{-12} = -frac{1}{3} )So, the roots are ( x = -frac{1}{2} ) and ( x = -frac{1}{3} ).Wait, let me double-check my calculations to make sure I didn't make any mistakes.Starting from the quadratic equation:( -6x^2 -5x -1 = 0 )Quadratic formula:( x = frac{-b pm sqrt{b^2 - 4ac}}{2a} )Here, ( a = -6 ), ( b = -5 ), ( c = -1 )So,( x = frac{-(-5) pm sqrt{(-5)^2 - 4*(-6)*(-1)}}{2*(-6)} )Simplify:( x = frac{5 pm sqrt{25 - 24}}{-12} )( x = frac{5 pm 1}{-12} )So,1. ( x = frac{5 + 1}{-12} = frac{6}{-12} = -frac{1}{2} )2. ( x = frac{5 - 1}{-12} = frac{4}{-12} = -frac{1}{3} )Yes, that seems correct.Alternatively, maybe I can factor the quadratic equation to see if it gives the same roots.So, ( -6x^2 -5x -1 = 0 ). Let me factor out a negative sign to make it easier:( - (6x^2 + 5x + 1) = 0 )So, ( 6x^2 + 5x + 1 = 0 )Now, let's try to factor this quadratic.Looking for two numbers that multiply to ( 6 times 1 = 6 ) and add up to 5.Hmm, 2 and 3 multiply to 6 and add up to 5. So, let's split the middle term:( 6x^2 + 2x + 3x + 1 = 0 )Group the terms:( (6x^2 + 2x) + (3x + 1) = 0 )Factor out common terms:( 2x(3x + 1) + 1(3x + 1) = 0 )Now, factor out ( (3x + 1) ):( (2x + 1)(3x + 1) = 0 )So, the solutions are:( 2x + 1 = 0 ) => ( x = -frac{1}{2} )and( 3x + 1 = 0 ) => ( x = -frac{1}{3} )Which matches the roots I found earlier using the quadratic formula. So, that confirms my answer.Therefore, the roots of ( g(x) ) are ( -frac{1}{2} ) and ( -frac{1}{3} ).**Final Answer**The roots of the function ( g(x) ) are boxed{-dfrac{1}{2}} and boxed{-dfrac{1}{3}}.Wait, hold on, the instructions say to put the final answer within a single box. Maybe I should present both roots in one box.**Final Answer**The roots of the function ( g(x) ) are boxed{-dfrac{1}{2}} and boxed{-dfrac{1}{3}}.Hmm, actually, looking back at the original problem, it says "the roots of the function g(x) are ______." So, maybe I should list both roots in a single box, separated by commas.**Final Answer**The roots of the function ( g(x) ) are boxed{-dfrac{1}{2}, -dfrac{1}{3}}.Yes, that looks better.

💡Alright, so I have this problem here about points in a plane, and I need to prove an inequality involving the greatest and smallest distances between any two points. Let me try to unpack this step by step.First, let me restate the problem to make sure I understand it. We have a set ( E ) of ( n geq 2 ) points in the plane. ( D ) is the greatest distance between any two points in ( E ), and ( d ) is the smallest distance between any two distinct points in ( E ). We need to prove that:[D geq frac{sqrt{3}}{2}(sqrt{n} - 1) d]Okay, so I need to relate the maximum distance ( D ) to the minimum distance ( d ) and the number of points ( n ). The inequality suggests that as ( n ) increases, the lower bound on ( D ) increases proportionally to ( sqrt{n} ). That makes sense intuitively because with more points, you'd expect at least one pair to be further apart.Let me think about how to approach this. Maybe I can use some geometric arguments or perhaps apply some known theorems in combinatorial geometry. I recall something about circle packing or sphere packing, but I'm not sure if that's directly applicable here.Wait, the problem mentions points in the plane, so maybe I can consider the smallest enclosing circle for all the points. If I can relate the radius of this circle to the distances ( D ) and ( d ), that might help. Let me try that.Let ( Gamma ) be the smallest circle that contains all the points in ( E ), and let ( k ) be its diameter. So, ( k ) is the maximum distance between any two points on the boundary of ( Gamma ). Since ( D ) is the greatest distance between any two points in ( E ), ( D ) must be less than or equal to ( k ). Hmm, but I need to relate ( D ) to ( d ) and ( n ), so maybe I can find a relationship between ( k ) and ( d ) using ( n ).If none of the points lie on the boundary of ( Gamma ), then I could reduce the radius of ( Gamma ) and still contain all the points, which contradicts the minimality of ( Gamma ). So, at least one point must lie on the boundary of ( Gamma ).Now, if all the points on the boundary of ( Gamma ) lie on a single semi-circle, I could shift ( Gamma ) in that direction and reduce its radius, again contradicting the minimality. Therefore, there must be at least two points diametrically opposed on ( Gamma ) or at least three points not all lying on any half-circle.Let me consider the case where there are three points ( A, B, C ) on ( Gamma ) such that they form a triangle containing the center of ( Gamma ). Since they are not all on the same semi-circle, one of the central angles must be at least ( 120^circ ). Without loss of generality, let's say the angle at the center between points ( A ) and ( B ) is ( alpha geq 120^circ ).The distance between ( A ) and ( B ) can be calculated using the chord length formula:[AB = 2r sinleft(frac{alpha}{2}right)]where ( r ) is the radius of ( Gamma ). Since ( alpha geq 120^circ ), we have:[AB geq 2r sin(60^circ) = 2r cdot frac{sqrt{3}}{2} = rsqrt{3}]But ( AB ) is one of the distances between points in ( E ), so ( AB leq D ). Therefore,[D geq rsqrt{3}]But ( k = 2r ), so ( r = frac{k}{2} ). Substituting back,[D geq frac{sqrt{3}}{2}k]Okay, so that gives me a relationship between ( D ) and ( k ). Now, I need to relate ( k ) to ( d ) and ( n ).Let me think about the area. If I consider each point in ( E ) as the center of a circle with diameter ( d ), then these circles must be disjoint because ( d ) is the minimum distance between any two points. So, the area covered by these ( n ) small circles must be less than or equal to the area of a larger circle that contains all of them.What's the radius of this larger circle? If the smallest enclosing circle has diameter ( k ), then the radius is ( frac{k}{2} ). But if I consider a slightly larger circle with diameter ( k + d ), then the radius is ( frac{k + d}{2} ). This larger circle should contain all the smaller circles centered at the points in ( E ).So, the area of the larger circle is:[pi left( frac{k + d}{2} right)^2]And the total area of the ( n ) smaller circles is:[n pi left( frac{d}{2} right)^2]Since the larger circle must contain all the smaller circles, their areas must satisfy:[pi left( frac{k + d}{2} right)^2 geq n pi left( frac{d}{2} right)^2]Simplifying this inequality by dividing both sides by ( pi ):[left( frac{k + d}{2} right)^2 geq n left( frac{d}{2} right)^2]Multiply both sides by 4 to eliminate the denominators:[(k + d)^2 geq n d^2]Expanding the left side:[k^2 + 2kd + d^2 geq n d^2]Subtract ( d^2 ) from both sides:[k^2 + 2kd geq (n - 1) d^2]Now, from earlier, we have ( D geq frac{sqrt{3}}{2}k ), which implies ( k leq frac{2}{sqrt{3}}D ). Let's substitute this into the inequality:[left( frac{2}{sqrt{3}}D right)^2 + 2 cdot frac{2}{sqrt{3}}D cdot d geq (n - 1) d^2]Calculating each term:[frac{4}{3}D^2 + frac{4}{sqrt{3}}D d geq (n - 1) d^2]Divide both sides by ( d^2 ) to normalize:[frac{4}{3} left( frac{D}{d} right)^2 + frac{4}{sqrt{3}} left( frac{D}{d} right) geq n - 1]Let me denote ( x = frac{D}{d} ) for simplicity. Then the inequality becomes:[frac{4}{3}x^2 + frac{4}{sqrt{3}}x geq n - 1]This is a quadratic inequality in terms of ( x ). To find the minimum value of ( x ) that satisfies this inequality, I can solve the equation:[frac{4}{3}x^2 + frac{4}{sqrt{3}}x - (n - 1) = 0]Using the quadratic formula:[x = frac{ -frac{4}{sqrt{3}} pm sqrt{ left( frac{4}{sqrt{3}} right)^2 + 4 cdot frac{4}{3} cdot (n - 1) } }{ 2 cdot frac{4}{3} }]Simplify the discriminant:[left( frac{4}{sqrt{3}} right)^2 = frac{16}{3}][4 cdot frac{4}{3} cdot (n - 1) = frac{16}{3}(n - 1)][text{Discriminant} = frac{16}{3} + frac{16}{3}(n - 1) = frac{16}{3}n]So,[x = frac{ -frac{4}{sqrt{3}} pm sqrt{ frac{16}{3}n } }{ frac{8}{3} }][x = frac{ -frac{4}{sqrt{3}} pm frac{4}{sqrt{3}}sqrt{n} }{ frac{8}{3} }][x = frac{ -1 pm sqrt{n} }{ 2 } cdot frac{3}{sqrt{3}}][x = frac{3}{sqrt{3}} cdot frac{ -1 pm sqrt{n} }{ 2 }][x = frac{sqrt{3}}{2} ( -1 pm sqrt{n} )]Since ( x ) must be positive, we discard the negative solution:[x = frac{sqrt{3}}{2} ( sqrt{n} - 1 )]Therefore,[frac{D}{d} geq frac{sqrt{3}}{2} ( sqrt{n} - 1 )]Which implies:[D geq frac{sqrt{3}}{2} ( sqrt{n} - 1 ) d]So, after going through all these steps, I've managed to derive the required inequality. I think this makes sense because as ( n ) increases, the minimum distance ( d ) would require the maximum distance ( D ) to be larger to accommodate all the points without violating the minimum distance constraint. The factor of ( sqrt{3}/2 ) comes from the geometric considerations involving the angles and chord lengths in the circle.I should probably double-check my steps to ensure I didn't make any mistakes, especially in the algebra when solving the quadratic equation. Let me go back and verify:1. Defined the smallest enclosing circle ( Gamma ) with diameter ( k ).2. Showed that at least two points must be at least ( frac{sqrt{3}}{2}k ) apart.3. Related ( k ) to ( D ) via ( D geq frac{sqrt{3}}{2}k ).4. Considered the area of the larger circle containing all smaller circles of diameter ( d ).5. Derived the inequality ( (k + d)^2 geq n d^2 ).6. Substituted ( k leq frac{2}{sqrt{3}}D ) into the inequality.7. Solved the resulting quadratic inequality for ( x = frac{D}{d} ).8. Found the positive solution ( x = frac{sqrt{3}}{2} ( sqrt{n} - 1 ) ).Everything seems to check out. I think I can confidently say that the inequality holds as required.**Final Answer**boxed{D geq frac{sqrt{3}}{2}(sqrt{n} - 1) d}

💡First, I need to find the value of the 5th observation, ( x_5 ), using the given information about the averages of different sets of observations.I start by converting the given average equations into total sums to simplify the calculations.From the first average:[frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = 10]Multiplying both sides by 5:[x_1 + x_2 + x_3 + x_4 + x_5 = 50 quad text{(Equation 4)}]From the second average:[frac{x_5 + x_6 + x_7 + x_8 + x_9}{5} = 8]Multiplying both sides by 5:[x_5 + x_6 + x_7 + x_8 + x_9 = 40 quad text{(Equation 5)}]From the overall average:[frac{x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9}{9} = 8]Multiplying both sides by 9:[x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9 = 72 quad text{(Equation 6)}]Next, I subtract Equation 5 from Equation 6 to find the sum of the first four observations:[(x_1 + x_2 + x_3 + x_4 + x_5 + x_6 + x_7 + x_8 + x_9) - (x_5 + x_6 + x_7 + x_8 + x_9) = 72 - 40][x_1 + x_2 + x_3 + x_4 = 32 quad text{(Equation 7)}]Finally, I subtract Equation 7 from Equation 4 to isolate ( x_5 ):[(x_1 + x_2 + x_3 + x_4 + x_5) - (x_1 + x_2 + x_3 + x_4) = 50 - 32][x_5 = 18]Therefore, the value of the 5th observation, ( x_5 ), is (boxed{18}).

💡Alright, so I have this problem about the Robinson family. Let me try to understand it step by step. The family consists of a mother, a father, and some children. The average age of all the family members is 22 years. The father is 50 years old, and the average age of just the mother and the children is 18. I need to find out how many children are in the family.Okay, let's break this down. First, the family has a mother, a father, and some children. So, the total number of family members is 2 (mother and father) plus the number of children. Let me denote the number of children as ( x ). So, the total number of family members is ( 2 + x ).The average age of the entire family is 22. That means if I take the sum of all their ages and divide it by the number of family members, I get 22. So, mathematically, that would be:[frac{text{Total age of all family members}}{2 + x} = 22]Which implies:[text{Total age of all family members} = 22 times (2 + x)]Now, the father is 50 years old. So, the total age of the family can also be expressed as the sum of the father's age, the mother's age, and the total age of all the children. Let me denote the mother's age as ( m ) and the total age of the children as ( c ). So, we have:[50 + m + c = 22 times (2 + x)]Simplifying the right side:[50 + m + c = 44 + 22x]Okay, so that's one equation. Now, the problem also states that the average age of the mother and the children is 18. That means if I take the sum of the mother's age and the total age of the children and divide it by the number of mother and children, which is ( 1 + x ), I get 18. So, mathematically:[frac{m + c}{1 + x} = 18]Which implies:[m + c = 18 times (1 + x)]Simplifying:[m + c = 18 + 18x]Now, I have two equations:1. ( 50 + m + c = 44 + 22x )2. ( m + c = 18 + 18x )I can substitute the second equation into the first one to eliminate ( m + c ). Let's do that.From equation 2, ( m + c = 18 + 18x ). So, substituting into equation 1:[50 + (18 + 18x) = 44 + 22x]Simplify the left side:[50 + 18 + 18x = 44 + 22x]Which is:[68 + 18x = 44 + 22x]Now, let's solve for ( x ). Subtract ( 18x ) from both sides:[68 = 44 + 4x]Then, subtract 44 from both sides:[24 = 4x]Divide both sides by 4:[x = 6]So, according to this, there are 6 children in the family. Hmm, let me double-check to make sure I didn't make a mistake.If there are 6 children, then the total number of family members is 2 (mother and father) + 6 = 8. The total age of the family is 22 * 8 = 176 years.The father is 50, so the total age of the mother and children is 176 - 50 = 126 years.The average age of the mother and children is 18, so the total age should be 18 * (1 + 6) = 18 * 7 = 126 years.Yes, that matches. So, the calculations seem correct. Therefore, the number of children is 6.**Final Answer**The number of children in the Robinson family is boxed{E}.