answer：First, I need to understand the given function and the condition provided. The function is ( f(x) = x + frac{1}{x} - 2 ), and it's given that ( f(a) = 3 ).To find ( f(-a) ), I'll start by using the condition ( f(a) = 3 ) to determine the value of ( a + frac{1}{a} ). Substituting ( a ) into the function:[f(a) = a + frac{1}{a} - 2 = 3]Solving for ( a + frac{1}{a} ):[a + frac{1}{a} = 3 + 2 = 5]Now, I'll compute ( f(-a) ) by substituting (-a) into the function:[f(-a) = -a + frac{1}{-a} - 2 = -left(a + frac{1}{a}right) - 2]Using the previously found value of ( a + frac{1}{a} = 5 ):[f(-a) = -5 - 2 = -7]Therefore, the value of ( f(-a) ) is (-7).

answer：First, I need to determine the relationship between the times it takes for each machine to produce ( X ) units.1. **Machine T** can produce ( X ) units in ( frac{3}{4} ) of the time it takes **Machine N**. If it takes **Machine N** ( t_N ) time to produce ( X ) units, then it takes **Machine T** ( t_T = frac{3}{4} t_N ) time.2. **Machine N** can produce ( X ) units in ( frac{2}{3} ) the time it takes **Machine O**. Therefore, if it takes **Machine N** ( t_N ) time, it takes **Machine O** ( t_O = frac{3}{2} t_N ) time.Next, I'll calculate the production rates for each machine. The production rate is the number of units produced per unit of time.- **Rate of Machine T** (( R_T )) = ( frac{X}{t_T} = frac{X}{frac{3}{4} t_N} = frac{4}{3} cdot frac{X}{t_N} )- **Rate of Machine N** (( R_N )) = ( frac{X}{t_N} )- **Rate of Machine O** (( R_O )) = ( frac{X}{t_O} = frac{X}{frac{3}{2} t_N} = frac{2}{3} cdot frac{X}{t_N} )Now, I'll find the total production rate when all three machines are working simultaneously:[R_{text{Total}} = R_T + R_N + R_O = frac{4}{3} cdot frac{X}{t_N} + frac{X}{t_N} + frac{2}{3} cdot frac{X}{t_N}]Combining the terms:[R_{text{Total}} = left( frac{4}{3} + 1 + frac{2}{3} right) cdot frac{X}{t_N} = left( frac{4}{3} + frac{3}{3} + frac{2}{3} right) cdot frac{X}{t_N} = frac{9}{3} cdot frac{X}{t_N} = 3 cdot frac{X}{t_N}]Finally, to determine the fraction of the total output produced by **Machine N**, I'll divide the rate of **Machine N** by the total rate:[text{Fraction by N} = frac{R_N}{R_{text{Total}}} = frac{frac{X}{t_N}}{3 cdot frac{X}{t_N}} = frac{1}{3}]Therefore, **Machine N** produces ( boxed{frac{1}{3}} ) of the total output when all three machines are working simultaneously.

answer：Okay, so I need to find the domain of this function: f(x) = sqrt(x + 1) divided by (x squared plus x). Hmm, let me think about how to approach this. First, I remember that the domain of a function includes all the real numbers x for which the function is defined. So, for this function, there are two main things I need to consider: the square root in the numerator and the denominator. Starting with the square root, sqrt(x + 1). I know that the expression inside a square root has to be greater than or equal to zero because you can't take the square root of a negative number in the set of real numbers. So, I'll set up the inequality:x + 1 >= 0Solving this, I subtract 1 from both sides:x >= -1Okay, so that tells me that x has to be greater than or equal to -1 for the square root to be defined. That's one condition.Now, moving on to the denominator: x squared plus x. I know that the denominator can't be zero because division by zero is undefined. So, I need to find the values of x that make x squared plus x equal to zero and exclude them from the domain.Let me factor the denominator:x^2 + x = x(x + 1)So, setting this equal to zero:x(x + 1) = 0This gives me two solutions:x = 0 or x + 1 = 0Which means:x = 0 or x = -1So, x can't be 0 or -1 because those would make the denominator zero, and the function would be undefined at those points.Wait a minute, but from the square root condition, I already have that x >= -1. So, x = -1 is already included in that condition, but since the denominator is zero at x = -1, I have to exclude it. Similarly, x = 0 is also excluded.So, putting it all together, x has to be greater than or equal to -1, but not equal to -1 or 0. So, in interval notation, that would be:(-1, 0) union (0, infinity)Let me double-check to make sure I didn't miss anything. The square root requires x >= -1, and the denominator can't be zero, which happens at x = 0 and x = -1. So, we exclude those two points. Therefore, the domain is all real numbers greater than -1 except for 0.Looking at the options given:A: [-1, infinity) – This includes x = -1, which is not allowed because the denominator would be zero there.B: (-1, 0) union (0, infinity) – This seems correct because it excludes -1 and 0.C: (-1, infinity) – This includes 0, which is not allowed because the denominator would be zero.D: [-1, 0) union (0, infinity) – This includes -1, which is not allowed.So, the correct answer should be option B.

answer：Alright, so I've got this system of inequalities to solve:left{begin{array}{l}11 x^{2}-10 x y+3 y^{2} leq 3 5 x+y leq -10end{array}right.Hmm, okay. Let's see. I need to find all the points (x, y) that satisfy both of these inequalities. The first one is a quadratic inequality, and the second one is linear. Maybe I can start by solving one of them and then substitute into the other.Let me look at the second inequality first because it's linear and might be easier to handle. So, the second inequality is:5x + y leq -10I can solve for y to make it easier to substitute into the first inequality. Let's do that:y leq -10 - 5xOkay, so y is less than or equal to -10 minus 5 times x. That gives me an expression for y in terms of x, which I can plug into the first inequality.Now, let's substitute this expression for y into the first inequality:11x^2 - 10x(-10 - 5x) + 3(-10 - 5x)^2 leq 3Alright, let's simplify this step by step. First, distribute the terms:Starting with the first part:-10x(-10 - 5x) = 100x + 50x^2Okay, so that part becomes 100x + 50x².Now, the second part is:3(-10 - 5x)^2Let's expand (-10 - 5x)² first:(-10 - 5x)^2 = (-10)^2 + 2(-10)(-5x) + (-5x)^2 = 100 + 100x + 25x^2So, multiplying by 3:3(100 + 100x + 25x^2) = 300 + 300x + 75x^2Now, let's put it all back into the inequality:11x^2 + 100x + 50x^2 + 300 + 300x + 75x^2 leq 3Combine like terms:First, the x² terms:11x^2 + 50x^2 + 75x^2 = 136x^2Next, the x terms:100x + 300x = 400xAnd the constant term:300So, the inequality becomes:136x^2 + 400x + 300 leq 3Subtract 3 from both sides to set it to zero:136x^2 + 400x + 297 leq 0Now, I have a quadratic inequality: 136x² + 400x + 297 ≤ 0.To solve this, I need to find the roots of the quadratic equation 136x² + 400x + 297 = 0. Let's calculate the discriminant to see if there are real roots.The discriminant D is given by:D = b^2 - 4acWhere a = 136, b = 400, and c = 297.So,D = 400^2 - 4 times 136 times 297Calculating 400²:400^2 = 160,000Now, calculating 4 × 136 × 297:First, 4 × 136 = 544.Then, 544 × 297. Let's compute that:544 × 200 = 108,800544 × 97 = ?Wait, 544 × 97:544 × 100 = 54,400Subtract 544 × 3 = 1,632So, 54,400 - 1,632 = 52,768So, total is 108,800 + 52,768 = 161,568Therefore,D = 160,000 - 161,568 = -1,568Hmm, the discriminant is negative, which means there are no real roots. That implies that the quadratic expression 136x² + 400x + 297 is always positive because the coefficient of x² is positive (136 > 0). So, the quadratic is always above zero, meaning 136x² + 400x + 297 > 0 for all real x.But our inequality is 136x² + 400x + 297 ≤ 0, which would require the quadratic to be less than or equal to zero. Since it's always positive, there are no real solutions for x that satisfy this inequality.Therefore, there are no points (x, y) that satisfy both inequalities in the system.So, the system has no solution.**Final Answer**The system of inequalities has no solution. boxed{text{No solutions}}

answer：First, I recognize that increasing the diameter of a circle by 100% means the diameter doubles. If the original diameter is ( D ), the new diameter becomes ( 2D ).Since the radius of a circle is half of its diameter, the original radius ( R ) becomes ( 2R ) after the diameter doubles.The area of a circle is calculated using the formula ( A = pi R^2 ). - The original area is ( A = pi R^2 ).- The new area with the doubled radius is ( A' = pi (2R)^2 = 4pi R^2 ).To find the percentage increase in area, I subtract the original area from the new area and then divide by the original area:[text{Percentage Increase} = left( frac{A' - A}{A} right) times 100% = left( frac{4pi R^2 - pi R^2}{pi R^2} right) times 100% = 300%]Therefore, the area of the circle has increased by 300%.

answer：Okay, so I have this problem about a rhombus ABCD with one interior angle of 60 degrees. Points A and B are on the line l: y = sqrt(3)x + 4, and points C and D are on the circle x² + y² = r², where r is between 1 and 2. I need to find the range of the area S of the rhombus as r varies.First, I remember that in a rhombus, all sides are equal, and the opposite angles are equal. Since one interior angle is 60 degrees, the adjacent angle must be 120 degrees. So, the rhombus can be thought of as two equilateral triangles joined together. But wait, is that always the case? Hmm, maybe not necessarily, but since one angle is 60 degrees, it might be helpful to consider that.Given that A and B are on the line l, which is y = sqrt(3)x + 4. This line has a slope of sqrt(3), which means it makes a 60-degree angle with the x-axis because tan(60°) = sqrt(3). That might be useful because the rhombus has a 60-degree angle, so maybe the sides are aligned in a way that relates to this slope.The circle is centered at the origin with radius r, which is between 1 and 2. So, points C and D lie on this circle. Since the circle is moving (r is varying), the positions of C and D will change, affecting the shape and area of the rhombus.I think I need to find the coordinates of points A, B, C, and D such that ABCD is a rhombus with the given conditions. Maybe I can parameterize points A and B on line l and then find points C and D on the circle such that ABCD is a rhombus.Let me consider the parametric equations for line l. Since it's y = sqrt(3)x + 4, any point on l can be written as (t, sqrt(3)t + 4), where t is a parameter. So, let me denote point A as (t, sqrt(3)t + 4) and point B as (s, sqrt(3)s + 4). Since AB is a side of the rhombus, the length AB should be equal to the lengths BC, CD, and DA.But wait, in a rhombus, all sides are equal, so AB = BC = CD = DA. Also, the angles are either 60 or 120 degrees. Since one angle is 60 degrees, the adjacent angle is 120 degrees.I think I need to use vectors to represent the sides. Let me denote vector AB as (s - t, sqrt(3)(s - t)). Then, vector BC should have the same magnitude as AB but rotated by 60 degrees because of the interior angle.Wait, actually, the angle between AB and BC is 60 degrees. So, if I have vector AB, then vector BC is vector AB rotated by 60 degrees. Similarly, vector CD is vector BC rotated by 60 degrees, and so on.But maybe it's better to think in terms of coordinates. Let me try to set up a coordinate system.Since line l has a slope of sqrt(3), which is 60 degrees, and the rhombus has a 60-degree angle, perhaps the sides of the rhombus are aligned with this line.Alternatively, maybe I can use the distance from the center of the circle to the line l. The distance from (0,0) to the line y = sqrt(3)x + 4 is |4| / sqrt(1 + 3) = 4 / 2 = 2. So, the distance is 2 units. Since the radius r is between 1 and 2, the circle does not intersect the line l because the distance is equal to the maximum radius (2). So, for r < 2, the circle is entirely on one side of the line l.This might help in determining the positions of points C and D on the circle.Now, considering the rhombus ABCD, with A and B on l, and C and D on the circle. Since ABCD is a rhombus, the diagonals bisect each other at right angles? Wait, no, that's a property of a square. In a rhombus, the diagonals bisect the angles but are not necessarily perpendicular unless it's a square.But since one angle is 60 degrees, the diagonals will split those angles into 30 degrees each.Alternatively, maybe I can use the area formula for a rhombus. The area S is given by S = ab sin(theta), where a and b are the lengths of the diagonals, but wait, no, that's not quite right. Actually, the area of a rhombus is base times height, or equivalently, (d1 * d2) / 2, where d1 and d2 are the lengths of the diagonals. But since it's a rhombus, all sides are equal, so maybe it's better to express the area in terms of side length and the sine of one of the angles.Yes, the area of a rhombus can also be expressed as S = s² sin(theta), where s is the side length and theta is one of the interior angles. Since theta is 60 degrees, S = s² * sin(60°) = (s² * sqrt(3)) / 2.So, if I can find the side length s in terms of r, then I can express the area S in terms of r and find its range.To find s, I need to relate the positions of points A, B, C, and D. Points A and B are on line l, and points C and D are on the circle. Since ABCD is a rhombus, the sides AB, BC, CD, and DA are equal.Let me denote the coordinates:Let A = (x1, y1) on l, so y1 = sqrt(3)x1 + 4.Similarly, B = (x2, y2) on l, so y2 = sqrt(3)x2 + 4.Points C and D are on the circle x² + y² = r².Since ABCD is a rhombus, vectors AB and AD should have the same magnitude and the angle between them is 60 degrees.Wait, maybe it's better to use vectors. Let me define vector AB and vector AD.Vector AB = (x2 - x1, y2 - y1) = (x2 - x1, sqrt(3)(x2 - x1)).Vector AD = (xC - x1, yC - y1), where C = (xC, yC) is on the circle.Since ABCD is a rhombus, |AB| = |AD|, and the angle between AB and AD is 60 degrees.So, |AB| = sqrt[(x2 - x1)^2 + (sqrt(3)(x2 - x1))^2] = sqrt[(x2 - x1)^2 + 3(x2 - x1)^2] = sqrt[4(x2 - x1)^2] = 2|x2 - x1|.Similarly, |AD| = sqrt[(xC - x1)^2 + (yC - y1)^2].Since |AB| = |AD|, we have 2|x2 - x1| = sqrt[(xC - x1)^2 + (yC - y1)^2].Also, the angle between AB and AD is 60 degrees, so the dot product of AB and AD is |AB||AD|cos(60°).Dot product AB · AD = (x2 - x1)(xC - x1) + (sqrt(3)(x2 - x1))(yC - y1).This should equal |AB||AD| * 0.5.But since |AB| = |AD|, let's denote |AB| = |AD| = s.So, AB · AD = s² * 0.5.But AB · AD = (x2 - x1)(xC - x1) + sqrt(3)(x2 - x1)(yC - y1).Factor out (x2 - x1):AB · AD = (x2 - x1)[(xC - x1) + sqrt(3)(yC - y1)].But since y1 = sqrt(3)x1 + 4, we can write yC - y1 = sqrt(3)(xC - x1) + (yC - sqrt(3)xC - 4 + 4) = sqrt(3)(xC - x1) + (yC - sqrt(3)xC).Wait, that might not be helpful. Alternatively, since C is on the circle, xC² + yC² = r².This is getting complicated. Maybe I need a different approach.Perhaps I can consider the distance from the center of the circle to the line l, which is 2 units, as calculated earlier. Since the radius r is less than 2, the circle is entirely on one side of the line l.Now, considering the rhombus, points C and D are on the circle, so the distance from the center to points C and D is r. The line CD is another side of the rhombus, so it should be parallel to AB because in a rhombus, opposite sides are parallel.Since AB is on line l, which has slope sqrt(3), line CD should also have slope sqrt(3). Therefore, line CD is parallel to l and can be written as y = sqrt(3)x + c, where c is a constant.The distance between lines l and CD is |4 - c| / sqrt(1 + 3) = |4 - c| / 2.Since CD is a side of the rhombus, the length of CD can be found using the distance between points C and D on the circle.But wait, CD is a chord of the circle. The length of chord CD is 2*sqrt(r² - d²), where d is the distance from the center to the chord.In this case, the distance from the center (0,0) to line CD is |c| / sqrt(1 + 3) = |c| / 2.Therefore, length CD = 2*sqrt(r² - (c/2)²).But since CD is a side of the rhombus, its length should be equal to AB.Earlier, we found that |AB| = 2|x2 - x1|. But AB is also a side of the rhombus, so |AB| = |CD|.Therefore, 2|x2 - x1| = 2*sqrt(r² - (c/2)²).Simplifying, |x2 - x1| = sqrt(r² - (c/2)²).But I also know that the distance between lines l and CD is |4 - c| / 2, and this distance should relate to the height of the rhombus.Wait, the height h of the rhombus can be expressed as h = |AB| * sin(theta), where theta is 60 degrees.So, h = |AB| * sin(60°) = |AB| * (sqrt(3)/2).But the distance between lines l and CD is also equal to h, because AB is on l and CD is on the parallel line. Therefore:|4 - c| / 2 = |AB| * (sqrt(3)/2).But |AB| = 2|x2 - x1|, so:|4 - c| / 2 = 2|x2 - x1| * (sqrt(3)/2) = |x2 - x1| * sqrt(3).But from earlier, |x2 - x1| = sqrt(r² - (c/2)²).So, substituting:|4 - c| / 2 = sqrt(r² - (c/2)²) * sqrt(3).Let me square both sides to eliminate the square roots:(|4 - c| / 2)^2 = (sqrt(r² - (c/2)²) * sqrt(3))^2( (4 - c)^2 ) / 4 = 3(r² - (c²)/4 )Multiply both sides by 4:(4 - c)^2 = 12r² - 3c²Expand the left side:16 - 8c + c² = 12r² - 3c²Bring all terms to one side:16 - 8c + c² - 12r² + 3c² = 0Combine like terms:4c² - 8c + 16 - 12r² = 0Divide the entire equation by 4:c² - 2c + 4 - 3r² = 0So, we have a quadratic equation in c:c² - 2c + (4 - 3r²) = 0Let me solve for c using the quadratic formula:c = [2 ± sqrt(4 - 4*(1)*(4 - 3r²))]/2Simplify the discriminant:sqrt(4 - 16 + 12r²) = sqrt(12r² - 12) = sqrt(12(r² - 1)) = 2*sqrt(3(r² - 1))So, c = [2 ± 2*sqrt(3(r² - 1))]/2 = 1 ± sqrt(3(r² - 1))Therefore, c = 1 + sqrt(3(r² - 1)) or c = 1 - sqrt(3(r² - 1))Since r is between 1 and 2, sqrt(3(r² - 1)) is real and positive. So, c can take two values.Now, let's find the distance between lines l and CD, which is |4 - c| / 2.Case 1: c = 1 + sqrt(3(r² - 1))Distance = |4 - (1 + sqrt(3(r² - 1)))| / 2 = |3 - sqrt(3(r² - 1))| / 2Case 2: c = 1 - sqrt(3(r² - 1))Distance = |4 - (1 - sqrt(3(r² - 1)))| / 2 = |3 + sqrt(3(r² - 1))| / 2Since sqrt(3(r² - 1)) is positive, in Case 1, the distance is (3 - sqrt(3(r² - 1)))/2, and in Case 2, it's (3 + sqrt(3(r² - 1)))/2.But since the distance must be positive, both cases are valid as long as 3 - sqrt(3(r² - 1)) > 0.So, 3 > sqrt(3(r² - 1)) => 9 > 3(r² - 1) => 3 > r² - 1 => r² < 4, which is true since r < 2.So, both cases are possible.Now, recalling that the distance between l and CD is equal to |AB| * sin(60°), which is |AB| * (sqrt(3)/2).But |AB| = 2|x2 - x1|, and from earlier, |x2 - x1| = sqrt(r² - (c/2)²).Wait, let's substitute c into this.From earlier, |x2 - x1| = sqrt(r² - (c/2)²).So, |AB| = 2*sqrt(r² - (c/2)²).But we also have that the distance between l and CD is |4 - c| / 2 = |AB| * (sqrt(3)/2).So, substituting |AB|:|4 - c| / 2 = 2*sqrt(r² - (c/2)²) * (sqrt(3)/2) = sqrt(r² - (c/2)²) * sqrt(3)Which is the same equation we had before, leading to the quadratic in c.So, now, having found c in terms of r, we can proceed to find the area S.The area S of the rhombus is |AB| * height, where height is the distance between l and CD.But we have:S = |AB| * distance = |AB| * (|4 - c| / 2)But |AB| = 2*sqrt(r² - (c/2)²), so:S = 2*sqrt(r² - (c/2)²) * (|4 - c| / 2) = sqrt(r² - (c/2)²) * |4 - c|But from earlier, we have |4 - c| / 2 = sqrt(r² - (c/2)²) * sqrt(3)So, sqrt(r² - (c/2)²) = |4 - c| / (2*sqrt(3))Therefore, S = (|4 - c| / (2*sqrt(3))) * |4 - c| = (|4 - c|²) / (2*sqrt(3))So, S = ( (4 - c)^2 ) / (2*sqrt(3))But c = 1 ± sqrt(3(r² - 1)), so let's substitute:Case 1: c = 1 + sqrt(3(r² - 1))Then, 4 - c = 3 - sqrt(3(r² - 1))So, S = ( (3 - sqrt(3(r² - 1)))^2 ) / (2*sqrt(3))Case 2: c = 1 - sqrt(3(r² - 1))Then, 4 - c = 3 + sqrt(3(r² - 1))So, S = ( (3 + sqrt(3(r² - 1)))^2 ) / (2*sqrt(3))Now, let's compute these expressions.First, expand (3 ± sqrt(3(r² - 1)))^2:= 9 ± 6*sqrt(3(r² - 1)) + 3(r² - 1)= 9 ± 6*sqrt(3(r² - 1)) + 3r² - 3= 6 ± 6*sqrt(3(r² - 1)) + 3r²So, S = (6 ± 6*sqrt(3(r² - 1)) + 3r²) / (2*sqrt(3))Factor numerator:= [3r² + 6 ± 6*sqrt(3(r² - 1))] / (2*sqrt(3))Factor 3:= 3[r² + 2 ± 2*sqrt(3(r² - 1))] / (2*sqrt(3))Simplify:= (3 / (2*sqrt(3))) [r² + 2 ± 2*sqrt(3(r² - 1))]= (sqrt(3)/2) [r² + 2 ± 2*sqrt(3(r² - 1))]So, S = (sqrt(3)/2)(r² + 2 ± 2*sqrt(3(r² - 1)))Now, let's analyze both cases.Case 1: S = (sqrt(3)/2)(r² + 2 - 2*sqrt(3(r² - 1)))Case 2: S = (sqrt(3)/2)(r² + 2 + 2*sqrt(3(r² - 1)))We need to find the range of S as r varies from 1 to 2.Let's consider r approaching 1 from above.When r approaches 1:sqrt(3(r² - 1)) approaches 0.So, Case 1: S approaches (sqrt(3)/2)(1 + 2 - 0) = (sqrt(3)/2)(3) = (3*sqrt(3))/2Case 2: S approaches (sqrt(3)/2)(1 + 2 + 0) = same as above, (3*sqrt(3))/2But wait, when r approaches 1, both cases give the same limit. However, for r > 1, the expressions differ.Wait, actually, when r approaches 1, sqrt(3(r² - 1)) approaches 0, so both cases give the same value. But for r > 1, we have two different expressions.Wait, let me check when r = 2.When r = 2:sqrt(3(r² - 1)) = sqrt(3*(4 - 1)) = sqrt(9) = 3Case 1: S = (sqrt(3)/2)(4 + 2 - 2*3) = (sqrt(3)/2)(6 - 6) = 0Case 2: S = (sqrt(3)/2)(4 + 2 + 2*3) = (sqrt(3)/2)(6 + 6) = (sqrt(3)/2)(12) = 6*sqrt(3)So, when r approaches 2, Case 1 gives S approaching 0, and Case 2 gives S approaching 6*sqrt(3).But wait, when r = 2, the circle has radius 2, which is equal to the distance from the center to line l, so the circle is tangent to line l. Therefore, points C and D would coincide at the point of tangency, making the rhombus collapse into a line, hence area 0. So, Case 1 corresponds to the area decreasing to 0 as r approaches 2.Case 2 corresponds to the area increasing to 6*sqrt(3) as r approaches 2.But wait, when r approaches 1, both cases give the same area, (3*sqrt(3))/2, which is approximately 2.598.But when r increases beyond 1, Case 1 gives a decreasing area from (3*sqrt(3))/2 to 0, and Case 2 gives an increasing area from (3*sqrt(3))/2 to 6*sqrt(3).Therefore, the area S can take any value between 0 and 6*sqrt(3), but actually, it's split into two intervals: from 0 to (3*sqrt(3))/2 and from (3*sqrt(3))/2 to 6*sqrt(3).Wait, but when r approaches 1, both cases give the same area, so the area doesn't jump; it smoothly transitions. But actually, for each r in (1,2), there are two possible rhombuses: one with smaller area and one with larger area.Therefore, the range of S is two intervals: (0, (3*sqrt(3))/2) and ((3*sqrt(3))/2, 6*sqrt(3)).But wait, when r approaches 1, the area approaches (3*sqrt(3))/2 from both sides. So, actually, the area can take any value greater than 0 up to 6*sqrt(3), but with a gap at (3*sqrt(3))/2? Or is it continuous?Wait, no. For each r in (1,2), there are two possible areas: one less than (3*sqrt(3))/2 and one greater. But when r approaches 1, both areas approach (3*sqrt(3))/2. So, the range is actually (0, 6*sqrt(3)), because as r increases from 1 to 2, the smaller area decreases from (3*sqrt(3))/2 to 0, and the larger area increases from (3*sqrt(3))/2 to 6*sqrt(3). Therefore, the union of these two intervals is (0, 6*sqrt(3)).But wait, that can't be right because when r is exactly 1, the area is (3*sqrt(3))/2, and as r increases, the area can be both smaller and larger than that value. So, the range should be from 0 to 6*sqrt(3), excluding the point where r=1, but since r is strictly greater than 1, the area can approach (3*sqrt(3))/2 but doesn't include it as a minimum or maximum.Wait, no. When r approaches 1 from above, the area approaches (3*sqrt(3))/2 from both sides. So, the area can be just above 0 up to 6*sqrt(3). But actually, for each r in (1,2), there are two possible areas: one less than (3*sqrt(3))/2 and one greater. So, the range is two intervals: (0, (3*sqrt(3))/2) and ((3*sqrt(3))/2, 6*sqrt(3)).But wait, when r is exactly 1, the area is (3*sqrt(3))/2, but r is strictly greater than 1, so the area can get arbitrarily close to (3*sqrt(3))/2 but doesn't actually reach it. Therefore, the range is (0, 6*sqrt(3)).But I think the correct interpretation is that for each r in (1,2), there are two possible rhombuses with areas in (0, (3*sqrt(3))/2) and ((3*sqrt(3))/2, 6*sqrt(3)). Therefore, the range of S is the union of these two intervals.So, the final answer is that the area S ranges from 0 to 6*sqrt(3), excluding the point (3*sqrt(3))/2, but since r is continuous, the area can take any value in (0, 6*sqrt(3)).Wait, but actually, when r approaches 1, the area approaches (3*sqrt(3))/2, and as r increases, the area can be both smaller and larger than that value. So, the range is two separate intervals: (0, (3*sqrt(3))/2) and ((3*sqrt(3))/2, 6*sqrt(3)).Therefore, the range of S is (0, (3*sqrt(3))/2) union ((3*sqrt(3))/2, 6*sqrt(3)).But let me double-check.When r approaches 1, the circle is very small, just slightly larger than radius 1. The rhombus would have points C and D close to the circle, but since the circle is close to the origin, and line l is far away (distance 2), the rhombus would be "compressed" towards line l, making the area approach (3*sqrt(3))/2.As r increases, the circle grows, allowing points C and D to move further away, creating a larger rhombus. However, there are two configurations: one where the rhombus is "above" the line l and one where it's "below". But since the circle is on one side of l, both configurations are on the same side, but with different orientations.Wait, actually, since the circle is on one side of l, both points C and D must be on the same side, so the rhombus can be either "above" or "below" in terms of orientation, but since l is fixed, it's more about the position relative to l.But in any case, the key point is that for each r, there are two possible areas, one smaller and one larger than (3*sqrt(3))/2.Therefore, the range of S is two intervals: from 0 up to (3*sqrt(3))/2 and from (3*sqrt(3))/2 up to 6*sqrt(3).Hence, the range is (0, (3*sqrt(3))/2) union ((3*sqrt(3))/2, 6*sqrt(3)).But let me confirm with r=1.5.At r=1.5, sqrt(3(r² -1)) = sqrt(3*(2.25 -1))=sqrt(3*1.25)=sqrt(3.75)=approx 1.936.So, c =1 ±1.936.Case 1: c=1 -1.936≈-0.936Then, 4 - c≈4 - (-0.936)=4.936So, S≈(4.936)^2/(2*sqrt(3))≈24.36/(3.464)≈7.03Case 2: c=1 +1.936≈2.936Then, 4 - c≈1.064So, S≈(1.064)^2/(2*sqrt(3))≈1.132/(3.464)≈0.327Wait, but 7.03 is less than 6*sqrt(3)≈10.392, so that's fine.Wait, but when r=1.5, S can be approximately 0.327 or 7.03.So, as r increases from 1 to 2, the smaller area goes from ~2.598 to 0, and the larger area goes from ~2.598 to ~10.392.Therefore, the range of S is from 0 to 10.392, but split into two intervals: (0, 2.598) and (2.598, 10.392).But 2.598 is approximately (3*sqrt(3))/2≈2.598, and 10.392≈6*sqrt(3).Therefore, the range is (0, (3*sqrt(3))/2) union ((3*sqrt(3))/2, 6*sqrt(3)).So, the final answer is that the area S of the rhombus varies between 0 and 6*sqrt(3), excluding the point (3*sqrt(3))/2, but since r is continuous, the area can take any value in the union of the two intervals.Hence, the range of S is (0, (3*sqrt(3))/2) union ((3*sqrt(3))/2, 6*sqrt(3)).