💡Okay, so I have this problem here: "The number of proper subsets of the set {0,3,4} is: A: 4, B: 6, C: 7, D: 8." Hmm, I need to figure out how many proper subsets this set has. I remember that subsets are sets that contain elements from the original set, but I'm a bit fuzzy on the exact definition of a proper subset. Let me think.Alright, a subset is any set where all its elements are also in the original set. So, for example, {0} is a subset of {0,3,4} because 0 is in the original set. Similarly, {3,4} is also a subset. But what's a proper subset? I think a proper subset is a subset that is not equal to the original set itself. So, it can't be the entire set; it has to be missing at least one element.Okay, so to find the number of proper subsets, I need to find all possible subsets of {0,3,4} and then subtract one because the set itself is not considered a proper subset. How do I find all the subsets?I remember that the number of subsets of a set with n elements is 2^n. So, since {0,3,4} has 3 elements, the total number of subsets should be 2^3, which is 8. Let me list them out to make sure:1. The empty set: {}2. Single-element subsets: {0}, {3}, {4}3. Two-element subsets: {0,3}, {0,4}, {3,4}4. The entire set: {0,3,4}So, that's 1 + 3 + 3 + 1 = 8 subsets in total. Now, since a proper subset can't be the entire set, I need to exclude that one. So, 8 total subsets minus 1 gives me 7 proper subsets.Wait, let me double-check. If I list all the proper subsets, they should be:1. {}2. {0}3. {3}4. {4}5. {0,3}6. {0,4}7. {3,4}Yes, that's 7 subsets. None of them are the entire set {0,3,4}, so they're all proper subsets. Therefore, the number of proper subsets is 7.Looking back at the options, C is 7, so that should be the correct answer. I think I got it right, but just to be thorough, let me consider if there's any other way to approach this.Another way is to use the formula for the number of proper subsets, which is 2^n - 1, where n is the number of elements in the set. Here, n is 3, so 2^3 - 1 = 8 - 1 = 7. Yep, that confirms it.I guess I was a bit unsure at first because sometimes people confuse subsets and proper subsets, but breaking it down step by step helped me understand better. It's also good to remember the formula, but listing them out is a solid method too, especially for smaller sets like this one.So, in conclusion, the number of proper subsets of {0,3,4} is 7, which is option C.

💡Okay, so I've got this problem here about McGregor and Floyd who were assigned to read 89 books for the week. McGregor managed to finish 34 books, and Floyd finished some number of books. Together, they still have 23 books left to read. The question is asking how many books Floyd finished.Alright, let's break this down. First, the total number of books they were supposed to read is 89. McGregor read 34 of them. So, if I subtract the number of books McGregor read from the total, that should give me the number of books that are either read by Floyd or still left to read. So, 89 minus 34 is... let's see, 89 minus 30 is 59, and then minus 4 more is 55. So, 55 books are either read by Floyd or still unread.Now, the problem also says that they still have 23 books left to read. That means out of those 55 books, 23 are still unread. So, to find out how many books Floyd read, I need to subtract the unread books from that 55. So, 55 minus 23 equals... 55 minus 20 is 35, and then minus 3 more is 32. So, Floyd must have read 32 books.Wait, let me double-check that. If McGregor read 34 and Floyd read 32, together they've read 66 books. Adding the 23 books they still have to read, that's 66 plus 23, which equals 89. That matches the total number of books they were assigned, so it makes sense.I think that's the right answer. Floyd finished 32 books.

💡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 ( C: frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ) where ( a > b > 0 ). The left focus is ( F_1(-1, 0) ), so that tells me something about the distance from the center to the focus. Also, the eccentricity is ( e ), and ( b ) is the geometric mean of ( 3e ) and ( a ). Alright, for part (1), I need to find the equation of the curve ( C ). So, I need to find the values of ( a ) and ( b ). Let's recall some properties of ellipses.The standard form of an ellipse is ( frac{x^2}{a^2} + frac{y^2}{b^2} = 1 ), where ( a ) is the semi-major axis, ( b ) is the semi-minor axis, and the relationship between ( a ), ( b ), and the eccentricity ( e ) is ( e = frac{c}{a} ), where ( c ) is the distance from the center to a focus. Also, ( c^2 = a^2 - b^2 ).Given that the left focus is at ( (-1, 0) ), the center of the ellipse must be at the origin because the foci are symmetric with respect to the center. So, the distance from the center to each focus is ( c = 1 ). Therefore, ( c = 1 ).Now, the eccentricity ( e ) is given by ( e = frac{c}{a} = frac{1}{a} ).Next, it says that ( b ) is the geometric mean of ( 3e ) and ( a ). The geometric mean of two numbers ( x ) and ( y ) is ( sqrt{xy} ). So, ( b = sqrt{3e cdot a} ).Let me write that down:( b = sqrt{3e cdot a} ).But since ( e = frac{1}{a} ), substitute that in:( b = sqrt{3 cdot frac{1}{a} cdot a} = sqrt{3} ).Wait, that simplifies nicely. So, ( b = sqrt{3} ).Now, I can use the relationship ( c^2 = a^2 - b^2 ) to find ( a ).We know ( c = 1 ) and ( b = sqrt{3} ), so:( 1^2 = a^2 - (sqrt{3})^2 )( 1 = a^2 - 3 )So, ( a^2 = 4 ), which means ( a = 2 ) since ( a > 0 ).Therefore, the equation of the ellipse is:( frac{x^2}{4} + frac{y^2}{3} = 1 ).Alright, that seems straightforward. Let me double-check my steps.1. Given the left focus at (-1, 0), so ( c = 1 ).2. Eccentricity ( e = c/a = 1/a ).3. ( b ) is the geometric mean of ( 3e ) and ( a ), so ( b = sqrt{3e cdot a} = sqrt{3 cdot (1/a) cdot a} = sqrt{3} ).4. Then, using ( c^2 = a^2 - b^2 ), we get ( 1 = a^2 - 3 ), so ( a^2 = 4 ), ( a = 2 ).5. Plugging back into the ellipse equation, we get ( x^2/4 + y^2/3 = 1 ).Yep, that seems correct. So, part (1) is done.Now, moving on to part (2). This seems more involved.We have a line with a slope angle ( alpha ) passing through the origin ( O ) and intersecting the ellipse at points ( A ) and ( B ). Another line with a slope angle ( beta ) passes through ( F_1 ) and intersects the ellipse at points ( D ) and ( E ). It's given that ( alpha + beta = pi ). We need to find the value of ( frac{|AB|^2}{|DE|} ).Hmm, okay. Let me parse this.First, a line with slope angle ( alpha ) through the origin. The slope angle is the angle the line makes with the positive x-axis, so the slope ( m ) is ( tan(alpha) ). Similarly, the line through ( F_1 ) with slope angle ( beta ) has slope ( tan(beta) ).Given that ( alpha + beta = pi ), that means ( beta = pi - alpha ). So, the slope of the second line is ( tan(beta) = tan(pi - alpha) = -tan(alpha) ). Therefore, the two lines have slopes that are negatives of each other.So, if the first line has slope ( k = tan(alpha) ), the second line has slope ( -k ).Let me denote the first line as ( l_1 ) and the second as ( l_2 ).So, ( l_1 ) passes through the origin and has slope ( k ), so its equation is ( y = kx ).( l_2 ) passes through ( F_1(-1, 0) ) and has slope ( -k ), so its equation is ( y = -k(x + 1) ).We need to find the points where these lines intersect the ellipse ( frac{x^2}{4} + frac{y^2}{3} = 1 ).First, let's find the points ( A ) and ( B ) where ( l_1 ) intersects the ellipse.Substitute ( y = kx ) into the ellipse equation:( frac{x^2}{4} + frac{(kx)^2}{3} = 1 )Simplify:( frac{x^2}{4} + frac{k^2 x^2}{3} = 1 )Factor out ( x^2 ):( x^2 left( frac{1}{4} + frac{k^2}{3} right) = 1 )Combine the terms:( x^2 left( frac{3 + 4k^2}{12} right) = 1 )Multiply both sides by 12:( x^2 (3 + 4k^2) = 12 )So,( x^2 = frac{12}{3 + 4k^2} )Therefore,( x = pm sqrt{frac{12}{3 + 4k^2}} )So, the points ( A ) and ( B ) have coordinates:( A left( sqrt{frac{12}{3 + 4k^2}}, k sqrt{frac{12}{3 + 4k^2}} right) )and( B left( -sqrt{frac{12}{3 + 4k^2}}, -k sqrt{frac{12}{3 + 4k^2}} right) )Wait, actually, since ( y = kx ), if ( x ) is positive, ( y ) is positive, and if ( x ) is negative, ( y ) is negative. So, points ( A ) and ( B ) are symmetric with respect to the origin.Now, let's compute the distance ( |AB| ).Since ( A ) and ( B ) are diametrically opposite points on the ellipse with respect to the origin, the distance between them is twice the distance from the origin to ( A ).So, the distance from the origin to ( A ) is:( sqrt{ left( sqrt{frac{12}{3 + 4k^2}} right)^2 + left( k sqrt{frac{12}{3 + 4k^2}} right)^2 } )Simplify:( sqrt{ frac{12}{3 + 4k^2} + frac{12k^2}{3 + 4k^2} } = sqrt{ frac{12(1 + k^2)}{3 + 4k^2} } )Therefore, the distance ( |AB| ) is twice that:( |AB| = 2 sqrt{ frac{12(1 + k^2)}{3 + 4k^2} } )So, ( |AB|^2 ) is:( 4 cdot frac{12(1 + k^2)}{3 + 4k^2} = frac{48(1 + k^2)}{3 + 4k^2} )Alright, that's ( |AB|^2 ).Now, let's find ( |DE| ), the distance between points ( D ) and ( E ) where line ( l_2 ) intersects the ellipse.The equation of ( l_2 ) is ( y = -k(x + 1) ).Substitute this into the ellipse equation:( frac{x^2}{4} + frac{(-k(x + 1))^2}{3} = 1 )Simplify:( frac{x^2}{4} + frac{k^2(x + 1)^2}{3} = 1 )Expand ( (x + 1)^2 ):( x^2 + 2x + 1 )So, substitute back:( frac{x^2}{4} + frac{k^2(x^2 + 2x + 1)}{3} = 1 )Multiply through by 12 to eliminate denominators:( 3x^2 + 4k^2(x^2 + 2x + 1) = 12 )Expand:( 3x^2 + 4k^2x^2 + 8k^2x + 4k^2 = 12 )Combine like terms:( (3 + 4k^2)x^2 + 8k^2x + (4k^2 - 12) = 0 )So, we have a quadratic in ( x ):( (3 + 4k^2)x^2 + 8k^2x + (4k^2 - 12) = 0 )Let me denote this as ( Ax^2 + Bx + C = 0 ), where:( A = 3 + 4k^2 )( B = 8k^2 )( C = 4k^2 - 12 )We can find the roots ( x_1 ) and ( x_2 ) using the quadratic formula:( x = frac{-B pm sqrt{B^2 - 4AC}}{2A} )But since we need the distance between points ( D ) and ( E ), which are the two intersection points, we can use the formula for the distance between two points on a line.Alternatively, since both points lie on the line ( l_2 ), the distance ( |DE| ) can be expressed in terms of the difference in ( x )-coordinates and the slope.But perhaps a better approach is to use the distance formula based on the roots of the quadratic.Let me recall that for a quadratic equation ( Ax^2 + Bx + C = 0 ), the difference between the roots is ( sqrt{(x_1 + x_2)^2 - 4x_1x_2} ).But actually, the distance between the two points ( D ) and ( E ) can be calculated as:( |DE| = sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} )But since both points lie on the line ( y = -k(x + 1) ), we can express ( y_1 - y_2 ) in terms of ( x_1 - x_2 ):( y_1 - y_2 = -k(x_1 + 1) + k(x_2 + 1) = -k(x_1 - x_2) )Therefore,( |DE| = sqrt{(x_1 - x_2)^2 + (-k(x_1 - x_2))^2} = |x_1 - x_2| sqrt{1 + k^2} )So, ( |DE| = sqrt{1 + k^2} cdot |x_1 - x_2| )Now, ( |x_1 - x_2| ) can be found using the quadratic formula:( |x_1 - x_2| = sqrt{(x_1 + x_2)^2 - 4x_1x_2} )From the quadratic equation, we know:( x_1 + x_2 = -frac{B}{A} = -frac{8k^2}{3 + 4k^2} )( x_1 x_2 = frac{C}{A} = frac{4k^2 - 12}{3 + 4k^2} )Therefore,( |x_1 - x_2| = sqrt{left(-frac{8k^2}{3 + 4k^2}right)^2 - 4 cdot frac{4k^2 - 12}{3 + 4k^2}} )Simplify inside the square root:First, square ( x_1 + x_2 ):( left(-frac{8k^2}{3 + 4k^2}right)^2 = frac{64k^4}{(3 + 4k^2)^2} )Then, compute ( 4x_1x_2 ):( 4 cdot frac{4k^2 - 12}{3 + 4k^2} = frac{16k^2 - 48}{3 + 4k^2} )So, the expression under the square root becomes:( frac{64k^4}{(3 + 4k^2)^2} - frac{16k^2 - 48}{3 + 4k^2} )To combine these, let's get a common denominator:Multiply the second term by ( frac{3 + 4k^2}{3 + 4k^2} ):( frac{64k^4}{(3 + 4k^2)^2} - frac{(16k^2 - 48)(3 + 4k^2)}{(3 + 4k^2)^2} )Now, expand the numerator:First term: ( 64k^4 )Second term: ( -(16k^2 - 48)(3 + 4k^2) )Let me compute ( (16k^2 - 48)(3 + 4k^2) ):Multiply term by term:( 16k^2 cdot 3 = 48k^2 )( 16k^2 cdot 4k^2 = 64k^4 )( -48 cdot 3 = -144 )( -48 cdot 4k^2 = -192k^2 )So, adding these together:( 48k^2 + 64k^4 - 144 - 192k^2 = (64k^4) + (48k^2 - 192k^2) - 144 = 64k^4 - 144k^2 - 144 )Therefore, the numerator becomes:( 64k^4 - (64k^4 - 144k^2 - 144) = 64k^4 - 64k^4 + 144k^2 + 144 = 144k^2 + 144 )Factor out 144:( 144(k^2 + 1) )So, the expression under the square root is:( frac{144(k^2 + 1)}{(3 + 4k^2)^2} )Therefore,( |x_1 - x_2| = sqrt{ frac{144(k^2 + 1)}{(3 + 4k^2)^2} } = frac{12sqrt{k^2 + 1}}{3 + 4k^2} )Thus, the distance ( |DE| ) is:( |DE| = sqrt{1 + k^2} cdot frac{12sqrt{k^2 + 1}}{3 + 4k^2} = frac{12(k^2 + 1)}{3 + 4k^2} )So, now we have both ( |AB|^2 ) and ( |DE| ):( |AB|^2 = frac{48(k^2 + 1)}{3 + 4k^2} )( |DE| = frac{12(k^2 + 1)}{3 + 4k^2} )Therefore, the ratio ( frac{|AB|^2}{|DE|} ) is:( frac{frac{48(k^2 + 1)}{3 + 4k^2}}{frac{12(k^2 + 1)}{3 + 4k^2}} = frac{48}{12} = 4 )So, the value is 4, regardless of the value of ( k ) (as long as the lines intersect the ellipse, which they do because we've considered the discriminant earlier).Wait, let me double-check the discriminant part. When I substituted ( l_2 ) into the ellipse equation, I ended up with a quadratic in ( x ). I should ensure that the discriminant is positive so that there are two real intersection points.The discriminant ( D ) of the quadratic ( (3 + 4k^2)x^2 + 8k^2x + (4k^2 - 12) = 0 ) is:( D = (8k^2)^2 - 4(3 + 4k^2)(4k^2 - 12) )Compute this:( D = 64k^4 - 4(3 + 4k^2)(4k^2 - 12) )First, compute ( 4(3 + 4k^2)(4k^2 - 12) ):Multiply ( (3 + 4k^2)(4k^2 - 12) ):( 3 cdot 4k^2 = 12k^2 )( 3 cdot (-12) = -36 )( 4k^2 cdot 4k^2 = 16k^4 )( 4k^2 cdot (-12) = -48k^2 )So, adding these:( 12k^2 - 36 + 16k^4 - 48k^2 = 16k^4 - 36k^2 - 36 )Multiply by 4:( 4 cdot (16k^4 - 36k^2 - 36) = 64k^4 - 144k^2 - 144 )Therefore, the discriminant is:( D = 64k^4 - (64k^4 - 144k^2 - 144) = 64k^4 - 64k^4 + 144k^2 + 144 = 144k^2 + 144 )Which is always positive since ( 144(k^2 + 1) > 0 ) for all real ( k ). So, the quadratic always has two real roots, meaning the line ( l_2 ) always intersects the ellipse at two distinct points ( D ) and ( E ).Therefore, the ratio ( frac{|AB|^2}{|DE|} ) is always 4, regardless of the value of ( alpha ) (as long as ( alpha + beta = pi )).Let me just recap to make sure I didn't make any mistakes.1. Found the equation of the ellipse in part (1) by using the given focus and the relationship involving the geometric mean. Got ( frac{x^2}{4} + frac{y^2}{3} = 1 ).2. For part (2), considered the two lines with slopes ( k ) and ( -k ) due to ( alpha + beta = pi ). Found the intersection points with the ellipse, calculated ( |AB|^2 ) and ( |DE| ), then took their ratio, which simplified to 4.I think that's solid. I don't see any errors in the calculations.**Final Answer**(1) The equation of the curve ( C ) is boxed{dfrac{x^2}{4} + dfrac{y^2}{3} = 1}.(2) The value of ( dfrac{|AB|^2}{|DE|} ) is boxed{4}.

💡First, I need to determine the total area of the sandbox. Since it's a square, the area can be calculated by squaring the length of one side.Next, I'll calculate the area that one 30-pound bag of sand can fill. This will help me find out how many bags are required to fill the entire sandbox.Finally, by multiplying the number of bags needed by the weight of each bag, I'll find the total weight of sand required to fill the sandbox completely.

💡Okay, so I have this problem where I'm given a set A which is {0, 1, 2, ..., 9}. That means A has 10 elements, right? And I need to find the maximum number of non-empty subsets, let's call them B1, B2, ..., Bk, such that for any two different subsets Bi and Bj, the size of their intersection is at most 2. So, |Bi ∩ Bj| ≤ 2 for all i ≠ j.First, I need to understand what exactly is being asked. It's about finding the largest possible family of subsets where no two subsets share more than two elements. So, each pair of subsets can overlap by at most two elements. Got it.I think the way to approach this is to consider all possible subsets of A and then figure out which ones can coexist without violating the intersection condition. But since A has 10 elements, the total number of subsets is 2^10, which is 1024. But we're only considering non-empty subsets, so that's 1023. But obviously, we can't have all of them because many of them will intersect in more than two elements.So, maybe I should think about how to construct such a family of subsets. One idea is to limit the size of the subsets. If all subsets are small enough, say size 3 or less, then their intersections can't be too big. Let me explore that.If I consider all subsets of size 1, 2, or 3, then the maximum intersection between any two subsets would be 2, right? Because two subsets of size 3 can share at most 2 elements without being identical. Wait, actually, two subsets of size 3 can share up to 3 elements if they are identical, but since all subsets are distinct, the maximum intersection would be 2. Hmm, that seems correct.So, if I take all subsets of size 1, 2, and 3, then any two subsets will intersect in at most 2 elements. That seems to satisfy the condition. So, the total number of such subsets would be the sum of the number of subsets of size 1, size 2, and size 3.Calculating that, the number of subsets of size 1 is C(10,1) = 10. The number of subsets of size 2 is C(10,2) = 45. The number of subsets of size 3 is C(10,3) = 120. So, adding those up: 10 + 45 + 120 = 175.Wait, so does that mean the maximum k is 175? That seems high, but maybe it's correct. Let me check if there's any issue with this approach.One potential issue is whether all these subsets actually satisfy the intersection condition. For example, if I have two subsets of size 3, say {1,2,3} and {1,2,4}, their intersection is {1,2}, which has size 2. That's fine. What if I have a subset of size 1 and a subset of size 3? Their intersection can be at most 1, which is also fine. Similarly, two subsets of size 2 can intersect in at most 2 elements, which is allowed.So, it seems that all these subsets indeed satisfy the condition. Therefore, the total number of such subsets is 175, which would be the maximum k.But wait, is there a way to include more subsets without violating the intersection condition? Maybe by including some larger subsets as well? Let's think about that.If I include a subset of size 4, say {1,2,3,4}, then any other subset that shares more than 2 elements with it would violate the condition. So, I would have to ensure that no other subset in the family shares 3 or 4 elements with this subset. But since we already have all subsets of size 1, 2, and 3, there might be some overlap.For example, the subset {1,2,3} is already in our family. If we include {1,2,3,4}, then {1,2,3} and {1,2,3,4} intersect in 3 elements, which violates the condition. So, we can't include any subsets larger than size 3 without potentially violating the intersection condition with the existing subsets.Therefore, it seems that including subsets larger than size 3 is not possible without violating the condition. So, the maximum family of subsets we can have is indeed all subsets of size 1, 2, and 3, which totals 175.Wait, but let me double-check. Is there a way to include some larger subsets without overlapping too much? Maybe if we carefully select larger subsets that don't share too many elements with the smaller ones. But given that we already have all possible smaller subsets, it's likely that any larger subset would share at least three elements with some smaller subset.For example, take a larger subset like {1,2,3,4}. It shares three elements with {1,2,3}, which is already in our family. So, that's a problem. Similarly, any larger subset will share at least three elements with some subset of size 3, which is already in our family. Therefore, we can't include any larger subsets without violating the intersection condition.So, it seems that 175 is indeed the maximum number of subsets we can have under the given condition. Therefore, the maximum value of k is 175.But just to be thorough, let me consider if there's any other approach or if I'm missing something. Maybe using some combinatorial design or something like that. But I think the approach of taking all subsets of size up to 3 is straightforward and satisfies the condition.Another way to think about it is that the problem is similar to a hypergraph where edges are subsets, and we want the hypergraph to be such that any two edges intersect in at most two vertices. In hypergraph terminology, this is called a 2-intersecting hypergraph. The maximum number of edges in such a hypergraph is what we're looking for.In this case, since our hypergraph is defined on 10 vertices, and edges can be of size 1, 2, or 3, the maximum number of edges is indeed the sum of the number of subsets of size 1, 2, and 3, which is 175.So, after considering all these points, I'm confident that the maximum value of k is 175.

💡Okay, so I have this inequality: 2x + a is greater than or equal to the square root of (2x - 1), and this has to hold true for all x in the interval [1/2, infinity). I need to find the range of values for a that satisfy this condition. Hmm, let me think about how to approach this.First, maybe I can rearrange the inequality to solve for a. If I subtract 2x from both sides, I get a is greater than or equal to sqrt(2x - 1) - 2x. So, a has to be at least as big as this expression sqrt(2x - 1) - 2x for all x in that interval. That means the smallest possible a that works is the maximum value of sqrt(2x - 1) - 2x over the interval [1/2, infinity). So, I need to find the maximum of this function f(x) = sqrt(2x - 1) - 2x.Alright, to find the maximum, I should probably take the derivative of f(x) and find its critical points. Let me compute f'(x). The derivative of sqrt(2x - 1) is (1/(2*sqrt(2x - 1)))*2, which simplifies to 1/sqrt(2x - 1). Then, the derivative of -2x is -2. So, putting it together, f'(x) = 1/sqrt(2x - 1) - 2.Now, I need to find where f'(x) is zero because that's where the function could have a maximum or minimum. Setting f'(x) equal to zero: 1/sqrt(2x - 1) - 2 = 0. Let's solve for x. So, 1/sqrt(2x - 1) = 2. Taking reciprocals on both sides, sqrt(2x - 1) = 1/2. Then, squaring both sides, 2x - 1 = 1/4. Adding 1 to both sides, 2x = 5/4, so x = 5/8.Okay, so x = 5/8 is a critical point. I need to check if this is a maximum or a minimum. Let's look at the sign of f'(x) around x = 5/8. For x less than 5/8, say x = 1/2, f'(x) would be 1/sqrt(2*(1/2) - 1) - 2. Wait, 2*(1/2) is 1, so sqrt(1 - 1) is sqrt(0), which is 0. But that would make the derivative undefined. Hmm, maybe I should pick a value just above 1/2, like x = 3/4. Then, f'(3/4) = 1/sqrt(2*(3/4) - 1) - 2 = 1/sqrt(1.5 - 1) - 2 = 1/sqrt(0.5) - 2 ≈ 1.414 - 2 ≈ -0.586, which is negative. Wait, but before x = 5/8, let me try x = 0.6. Then, 2x -1 = 1.2 -1 = 0.2, so sqrt(0.2) ≈ 0.447. Then, 1/0.447 ≈ 2.236. So, f'(0.6) ≈ 2.236 - 2 ≈ 0.236, which is positive. So, before x = 5/8, the derivative is positive, and after x = 5/8, it becomes negative. That means the function f(x) is increasing before x = 5/8 and decreasing after x = 5/8, so x = 5/8 is a maximum point.Therefore, the maximum value of f(x) is at x = 5/8. Let me compute f(5/8). So, sqrt(2*(5/8) -1) - 2*(5/8). Simplify inside the square root: 10/8 -1 = 10/8 - 8/8 = 2/8 = 1/4. So, sqrt(1/4) is 1/2. Then, 2*(5/8) is 10/8 = 5/4. So, f(5/8) = 1/2 - 5/4 = (2/4 - 5/4) = -3/4.So, the maximum value of f(x) is -3/4. That means a must be greater than or equal to -3/4 for the inequality to hold for all x in [1/2, infinity). So, the range of a is [-3/4, infinity). Looking at the options, that's option D.Wait, let me double-check. If a is -3/4, then the inequality becomes 2x - 3/4 >= sqrt(2x -1). At x = 5/8, we have 2*(5/8) - 3/4 = 5/4 - 3/4 = 2/4 = 1/2, and sqrt(2*(5/8) -1) is sqrt(1/4) = 1/2. So, equality holds there. For x > 5/8, since f(x) is decreasing, sqrt(2x -1) - 2x becomes less than -3/4, so 2x + a >= sqrt(2x -1) would still hold because a is at least -3/4. For x approaching infinity, 2x dominates, so the inequality definitely holds. For x just above 1/2, let's say x = 1/2, then 2x = 1, sqrt(2x -1) = 0, so 1 + a >= 0. If a = -3/4, then 1 - 3/4 = 1/4 >= 0, which is true. So, yeah, it seems correct.**Final Answer**boxed{D}