answer：Okay, so Jim gets paid every two weeks, and his gross pay is 1120. From this amount, they take out some money for taxes and some for his retirement account. After these deductions, Jim is left with 740 in his paycheck. I need to figure out what percentage of his paycheck goes into his retirement account.First, I should probably figure out how much money is being deducted in total from his gross pay. If his gross pay is 1120 and he's left with 740, then the total deductions must be 1120 minus 740. Let me write that down:Total deductions = Gross pay - Net payTotal deductions = 1120 - 740Total deductions = 380So, 380 is being taken out of his paycheck each bi-weekly period for taxes and retirement. I know that 100 of this is for taxes, so the rest must be for his retirement account. Let me subtract the tax amount from the total deductions to find out how much goes to retirement:Retirement deduction = Total deductions - Tax deductionRetirement deduction = 380 - 100Retirement deduction = 280Now, I need to find out what percentage 280 is of his gross pay of 1120. To find a percentage, I can use the formula:Percentage = (Part / Whole) * 100In this case, the "Part" is the retirement deduction (280), and the "Whole" is the gross pay (1120). Plugging in the numbers:Percentage = (280 / 1120) * 100Let me calculate that. First, divide 280 by 1120:280 ÷ 1120 = 0.25Then, multiply by 100 to get the percentage:0.25 * 100 = 25%So, 25% of Jim's paycheck goes into his retirement account. That seems straightforward, but let me double-check my calculations to make sure I didn't make any mistakes.Starting with the gross pay of 1120, if 25% goes to retirement, that's:25% of 1120 = 0.25 * 1120 = 280Then, adding the tax deduction of 100, the total deductions are 280 + 100 = 380. Subtracting that from the gross pay:1120 - 380 = 740Which matches the net pay given in the problem. So, my calculations seem correct. Therefore, the percentage of Jim's paycheck that goes into his retirement account is 25%.

answer：Okay, so I have this triangle ABC, and the angles A, B, and C form an arithmetic sequence. That means each angle increases by the same amount. Also, I know the side lengths a = 8 and b = 7. I need to find cos C. Hmm, let me think about how to approach this.First, since the angles are in an arithmetic sequence, they can be expressed as A, B, and C where B is the middle term. So, if I let the common difference be d, then A = B - d and C = B + d. Since the sum of angles in a triangle is 180 degrees, I can write:A + B + C = 180Substituting A and C in terms of B:(B - d) + B + (B + d) = 180Simplifying that:3B = 180So, B = 60 degrees. That makes sense because in an arithmetic sequence with three terms, the middle term is the average. So, angle B is 60 degrees.Now, I need to relate the sides and angles. I know sides a = 8 and b = 7. In triangle ABC, side a is opposite angle A, and side b is opposite angle B. So, using the Law of Sines, which states:a / sin A = b / sin B = c / sin CI can set up the ratio for sides a and b:8 / sin A = 7 / sin 60I know that sin 60 is √3 / 2, so plugging that in:8 / sin A = 7 / (√3 / 2) = 14 / √3So, 8 / sin A = 14 / √3Cross-multiplying:8 * √3 = 14 * sin ASo, sin A = (8 * √3) / 14 = (4 * √3) / 7Okay, so sin A is (4√3)/7. Now, I can find cos A using the Pythagorean identity:sin² A + cos² A = 1So, cos² A = 1 - sin² A = 1 - (16 * 3) / 49 = 1 - 48/49 = 1/49Therefore, cos A = ±1/7Hmm, so cos A could be positive or negative. That depends on whether angle A is acute or obtuse. Since the sum of angles in a triangle is 180, and angle B is 60, angles A and C must add up to 120. If angle A is obtuse, then angle C would be less than 60, which might not make sense because side lengths are given. Let me think about that.Wait, side a is 8, which is longer than side b, which is 7. In a triangle, the longer side is opposite the larger angle. So, since a > b, angle A > angle B. Since angle B is 60, angle A must be greater than 60. But angle A is also part of an arithmetic sequence with angle B and angle C.Given that the angles are in arithmetic progression, and angle B is 60, if angle A is greater than 60, then angle C must be greater than 60 as well. Wait, but angle A is greater than 60, angle B is 60, and angle C is greater than 60. But their sum is 180, so 60 + something + something else = 180. If both A and C are greater than 60, then 60 + 60 + 60 = 180, but A and C are more than 60, which would make the total more than 180. That can't be.Wait, that doesn't make sense. So, maybe angle A is less than 60? But side a is longer than side b, so angle A should be larger than angle B. Hmm, this is confusing.Wait, maybe I made a mistake in assuming the order of the arithmetic sequence. If the angles are in arithmetic progression, they could be increasing or decreasing. So, perhaps angle A is the smallest, angle B is in the middle, and angle C is the largest. So, A < B < C.Given that, since side a is opposite angle A, and side a is 8, which is longer than side b, which is 7, opposite angle B. So, if a > b, then angle A > angle B. But if angle A is greater than angle B, and angle B is 60, then angle A is more than 60, and angle C is even more. But that would make the sum of angles exceed 180, which is impossible.Wait, that can't be. So, maybe the arithmetic sequence is decreasing? So, angle A is the largest, angle B is in the middle, and angle C is the smallest. So, A > B > C.In that case, side a is opposite angle A, which is the largest, so a is the longest side. But in the problem, side a is 8, and side b is 7. So, if a is the longest side, then side c must be less than 8. But we don't know side c yet.Wait, but in this case, angle A is the largest, angle B is 60, and angle C is the smallest. So, angle A = 60 + d, angle B = 60, angle C = 60 - d. Then, the sum is 180, which is correct.So, angle A is 60 + d, angle C is 60 - d. Since side a is opposite angle A, which is larger than angle B, so a > b, which is given as 8 > 7. So, that makes sense.So, angle A is 60 + d, angle C is 60 - d. Then, using the Law of Sines, we can relate sides a, b, and c.We have:a / sin A = b / sin B = c / sin CWe already found sin A = (4√3)/7, which is approximately 0.9798. Wait, but sin A can't be more than 1. Let me check my calculation.Wait, sin A = (4√3)/7. Let's compute that:√3 ≈ 1.732, so 4 * 1.732 ≈ 6.9286.928 / 7 ≈ 0.9897, which is less than 1, so that's okay.So, sin A ≈ 0.9897, which is close to 1, so angle A is close to 90 degrees. So, angle A is 60 + d ≈ 90, so d ≈ 30. Then, angle C would be 60 - 30 = 30 degrees. Let me check if that works.If angle A is 90, angle B is 60, angle C is 30, sum is 180, which is correct. Then, sides would be in the ratio of sin 90 : sin 60 : sin 30, which is 1 : √3/2 : 1/2. So, multiplying by 2, we get 2 : √3 : 1. So, sides opposite 90, 60, 30 would be 2, √3, 1.But in our case, sides a = 8 and b = 7. So, let's see if 8 / 7 is approximately equal to 2 / √3. 2 / √3 ≈ 1.1547, and 8 / 7 ≈ 1.1429, which is close but not exact. So, maybe angle A is slightly less than 90, and angle C slightly more than 30.But let's proceed with exact values.We have sin A = (4√3)/7, so angle A is arcsin((4√3)/7). Let me compute that.But maybe I don't need to compute the angle itself. Instead, since I need cos C, perhaps I can find it using some trigonometric identities.Since angle A + angle B + angle C = 180, and angle B = 60, so angle A + angle C = 120. So, angle C = 120 - angle A.Therefore, cos C = cos(120 - A). Using the cosine of difference identity:cos(120 - A) = cos 120 cos A + sin 120 sin AWe know cos 120 = -1/2, sin 120 = √3/2.So, cos C = (-1/2) cos A + (√3/2) sin AWe already have sin A = (4√3)/7, and we found cos A = ±1/7.But we need to determine whether cos A is positive or negative.Since angle A is in a triangle, it must be between 0 and 180 degrees. Given that angle A is part of an arithmetic sequence with angle B = 60 and angle C, and angle A is larger than angle B (since side a > side b), angle A must be greater than 60 degrees. So, angle A is between 60 and 120 degrees.In this range, cosine is positive from 0 to 90 degrees and negative from 90 to 180 degrees. Since angle A is greater than 60, it could be in either the first or second quadrant. But given that sin A is (4√3)/7 ≈ 0.9897, which is close to 1, angle A is close to 90 degrees. So, cos A would be positive if angle A is less than 90, and negative if it's more than 90.But wait, if angle A is 60 + d, and angle C is 60 - d, and angle A + angle C = 120, then if angle A is greater than 90, angle C would be less than 30. But let's see.If angle A is greater than 90, then angle C = 120 - angle A would be less than 30. But side c is opposite angle C, and side b is 7 opposite angle B = 60. So, if angle C is less than 30, side c would be less than side b, which is 7. But we don't know side c yet.Alternatively, if angle A is less than 90, then angle C is greater than 30, and side c would be longer than some value.But let's see, if angle A is 60 + d, and sin A = (4√3)/7 ≈ 0.9897, then angle A is approximately arcsin(0.9897) ≈ 81.79 degrees. So, angle A is approximately 81.79 degrees, which is less than 90. Therefore, cos A is positive.So, cos A = 1/7.Therefore, cos C = (-1/2)(1/7) + (√3/2)(4√3/7)Let me compute that:First term: (-1/2)(1/7) = -1/14Second term: (√3/2)(4√3/7) = (4 * 3)/14 = 12/14 = 6/7So, cos C = -1/14 + 6/7 = -1/14 + 12/14 = 11/14Wait, but that would make cos C positive, which would mean angle C is acute. But angle C is 60 - d, and if angle A is approximately 81.79, then d ≈ 21.79, so angle C ≈ 38.21 degrees, which is acute. So, cos C is positive, which matches.But wait, in the earlier step, I thought angle C was 60 - d, which would be less than 60, but in this case, it's approximately 38.21, which is still acute. So, cos C is positive.But wait, in the initial problem, the angles are in arithmetic progression, so if angle A is 81.79, angle B is 60, angle C is 38.21, which is an arithmetic sequence with common difference approximately -21.79. So, that works.But wait, earlier I thought that if angle A is greater than 60, angle C would have to be less than 60, but in this case, it's still okay because angle C is 38.21, which is less than 60, but the sum is still 180.Wait, but in my initial calculation, I got cos C = 11/14, which is positive. But in the problem, the user initially thought it might be negative. Let me double-check.Wait, in the problem, the user wrote:"Using the identity for the cosine of the sum of angles. Since C = 180 - (A + 60) and cos(180 - θ) = -cos(θ), we have: cos C = cos(A + 60)"Wait, that's a different approach. So, if C = 180 - (A + 60), then cos C = -cos(A + 60). So, cos C = -cos(A + 60). Hmm, that's another way to look at it.So, let's try that approach.Since C = 180 - (A + B) = 180 - (A + 60) = 120 - A.Wait, but earlier I used C = 120 - A, and then expressed cos C as cos(120 - A). But the user is saying cos C = -cos(A + 60). Let me see.Wait, 180 - (A + 60) = 120 - A, so cos(180 - (A + 60)) = -cos(A + 60). So, cos C = -cos(A + 60). So, that's another way to write it.So, cos C = -cos(A + 60). Let's compute that.Using the cosine addition formula:cos(A + 60) = cos A cos 60 - sin A sin 60We have cos A = 1/7, sin A = 4√3/7, cos 60 = 1/2, sin 60 = √3/2.So,cos(A + 60) = (1/7)(1/2) - (4√3/7)(√3/2) = (1/14) - (12/14) = (1 - 12)/14 = -11/14Therefore, cos C = -cos(A + 60) = -(-11/14) = 11/14Wait, so that's consistent with my earlier result. So, cos C = 11/14.But in the initial problem, the user had two possible values, -11/14 and -13/14, but that was because they considered cos A as ±1/7. But in reality, since angle A is less than 90, cos A is positive, so only 1/7 is valid. Therefore, cos C = 11/14.Wait, but in the initial problem, the user concluded that cos C could be -11/14 or -13/14, but that was based on considering both positive and negative cos A, which isn't correct because angle A is less than 90, so cos A is positive. Therefore, the correct value is 11/14.But wait, let me double-check.If angle A is approximately 81.79 degrees, then angle C is 120 - 81.79 ≈ 38.21 degrees. So, cos C ≈ cos(38.21) ≈ 0.785, which is approximately 11/14 ≈ 0.7857. So, that matches.Therefore, the correct value is 11/14.But wait, in the initial problem, the user had a different approach and got two possible values, but that was because they didn't consider the correct sign for cos A. So, the correct answer is 11/14.Wait, but in the initial problem, the user wrote:"Using the identity for the cosine of the sum of angles. Since C = 180 - (A + 60) and cos(180 - θ) = -cos(θ), we have: cos C = cos(A + 60)"Wait, that's incorrect. Because C = 180 - (A + 60), so cos C = cos(180 - (A + 60)) = -cos(A + 60). So, cos C = -cos(A + 60). Therefore, the user made a mistake in their initial step by writing cos C = cos(A + 60), but it should be cos C = -cos(A + 60).Therefore, in their calculation, they should have:cos C = - [ (1/7)(1/2) - (4√3/7)(√3/2) ] = - [ (1/14) - (12/14) ] = - [ -11/14 ] = 11/14So, the correct value is 11/14.Therefore, the initial problem's conclusion of -11/14 and -13/14 was incorrect because they didn't account for the negative sign properly and considered both positive and negative cos A, which isn't valid in this case.So, the correct answer is 11/14.

answer：Okay, so I have this equation to solve: 16 raised to the power of (2x minus 3) equals (1/2) raised to the power of (x plus 8). Hmm, that looks a bit complicated, but maybe I can simplify it by expressing both sides with the same base. Let me think. I know that 16 is a power of 2. Specifically, 16 is 2 to the fourth power because 2 times 2 is 4, times 2 is 8, times 2 is 16. So, 16 is 2^4. That means 16^(2x - 3) can be rewritten as (2^4)^(2x - 3). Now, when you raise a power to another power, you multiply the exponents. So, (2^4)^(2x - 3) becomes 2^(4*(2x - 3)). Let me calculate that exponent: 4 times 2x is 8x, and 4 times -3 is -12. So, the left side simplifies to 2^(8x - 12).Alright, now the right side of the equation is (1/2)^(x + 8). I remember that 1/2 is the same as 2 to the power of -1 because 2^-1 equals 1/2. So, (1/2)^(x + 8) can be rewritten as (2^-1)^(x + 8). Again, when raising a power to another power, I multiply the exponents. So, (2^-1)^(x + 8) becomes 2^(-1*(x + 8)). Let me compute that exponent: -1 times x is -x, and -1 times 8 is -8. So, the right side simplifies to 2^(-x - 8).Now, my equation looks like this: 2^(8x - 12) equals 2^(-x - 8). Since both sides have the same base, which is 2, I can set the exponents equal to each other. That gives me the equation 8x - 12 = -x - 8.Okay, time to solve for x. Let me get all the x terms on one side and the constants on the other. I'll start by adding x to both sides to get rid of the -x on the right. Adding x to both sides gives me 8x + x - 12 = -8. Simplifying that, 9x - 12 = -8.Now, I'll add 12 to both sides to isolate the term with x. Adding 12 to both sides gives me 9x = 4. To solve for x, I'll divide both sides by 9. So, x equals 4 divided by 9, which is 4/9.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the beginning, I converted both sides to base 2 correctly. Then, I set the exponents equal because the bases were the same. Solving the linear equation, I added x to both sides, then added 12, and finally divided by 9. It seems correct. Just to be thorough, I'll plug x = 4/9 back into the original equation to verify. Let's compute both sides:Left side: 16^(2*(4/9) - 3) = 16^(8/9 - 27/9) = 16^(-19/9). Since 16 is 2^4, this becomes (2^4)^(-19/9) = 2^(-76/9).Right side: (1/2)^(4/9 + 8) = (1/2)^(4/9 + 72/9) = (1/2)^(76/9) = (2^-1)^(76/9) = 2^(-76/9).Both sides equal 2^(-76/9), so the solution x = 4/9 is correct.I think I've got it. The key was expressing both sides with the same base and then setting the exponents equal. It's important to remember the rules of exponents, especially when dealing with equations like this. I was a bit nervous about handling the negative exponents and fractions, but breaking it down step by step helped me manage it without getting confused.

answer：Okay, so I have this problem here where I'm given the universal set U, which is {1, 3}, and another set A, which is also {1, 3}. I need to find the complement of A with respect to U, denoted as C_UA. Hmm, let's break this down step by step.First, I remember that the complement of a set A with respect to the universal set U is the set of all elements that are in U but not in A. So, basically, it's like taking everything in U and removing the elements that are also in A. The result should be the elements that are exclusively in U and not shared with A.Now, looking at the sets given: U is {1, 3} and A is also {1, 3}. Wait a minute, they're exactly the same. So, if A has all the elements that U has, then there shouldn't be any elements left in U that aren't in A, right? That means when I try to find the complement, I'm essentially trying to find elements that are in U but not in A, but since A has everything U has, there are no such elements.Let me think about this more carefully. If I list out the elements of U, they are 1 and 3. Now, if I list out the elements of A, they are also 1 and 3. So, when I try to subtract A from U, I'm subtracting the same elements. Mathematically, that would look like U A = {1, 3} {1, 3}. Since both sets have the same elements, subtracting them would leave me with an empty set.Is there a formal way to represent this? I recall that the complement of A with respect to U is written as C_UA or sometimes as A^c. The formula for the complement is C_UA = U A. So, plugging in the values, C_UA = {1, 3} {1, 3} = empty set.Wait, but what exactly is the empty set? It's a set that contains no elements, right? It's like a set with nothing in it. So, in this case, since there are no elements in U that aren't in A, the complement is just an empty set.Let me double-check to make sure I'm not missing something. If A were a subset of U, then the complement would be all elements in U not in A. But since A is equal to U, there are no elements in U that aren't in A. Therefore, the complement must indeed be empty.Is there any other way to think about this? Maybe using Venn diagrams? If I imagine a Venn diagram where the universal set U is represented by a circle containing elements 1 and 3, and set A is another circle overlapping completely with U because they have the same elements. The complement of A would be the area inside U but outside A. But since A covers the entire U, there's no area left outside of A within U. So, visually, the complement would be empty.Another way to look at it is through the definition of a complement. The complement of A in U is the set of all elements that are in U and not in A. Symbolically, that's {x ∈ U | x ∉ A}. Since every element in U is also in A, there are no elements x that satisfy x ∈ U and x ∉ A. Therefore, the set is empty.I think I've covered this from multiple angles: set operations, Venn diagrams, and formal definitions. All of them lead me to the conclusion that the complement of A with respect to U is the empty set.Just to be thorough, let's consider what would happen if A were not equal to U. Suppose A was {1}, then the complement would be {3}. If A was {3}, the complement would be {1}. But in this specific case, since A is exactly U, the complement has to be empty.So, in summary, because A and U are identical sets, there are no elements in U that are not in A. Therefore, the complement of A with respect to U is the empty set.

answer：First, determine the number of crabs Tom catches each day by multiplying the number of crabs per bucket by the number of buckets: 12 crabs/bucket * 8 buckets = 96 crabs/day.Next, calculate the daily earnings by multiplying the number of crabs caught each day by the selling price per crab: 96 crabs/day * 5/crab = 480/day.Finally, find the weekly earnings by multiplying the daily earnings by the number of days in a week: 480/day * 7 days = 3,360.

answer：Alright, so I've got this problem about a hyperbola. Let me try to figure it out step by step. First, part (Ⅰ) asks for the standard equation of the hyperbola C. They've given me the asymptotic equations, which are y = ±(√3)/3 x. Hmm, okay, so for a hyperbola centered at the origin, the standard form is either x²/a² - y²/b² = 1 or y²/a² - x²/b² = 1. Since the asymptotes are y = ±(b/a)x, right? So here, the asymptotes are y = ±(√3)/3 x, which means that b/a = √3/3. So, if I let b/a = √3/3, that implies that b = (√3/3) a. That's one equation. They also told me that the right focus is at (2, 0). For a hyperbola, the distance from the center to each focus is c, and c² = a² + b². Since the focus is at (2, 0), that means c = 2. So, c² = 4 = a² + b². Now, since I have b in terms of a, I can substitute that into the equation. So, b = (√3/3)a, so b² = (3/9)a² = (1/3)a². Therefore, plugging into c² = a² + b², we get 4 = a² + (1/3)a². Combining like terms, that's 4 = (4/3)a². So, a² = 4 * (3/4) = 3. Therefore, a² = 3, so a = √3. Then, since b² = (1/3)a², that would be b² = (1/3)*3 = 1. So, b² = 1, meaning b = 1. Therefore, the standard equation of the hyperbola is x²/3 - y²/1 = 1, which simplifies to x²/3 - y² = 1. Okay, that seems straightforward. Let me just double-check. The asymptotes are y = ±(b/a)x, which would be y = ±(1/√3)x, which is the same as y = ±(√3/3)x. Yep, that matches. And c² = a² + b² = 3 + 1 = 4, so c = 2, which is the focus. Perfect. So, part (Ⅰ) is done. The standard equation is x²/3 - y² = 1.Now, moving on to part (Ⅱ). It says that the line l: y = kx + √2 always intersects the hyperbola C at two distinct points A and B, and the dot product of vectors OA and OB is greater than 0. I need to find the range of the real number k.Alright, let's break this down. First, the line intersects the hyperbola at two distinct points. So, substituting y = kx + √2 into the hyperbola equation x²/3 - y² = 1 should give me a quadratic equation in x, which will have two distinct real roots. So, substituting y into the hyperbola equation:x²/3 - (kx + √2)² = 1Let me expand that:x²/3 - [k²x² + 2k√2 x + 2] = 1Simplify term by term:x²/3 - k²x² - 2k√2 x - 2 = 1Combine like terms:(1/3 - k²)x² - 2k√2 x - 3 = 0So, that's a quadratic in x: (1/3 - k²)x² - 2k√2 x - 3 = 0For this quadratic to have two distinct real roots, the discriminant must be positive. The discriminant D is [(-2k√2)^2 - 4*(1/3 - k²)*(-3)].Calculating D:D = (8k²) - 4*(1/3 - k²)*(-3)Simplify the second term:4*(1/3 - k²)*(-3) = -12*(1/3 - k²) = -4 + 12k²So, D = 8k² - (-4 + 12k²) = 8k² + 4 - 12k² = -4k² + 4For two distinct real roots, D > 0:-4k² + 4 > 0Divide both sides by -4 (remembering to flip the inequality):k² - 1 < 0Which implies:k² < 1So, k must be between -1 and 1.But wait, also, the coefficient of x² in the quadratic equation is (1/3 - k²). For the quadratic to be valid (i.e., not a linear equation), this coefficient must not be zero. So,1/3 - k² ≠ 0 => k² ≠ 1/3 => k ≠ ±√(1/3) = ±(√3)/3So, combining these, k must satisfy |k| < 1 and k ≠ ±√3/3.So, the possible k values are (-1, -√3/3) ∪ (-√3/3, √3/3) ∪ (√3/3, 1). But wait, the problem says the line always intersects the hyperbola at two distinct points. So, we have that condition.But there's another condition: the dot product OA · OB > 0. So, I need to compute OA · OB and set that greater than 0.Let me denote points A and B as (x₁, y₁) and (x₂, y₂). Then, OA · OB = x₁x₂ + y₁y₂.Since both points lie on the line y = kx + √2, we can express y₁ and y₂ in terms of x₁ and x₂:y₁ = kx₁ + √2y₂ = kx₂ + √2So, OA · OB = x₁x₂ + (kx₁ + √2)(kx₂ + √2)Let me expand that:= x₁x₂ + [k²x₁x₂ + k√2 x₁ + k√2 x₂ + 2]Combine like terms:= x₁x₂ + k²x₁x₂ + k√2(x₁ + x₂) + 2Factor:= (1 + k²)x₁x₂ + k√2(x₁ + x₂) + 2Now, from the quadratic equation earlier, we can find x₁ + x₂ and x₁x₂ using Vieta's formulas.Given quadratic equation: (1/3 - k²)x² - 2k√2 x - 3 = 0So, sum of roots x₁ + x₂ = [2k√2] / (1/3 - k²)Product of roots x₁x₂ = (-3) / (1/3 - k²)Let me write that down:x₁ + x₂ = (2k√2) / (1/3 - k²)x₁x₂ = (-3) / (1/3 - k²)So, plugging these into OA · OB:= (1 + k²)(-3)/(1/3 - k²) + k√2*(2k√2)/(1/3 - k²) + 2Simplify each term:First term: (1 + k²)(-3)/(1/3 - k²) = (-3)(1 + k²)/(1/3 - k²)Second term: k√2*(2k√2)/(1/3 - k²) = (2k²*2)/(1/3 - k²) = (4k²)/(1/3 - k²)Third term: +2So, putting it all together:= [ (-3)(1 + k²) + 4k² ] / (1/3 - k²) + 2Simplify numerator:-3 - 3k² + 4k² = -3 + k²So, now:= ( -3 + k² ) / (1/3 - k² ) + 2Let me write 2 as 2*(1/3 - k²)/(1/3 - k²) to combine the terms:= [ (-3 + k² ) + 2*(1/3 - k²) ] / (1/3 - k² )Compute numerator:-3 + k² + 2/3 - 2k² = (-3 + 2/3) + (k² - 2k²) = (-7/3) - k²So, numerator is (-7/3 - k²), denominator is (1/3 - k²)Thus, OA · OB = (-7/3 - k²)/(1/3 - k²)We need OA · OB > 0, so:(-7/3 - k²)/(1/3 - k²) > 0Let me rewrite the inequality:( -7/3 - k² ) / (1/3 - k² ) > 0Factor out a negative sign from numerator and denominator:= [ - (7/3 + k²) ] / [ - (k² - 1/3) ] > 0The negatives cancel out:= (7/3 + k²) / (k² - 1/3) > 0So, the inequality becomes:(7/3 + k²)/(k² - 1/3) > 0Now, let's analyze this inequality. The numerator is 7/3 + k², which is always positive because k² is non-negative. So, numerator > 0 for all real k.Therefore, the sign of the entire expression depends on the denominator: k² - 1/3.So, for the fraction to be positive, denominator must be positive as well. Because numerator is always positive.Thus:k² - 1/3 > 0 => k² > 1/3 => |k| > √(1/3) => |k| > √3/3So, combining this with the earlier condition that |k| < 1 and k ≠ ±√3/3.Therefore, the range of k is:(-1, -√3/3) ∪ (√3/3, 1)Because k must satisfy both |k| < 1 and |k| > √3/3.Wait, let me confirm that. So, from the discriminant condition, we have |k| < 1 and k ≠ ±√3/3. From the dot product condition, we have |k| > √3/3. So, combining these, k must be in (-1, -√3/3) or (√3/3, 1).Yes, that makes sense.So, the range of k is (-1, -√3/3) ∪ (√3/3, 1).Let me just recap:1. Found the standard equation of the hyperbola by using asymptotes and foci.2. For part (Ⅱ), substituted the line into the hyperbola to get a quadratic equation, found conditions for two distinct intersections (discriminant > 0 and coefficient ≠ 0).3. Then, computed the dot product OA · OB in terms of the roots of the quadratic, used Vieta's formulas to express it in terms of k.4. Simplified the expression and found the inequality condition, leading to |k| > √3/3.5. Combined with the earlier condition |k| < 1, resulting in the range (-1, -√3/3) ∪ (√3/3, 1).I think that's solid. I don't see any mistakes in the steps, so I feel confident about this solution.**Final Answer**Ⅰ. The standard equation of the hyperbola is boxed{dfrac{x^2}{3} - y^2 = 1}.Ⅱ. The range of the real number ( k ) is boxed{(-1, -dfrac{sqrt{3}}{3}) cup (dfrac{sqrt{3}}{3}, 1)}.