answer：Okay, so I have this problem here where I need to figure out the positional relationship between a line given in polar coordinates and a circle given in Cartesian coordinates. The options are that they intersect, are tangent, are disjoint, or it's uncertain. Let me see how I can approach this.First, the line is given in polar form: ρ sin(θ + π/4) = √2. I remember that polar equations can sometimes be converted to Cartesian coordinates to make them easier to work with, especially when dealing with circles and lines. Maybe I should try converting this polar equation to Cartesian form.I recall that in polar coordinates, ρ sin(θ + α) can be expanded using the sine addition formula. So, sin(θ + π/4) would be sinθ cos(π/4) + cosθ sin(π/4). Since cos(π/4) and sin(π/4) are both √2/2, this simplifies to (√2/2)(sinθ + cosθ). So, substituting back into the equation, we have ρ*(√2/2)(sinθ + cosθ) = √2. If I multiply both sides by 2/√2 to simplify, that cancels out the √2 on the right side, leaving us with ρ(sinθ + cosθ) = 2.Now, in Cartesian coordinates, ρ sinθ is y and ρ cosθ is x. So, substituting those in, the equation becomes x + y = 2. Okay, so the line l in Cartesian coordinates is x + y = 2.Next, the circle C is given by (x - 1)^2 + y^2 = 1. That's a standard circle equation centered at (1, 0) with a radius of 1.Now, to find the positional relationship between the line and the circle, I need to determine if they intersect, are tangent, or are disjoint. For that, I can calculate the distance from the center of the circle to the line and compare it with the radius.The formula for the distance from a point (x0, y0) to the line Ax + By + C = 0 is |Ax0 + By0 + C| / sqrt(A^2 + B^2). In our case, the line is x + y = 2, which can be rewritten as x + y - 2 = 0. So, A = 1, B = 1, and C = -2. The center of the circle is (1, 0), so x0 = 1 and y0 = 0.Plugging these into the distance formula: |1*1 + 1*0 - 2| / sqrt(1^2 + 1^2) = |1 - 2| / sqrt(2) = | -1 | / sqrt(2) = 1 / sqrt(2). Simplifying that, 1 / sqrt(2) is approximately 0.707. The radius of the circle is 1. Since 0.707 is less than 1, the distance from the center to the line is less than the radius. That means the line intersects the circle at two points.Wait, but just to make sure I didn't make a mistake. Let me verify the calculations again. The distance is |1 + 0 - 2| / sqrt(2) which is |-1| / sqrt(2) = 1 / sqrt(2). Yep, that's correct. And since 1 / sqrt(2) is about 0.707, which is less than 1, the radius, so the line must intersect the circle.Alternatively, I could solve the system of equations to see if there are real solutions. Let's try that too for confirmation.We have the line x + y = 2, so y = 2 - x. Substitute that into the circle equation: (x - 1)^2 + (2 - x)^2 = 1.Expanding both terms: (x^2 - 2x + 1) + (4 - 4x + x^2) = 1.Combine like terms: x^2 - 2x + 1 + 4 - 4x + x^2 = 1.So, 2x^2 - 6x + 5 = 1.Subtract 1 from both sides: 2x^2 - 6x + 4 = 0.Divide the entire equation by 2: x^2 - 3x + 2 = 0.Factor: (x - 1)(x - 2) = 0.So, x = 1 or x = 2.Substituting back into y = 2 - x: If x = 1, y = 1; if x = 2, y = 0.So, the points of intersection are (1, 1) and (2, 0). That confirms that there are two points where the line intersects the circle, so they are indeed intersecting.Therefore, the correct answer should be A: Intersect.**Final Answer**boxed{A}

answer：To find the area of the triangle formed by the lines ( y = 4 ), ( y = 2 + x ), and ( y = 2 - x ), we first determine the points of intersection of these lines.1. **Intersection of ( y = 4 ) and ( y = 2 + x ):** [ 4 = 2 + x implies x = 2 ] Thus, the intersection point is ( (2, 4) ).2. **Intersection of ( y = 4 ) and ( y = 2 - x ):** [ 4 = 2 - x implies x = -2 ] Thus, the intersection point is ( (-2, 4) ).3. **Intersection of ( y = 2 + x ) and ( y = 2 - x ):** [ 2 + x = 2 - x implies x = 0 ] Substituting ( x = 0 ) into ( y = 2 + x ), we get ( y = 2 ). Thus, the intersection point is ( (0, 2) ).The vertices of the triangle are ( (2, 4) ), ( (-2, 4) ), and ( (0, 2) ).Next, we use the Shoelace Theorem to find the area of the triangle. The Shoelace Theorem states that the area of a polygon with vertices ((x_1, y_1), (x_2, y_2), ldots, (x_n, y_n)) is given by:[text{Area} = frac{1}{2} left| sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) right|]Applying this to our vertices:[text{Area} = frac{1}{2} left| (2 cdot 4 + (-2) cdot 2 + 0 cdot 4) - (4 cdot (-2) + 2 cdot 0 + 4 cdot 2) right|][= frac{1}{2} left| (8 - 4 + 0) - (-8 + 0 + 8) right|][= frac{1}{2} left| 4 - 0 right|][= frac{1}{2} times 4 = 2]However, upon re-evaluation, we can also calculate the area using base and height. The base of the triangle is the distance between ( (2, 4) ) and ( (-2, 4) ), which is ( 4 ). The height is the vertical distance from ( (0, 2) ) to the line ( y = 4 ), which is ( 2 ). Thus, the area is:[text{Area} = frac{1}{2} times 4 times 2 = 4]Therefore, the area of the triangle is (boxed{textbf{(B)}}).

answer：Okay, so I have this problem about a rhombus EFGH, and I need to find the measure of angle H when angle F is 130 degrees. Hmm, let me think about what I know regarding rhombuses.First off, I remember that a rhombus is a type of quadrilateral where all four sides are equal in length. That's one of the defining characteristics. But what about the angles? I think in a rhombus, opposite angles are equal. So, if angle F is 130 degrees, then angle H, which is opposite to angle F, should also be 130 degrees. That seems straightforward.Wait, but just to make sure I'm not making a mistake, let me recall more properties of rhombuses. I know that in a rhombus, the adjacent angles are supplementary. That means they add up to 180 degrees. So, if angle F is 130 degrees, then the adjacent angles, which would be angles E and G, should each be 50 degrees because 130 plus 50 equals 180. Let me visualize the rhombus EFGH. Let's say the vertices are labeled in order: E, F, G, H. So, angle F is at vertex F, and angle H is at vertex H. Since it's a rhombus, the sides EF, FG, GH, and HE are all equal. Now, if angle F is 130 degrees, then angle H, being opposite, should indeed be 130 degrees. But just to double-check, let's consider the sum of all interior angles in a quadrilateral. I remember that the sum of the interior angles in any quadrilateral is 360 degrees. So, if angle F is 130 degrees and angle H is also 130 degrees, then the remaining two angles, E and G, should each be 50 degrees because 130 plus 130 is 260, and 360 minus 260 leaves 100 degrees, which is split equally between angles E and G, giving 50 degrees each. That makes sense because in a rhombus, not only are opposite angles equal, but also the adjacent angles are supplementary. So, angle F (130 degrees) and angle E (50 degrees) add up to 180 degrees, which fits the definition. Similarly, angle G (50 degrees) and angle H (130 degrees) also add up to 180 degrees. I think I'm confident now that angle H is 130 degrees. But just to make sure I haven't overlooked anything, let me think about the diagonals of a rhombus. I remember that the diagonals of a rhombus bisect each other at right angles. So, if I were to draw the diagonals, they would intersect at 90 degrees and split each other into two equal parts. But does that affect the angles? Well, the diagonals also bisect the angles of the rhombus. So, if angle F is 130 degrees, then the diagonal from F would split it into two angles of 65 degrees each. Similarly, angle H, being 130 degrees, would also be split into two angles of 65 degrees each by the diagonal. This further confirms that angle H is indeed 130 degrees because the properties of the diagonals align with the angle measures we've calculated. The adjacent angles being supplementary and the opposite angles being equal both point to the same conclusion. I guess another way to think about it is by considering the properties of a parallelogram, since a rhombus is a special type of parallelogram. In a parallelogram, opposite angles are equal, and adjacent angles are supplementary. Since a rhombus has all sides equal, it maintains these properties but with the added feature of equal sides. So, applying the parallelogram properties, if angle F is 130 degrees, angle H, being opposite, must also be 130 degrees. The adjacent angles, E and G, would each be 50 degrees to satisfy the supplementary condition. I think I've covered all the bases here. The properties of a rhombus, the relationships between its angles, and even the behavior of its diagonals all support the conclusion that angle H is 130 degrees. I don't see any contradictions or errors in my reasoning, so I'm pretty sure that's the correct answer.

answer：Okay, so I have this problem about a rectangular box container that's partially filled with water. The water volume is 120 cubic meters, and depending on which side of the box is on the ground, the depth of the water changes—it can be 2 meters, 3 meters, or 5 meters. I need to find the total volume of the container.Alright, let me try to visualize this. A rectangular box has three dimensions: length, width, and height. Depending on which side is considered the base (the side on the ground), the depth of the water will correspond to one of these dimensions. So, if I denote the three dimensions as x, y, and z, then the depth of the water when each dimension is the height would be x, y, or z respectively.Given that the volume of water is 120 m³, and the depth can be 2 m, 3 m, or 5 m, I can set up equations based on the formula for the volume of a rectangular prism, which is length × width × height.Let me denote the dimensions as follows:- When the depth is 2 m, the volume of water is 120 m³. So, if 2 m is the height, then the base area (length × width) would be 120 / 2 = 60 m².- Similarly, when the depth is 3 m, the base area would be 120 / 3 = 40 m².- And when the depth is 5 m, the base area would be 120 / 5 = 24 m².So, I have three different base areas: 60 m², 40 m², and 24 m². These correspond to the different pairs of dimensions when one of them is considered the height.Let me denote the three dimensions as a, b, and c, where a < b < c. Then, the base areas would be:- a × b = 60 m² (when c is the height)- a × c = 40 m² (when b is the height)- b × c = 24 m² (when a is the height)Wait, that doesn't seem right. If a < b < c, then when a is the height, the base area should be b × c, which is the largest, but 24 m² is the smallest. Hmm, maybe I got that mixed up.Let me think again. If the depth is 2 m, which is the smallest, then the base area should be the largest because the same volume of water spread over a larger base would result in a smaller depth. Similarly, when the depth is 5 m, the base area should be the smallest.So, if depth = 2 m corresponds to base area = 60 m²,depth = 3 m corresponds to base area = 40 m²,and depth = 5 m corresponds to base area = 24 m².Therefore, the base areas are decreasing as the depth increases, which makes sense because a larger depth means the water is spread over a smaller base area.So, let's assign:- When depth = 2 m, base area = 60 m², so the dimensions are a × b = 60- When depth = 3 m, base area = 40 m², so the dimensions are a × c = 40- When depth = 5 m, base area = 24 m², so the dimensions are b × c = 24Now, I have three equations:1. a × b = 602. a × c = 403. b × c = 24I need to solve for a, b, and c.Let me see. If I take the first equation, a × b = 60, and the second equation, a × c = 40, I can divide the first equation by the second to eliminate a.So, (a × b) / (a × c) = 60 / 40 => b / c = 3 / 2 => b = (3/2)cSimilarly, from the third equation, b × c = 24, and since b = (3/2)c, substituting:(3/2)c × c = 24 => (3/2)c² = 24 => c² = (24 × 2)/3 = 16 => c = 4So, c = 4 mThen, from b = (3/2)c = (3/2) × 4 = 6 mNow, from the first equation, a × b = 60 => a × 6 = 60 => a = 10 mWait, but if a = 10 m, b = 6 m, c = 4 m, then a > b > c, which contradicts my initial assumption that a < b < c.Hmm, maybe I should have assigned the base areas differently.Alternatively, perhaps I made a mistake in assigning which depth corresponds to which base area.Let me try again.If the depth is 2 m, then the base area is 60 m², which is the largest base area, so the corresponding height is the smallest dimension.Similarly, depth = 5 m corresponds to the smallest base area, 24 m², which would mean the height is the largest dimension.So, perhaps:- Depth = 2 m (smallest height) corresponds to base area = 60 m² (a × b)- Depth = 3 m corresponds to base area = 40 m² (a × c)- Depth = 5 m (largest height) corresponds to base area = 24 m² (b × c)So, assigning:1. a × b = 602. a × c = 403. b × c = 24Now, solving these equations.From equation 1: a = 60 / bSubstitute into equation 2: (60 / b) × c = 40 => (60c) / b = 40 => 60c = 40b => 3c = 2b => c = (2/3)bFrom equation 3: b × c = 24, and c = (2/3)b, so:b × (2/3)b = 24 => (2/3)b² = 24 => b² = 24 × (3/2) = 36 => b = 6Then, c = (2/3) × 6 = 4From equation 1: a × 6 = 60 => a = 10So, dimensions are a = 10 m, b = 6 m, c = 4 mBut again, a > b > c, which contradicts the initial assumption that a < b < c.Wait, maybe the problem doesn't specify the order of the dimensions, so perhaps it's okay.So, the dimensions are 10 m, 6 m, and 4 m.Therefore, the total volume of the container is a × b × c = 10 × 6 × 4 = 240 m³.So, the volume of the container is 240 cubic meters.

answer：Okay, so I have this problem here where I'm given a sequence {a_n}, and the sum of the first n terms is S_n = n² + 2n, where n is a positive integer. I need to find the general formula for the term a_n. Hmm, let me think about how to approach this.First, I remember that the sum of the first n terms of a sequence is related to the individual terms. Specifically, the nth term of the sequence can be found by subtracting the sum of the first (n-1) terms from the sum of the first n terms. So, in other words, a_n = S_n - S_{n-1}. That makes sense because S_n includes all terms up to a_n, and S_{n-1} includes all terms up to a_{n-1}, so their difference should be just a_n.Alright, so let's write that down. For n ≥ 2, a_n = S_n - S_{n-1}. Since S_n is given as n² + 2n, let's substitute that in. So, S_n = n² + 2n and S_{n-1} would be (n-1)² + 2(n-1). Let me compute that.First, expand (n-1)². That's n² - 2n + 1. Then, 2(n-1) is 2n - 2. So, adding those together, S_{n-1} = (n² - 2n + 1) + (2n - 2). Let me simplify that. The -2n and +2n cancel out, leaving n² + (1 - 2) which is n² - 1. So, S_{n-1} = n² - 1.Wait, that seems a bit off. Let me double-check that. (n-1)² is n² - 2n + 1, and 2(n-1) is 2n - 2. So, adding those together: n² - 2n + 1 + 2n - 2. The -2n and +2n do cancel, leaving n² + (1 - 2) = n² - 1. Yeah, that's correct.So, now, a_n = S_n - S_{n-1} = (n² + 2n) - (n² - 1). Let's compute that. Subtracting n² - 1 from n² + 2n gives n² + 2n - n² + 1. The n² terms cancel out, leaving 2n + 1. So, a_n = 2n + 1 for n ≥ 2.But wait, what about when n = 1? Because the formula for S_n is given for all positive integers, including n = 1. So, let's check what a_1 is. S_1 is the sum of the first 1 term, which is just a_1. So, S_1 = 1² + 2*1 = 1 + 2 = 3. Therefore, a_1 = 3.Now, let's see if this fits into the formula we found for a_n when n ≥ 2. If we plug n = 1 into 2n + 1, we get 2*1 + 1 = 3, which matches a_1. So, actually, the formula a_n = 2n + 1 works for all positive integers n, including n = 1.Let me just verify this with another term to be sure. Let's compute a_2 using both the sum formula and the general term. S_2 = 2² + 2*2 = 4 + 4 = 8. So, a_2 = S_2 - S_1 = 8 - 3 = 5. Using the formula a_n = 2n + 1, a_2 should be 2*2 + 1 = 5. Perfect, that matches.Let's try n = 3. S_3 = 3² + 2*3 = 9 + 6 = 15. a_3 = S_3 - S_2 = 15 - 8 = 7. Using the formula, a_3 = 2*3 + 1 = 7. Again, it works.So, it seems like the general formula a_n = 2n + 1 correctly gives the terms of the sequence for all positive integers n. Therefore, I can confidently say that the general term of the sequence is 2n + 1.Just to recap, the steps I took were:1. Recognize that the nth term can be found by subtracting the sum of the first (n-1) terms from the sum of the first n terms.2. Substitute the given sum formula into this relationship.3. Simplify the expression to find the general term.4. Verify the formula with specific values to ensure its correctness.This method worked well, and the formula holds true for all tested cases. I don't see any mistakes in my calculations, so I think I'm good.

answer：Alright, so I have this trigonometric expression to evaluate:frac{sin(330^circ) cdot tanleft(-frac{13}{3}piright)}{cosleft(-frac{19}{6}piright) cdot cos(690^circ)}Hmm, okay. Let me break this down step by step. I remember that trigonometric functions have periodic properties, so maybe I can simplify these angles by finding their reference angles or using periodicity.First, let's tackle each function one by one.Starting with sin(330^circ). I know that 330 degrees is in the fourth quadrant, and sine is negative in the fourth quadrant. Also, 330 degrees is 30 degrees less than 360 degrees. So, sin(330^circ) = sin(360^circ - 30^circ) = -sin(30^circ). I remember that sin(30^circ) = frac{1}{2}, so sin(330^circ) = -frac{1}{2}.Next, tanleft(-frac{13}{3}piright). Tangent has a period of pi, so I can add or subtract multiples of pi to find an equivalent angle between 0 and 2pi. Let's see, -frac{13}{3}pi is the same as -frac{13}{3}pi + 4pi = -frac{13}{3}pi + frac{12}{3}pi = -frac{1}{3}pi. So, tanleft(-frac{13}{3}piright) = tanleft(-frac{pi}{3}right). Tangent is an odd function, so tan(-x) = -tan(x). Therefore, tanleft(-frac{pi}{3}right) = -tanleft(frac{pi}{3}right). I remember that tanleft(frac{pi}{3}right) = sqrt{3}, so this becomes -sqrt{3}.Moving on to the denominator, first cosleft(-frac{19}{6}piright). Cosine is an even function, so cos(-x) = cos(x). Therefore, cosleft(-frac{19}{6}piright) = cosleft(frac{19}{6}piright). Now, let's simplify frac{19}{6}pi. Since 2pi = frac{12}{6}pi, subtracting 2pi three times: frac{19}{6}pi - 3 times 2pi = frac{19}{6}pi - frac{18}{6}pi = frac{1}{6}pi. So, cosleft(frac{19}{6}piright) = cosleft(frac{pi}{6}right). I remember that cosleft(frac{pi}{6}right) = frac{sqrt{3}}{2}.Lastly, cos(690^circ). Let's convert 690 degrees to a reference angle by subtracting multiples of 360 degrees. 690 - 360 = 330, so cos(690^circ) = cos(330^circ). As before, 330 degrees is in the fourth quadrant, and cosine is positive there. So, cos(330^circ) = cos(360^circ - 30^circ) = cos(30^circ) = frac{sqrt{3}}{2}.Now, let's substitute all these back into the original expression:frac{sin(330^circ) cdot tanleft(-frac{13}{3}piright)}{cosleft(-frac{19}{6}piright) cdot cos(690^circ)} = frac{ -frac{1}{2} cdot (-sqrt{3}) }{ frac{sqrt{3}}{2} cdot frac{sqrt{3}}{2} }Simplifying the numerator: -frac{1}{2} cdot (-sqrt{3}) = frac{sqrt{3}}{2}.Simplifying the denominator: frac{sqrt{3}}{2} cdot frac{sqrt{3}}{2} = frac{3}{4}.So now, the expression becomes:frac{ frac{sqrt{3}}{2} }{ frac{3}{4} } = frac{sqrt{3}}{2} times frac{4}{3} = frac{4sqrt{3}}{6} = frac{2sqrt{3}}{3}Wait, but hold on. I think I missed something. The original expression had two negative signs: one from sin(330^circ) and one from tanleft(-frac{13}{3}piright). When I multiplied them together, they became positive. So, the final result should be positive frac{2sqrt{3}}{3}. But let me double-check.Looking back, sin(330^circ) = -frac{1}{2}, and tanleft(-frac{13}{3}piright) = -sqrt{3}. Multiplying these gives (-frac{1}{2}) times (-sqrt{3}) = frac{sqrt{3}}{2}, which is positive. The denominator was all positive, so the overall result is positive frac{2sqrt{3}}{3}.But wait, in the initial steps, I thought the final answer might be negative. Did I make a mistake there? Let me check again.No, actually, both the numerator and denominator are positive after simplifying, so the result should indeed be positive. Maybe I confused myself earlier, but now it seems correct.So, after carefully simplifying each part and substituting the values, the final result is frac{2sqrt{3}}{3}.