answer：Alright, so I have this question here asking which plane figure is the best analogy for a parallelepiped in space. The options are Triangle, Parallelogram, Trapezoid, and Rectangle. Hmm, okay, let me think about this step by step.First, I need to recall what a parallelepiped is. From what I remember, a parallelepiped is a three-dimensional figure formed by six parallelograms. It's like a 3D shape where all the faces are parallelograms. So, it's similar to a cube or a rectangular prism, but instead of all faces being rectangles, they're parallelograms. That means opposite faces are equal and parallel, and the edges are all congruent in pairs.Now, the question is asking for a plane figure that serves as an analogy for a parallelepiped. So, I need to think about what shape in two dimensions would correspond to a parallelepiped in three dimensions.Let's look at the options:A: TriangleB: ParallelogramC: TrapezoidD: RectangleOkay, starting with option A: Triangle. A triangle is a three-sided polygon, and while it's a plane figure, I don't see a direct analogy to a parallelepiped. A parallelepiped has six faces, all parallelograms, so a triangle seems too simple and doesn't capture the complexity of the 3D shape.Option B: Parallelogram. A parallelogram is a four-sided figure with opposite sides equal and parallel. This seems more promising because a parallelepiped is made up of parallelograms. So, in 2D, a parallelogram is like one face of the parallelepiped. But is it the best analogy?Option C: Trapezoid. A trapezoid is a four-sided figure with at least one pair of parallel sides. While it's also a quadrilateral, it doesn't have the same properties as a parallelogram, especially since only one pair of sides is parallel. This seems less likely to be the analogy.Option D: Rectangle. A rectangle is a special type of parallelogram where all angles are right angles. While rectangles are related to parallelograms, a parallelepiped doesn't necessarily have right angles unless it's a rectangular prism. So, a rectangle might be too specific.Going back to the parallelepiped, it's essentially a 3D figure where all faces are parallelograms. So, if I think about the 2D analogy, it would make sense that the corresponding figure is a parallelogram because it's the base shape that makes up the 3D figure.But wait, is there a better way to think about this? Maybe considering that a parallelepiped is to a parallelogram as a cube is to a square? In that case, a cube is a special type of parallelepiped where all angles are right angles, and a square is a special type of parallelogram with all angles right angles. So, if a cube is analogous to a square, then a general parallelepiped would be analogous to a general parallelogram.That makes sense. So, the parallelogram is the 2D figure that best represents the structure of a parallelepiped in 3D space.I should also consider if any of the other options could be better. For example, a rectangle is a type of parallelogram, but it's more specific. Since a parallelepiped doesn't require right angles, a rectangle might not be the best fit. Similarly, a trapezoid doesn't have the necessary properties to align with a parallelepiped.Therefore, after thinking through this, I believe the best answer is B: Parallelogram.

answer：Alright, so I need to figure out the horizontal length of a widescreen television that has a 50-inch diagonal. I remember that widescreen TVs usually have a 16:9 aspect ratio, which means the width to height ratio is 16 to 9. First, I think I need to recall what aspect ratio means. It's the proportion of the width to the height of the screen. So, for every 16 units of width, there are 9 units of height. This ratio helps in maintaining the shape of the image without distortion.Now, since the TV is a rectangle, I can use the Pythagorean theorem to relate the width, height, and diagonal. The theorem states that in a right-angled triangle, the square of the hypotenuse (which would be the diagonal in this case) is equal to the sum of the squares of the other two sides (width and height).Let me denote the width as 16x and the height as 9x, where x is a common multiplier. This way, the ratio of width to height remains 16:9.So, according to the Pythagorean theorem:(16x)^2 + (9x)^2 = (50)^2Let me calculate the squares:256x^2 + 81x^2 = 2500Adding the terms on the left:337x^2 = 2500Now, I need to solve for x:x^2 = 2500 / 337x = sqrt(2500 / 337)Calculating the square root:x ≈ sqrt(7.418) ≈ 2.7227Now, to find the horizontal length (width), I multiply 16 by x:Width = 16 * 2.7227 ≈ 43.56 inchesSo, the horizontal length of the television is approximately 43.56 inches.

answer：Alright, so I've got this problem here with two parts. Let me take it step by step.Starting with part (1): We have the function f(x) = cos(2x) + 4 sin x sin²(x/2 + π/4). Then, we're told that the graph of f(2x) is translated to the right by π/6 units to get g(x). We need to find the range of g(x) when x is in [π/12, π/2].First, I think I need to simplify the function f(x). Let me write it down again:f(x) = cos(2x) + 4 sin x sin²(x/2 + π/4)Hmm, maybe I can use some trigonometric identities to simplify this. Let's see.I remember that sin²θ can be written as (1 - cos(2θ))/2. So, let me apply that to the second term:sin²(x/2 + π/4) = [1 - cos(2*(x/2 + π/4))]/2 = [1 - cos(x + π/2)]/2So, substituting back into f(x):f(x) = cos(2x) + 4 sin x * [1 - cos(x + π/2)]/2Simplify the constants:4 sin x * [1 - cos(x + π/2)]/2 = 2 sin x [1 - cos(x + π/2)]So now, f(x) = cos(2x) + 2 sin x [1 - cos(x + π/2)]Let me expand that:f(x) = cos(2x) + 2 sin x - 2 sin x cos(x + π/2)Hmm, that term with cos(x + π/2) can be simplified. I recall that cos(x + π/2) is equal to -sin x. Because cos(θ + π/2) = -sin θ. Let me confirm:Yes, cos(θ + π/2) = cos θ cos(π/2) - sin θ sin(π/2) = cos θ * 0 - sin θ * 1 = -sin θ.So, cos(x + π/2) = -sin x. Therefore, the term becomes:-2 sin x cos(x + π/2) = -2 sin x (-sin x) = 2 sin²xSo, now f(x) becomes:f(x) = cos(2x) + 2 sin x + 2 sin²xNow, let's recall that cos(2x) can be written as 1 - 2 sin²x. Let me substitute that:f(x) = (1 - 2 sin²x) + 2 sin x + 2 sin²xSimplify:1 - 2 sin²x + 2 sin x + 2 sin²x = 1 + 2 sin xOh, that's nice! So, f(x) simplifies to 1 + 2 sin x.So, f(x) = 1 + 2 sin x.Now, we need f(2x). So, replacing x with 2x:f(2x) = 1 + 2 sin(2x)Then, the graph of f(2x) is translated to the right by π/6 units to get g(x). Translating a function to the right by c units means replacing x with x - c. So, g(x) = f(2(x - π/6)).Let me compute that:g(x) = f(2x - π/3) = 1 + 2 sin(2x - π/3)So, g(x) = 1 + 2 sin(2x - π/3)Now, we need to find the range of g(x) when x is in [π/12, π/2].First, let's find the interval for the argument of the sine function, which is 2x - π/3.When x = π/12:2*(π/12) - π/3 = π/6 - π/3 = -π/6When x = π/2:2*(π/2) - π/3 = π - π/3 = 2π/3So, the argument 2x - π/3 ranges from -π/6 to 2π/3.Now, let's consider the sine function over this interval. The sine function reaches its maximum at π/2 and its minimum at -π/6 in this interval.Let me recall the values:sin(-π/6) = -1/2sin(π/2) = 1sin(2π/3) = sin(π - π/3) = sin(π/3) = √3/2 ≈ 0.866So, the sine function increases from -π/6 to π/2, reaching 1, then decreases to √3/2 at 2π/3.Therefore, the maximum value of sin(2x - π/3) is 1, and the minimum is -1/2.Thus, the range of sin(2x - π/3) is [-1/2, 1].Therefore, g(x) = 1 + 2 sin(2x - π/3) will have:Minimum value: 1 + 2*(-1/2) = 1 - 1 = 0Maximum value: 1 + 2*(1) = 3Hence, the range of g(x) is [0, 3].Wait, let me double-check. When x = π/12, 2x - π/3 = -π/6, so sin(-π/6) = -1/2, so g(π/12) = 1 + 2*(-1/2) = 0.When x = 5π/12, 2x - π/3 = 5π/6 - π/3 = 5π/6 - 2π/6 = 3π/6 = π/2, so sin(π/2) = 1, so g(5π/12) = 1 + 2*1 = 3.But wait, x is only up to π/2. Let's check at x = π/2:2x - π/3 = π - π/3 = 2π/3, so sin(2π/3) = √3/2, so g(π/2) = 1 + 2*(√3/2) = 1 + √3 ≈ 2.732.But since 3 is higher than that, the maximum is indeed 3, achieved at x = 5π/12, which is within the interval [π/12, π/2].So, the range is from 0 to 3.Alright, that seems solid.Now, moving on to part (2):In triangle ABC, sides a, b, c opposite angles A, B, C respectively.Given:- b = 2- f(A) = √2 + 1- √3 a = 2b sin A- B ∈ (0, π/2)Find the area of triangle ABC.First, from part (1), we know that f(x) = 1 + 2 sin x. So, f(A) = 1 + 2 sin A = √2 + 1.So, 1 + 2 sin A = √2 + 1Subtract 1: 2 sin A = √2Divide by 2: sin A = √2 / 2So, sin A = √2 / 2, which implies that A is either π/4 or 3π/4. But since in a triangle, angles are less than π, and given that B is in (0, π/2), so A can't be 3π/4 because then the sum of angles would exceed π. So, A = π/4.So, angle A is π/4.Next, given √3 a = 2b sin A.We know b = 2, sin A = √2 / 2.So, plug in:√3 a = 2*2*(√2 / 2) = 2*√2Simplify:√3 a = 2√2Thus, a = (2√2) / √3 = (2√6) / 3So, side a is 2√6 / 3.Now, we need to find the area of triangle ABC.We can use the formula:Area = (1/2)ab sin CBut we don't know angle C or side c.Alternatively, since we know sides a and b, and angles A and B, perhaps we can find angle C and then use the formula.Wait, we know angle A = π/4, and we can find angle B.From the Law of Sines:a / sin A = b / sin BWe have a = 2√6 / 3, sin A = √2 / 2, b = 2.So,(2√6 / 3) / (√2 / 2) = 2 / sin BSimplify the left side:(2√6 / 3) * (2 / √2) = (4√6) / (3√2) = (4√3) / 3So,(4√3) / 3 = 2 / sin BSolve for sin B:sin B = 2 / (4√3 / 3) = (2 * 3) / (4√3) = 6 / (4√3) = (3) / (2√3) = √3 / 2So, sin B = √3 / 2, which implies that B is either π/3 or 2π/3. But since B is in (0, π/2), it must be π/3.So, angle B = π/3.Therefore, angle C = π - A - B = π - π/4 - π/3Convert to common denominator:π = 12π/12π/4 = 3π/12π/3 = 4π/12So, C = 12π/12 - 3π/12 - 4π/12 = 5π/12So, angle C is 5π/12.Now, to find the area, we can use:Area = (1/2)ab sin CWe have a = 2√6 / 3, b = 2, and sin C = sin(5π/12)We need to compute sin(5π/12). Let me recall that 5π/12 is 75 degrees.We can use the sine addition formula:sin(75°) = sin(45° + 30°) = sin45 cos30 + cos45 sin30Compute each term:sin45 = √2/2, cos30 = √3/2, cos45 = √2/2, sin30 = 1/2So,sin75 = (√2/2)(√3/2) + (√2/2)(1/2) = (√6)/4 + (√2)/4 = (√6 + √2)/4Therefore, sin(5π/12) = (√6 + √2)/4Now, plug into the area formula:Area = (1/2) * (2√6 / 3) * 2 * (√6 + √2)/4Simplify step by step:First, multiply constants:(1/2) * 2 * (√6 + √2)/4 = (1/2) * 2 * (√6 + √2)/4 = (√6 + √2)/4Then, multiply by (2√6 / 3):Wait, hold on. Let me write it all out:Area = (1/2) * (2√6 / 3) * 2 * (√6 + √2)/4Multiply the constants:(1/2) * 2 * (√6 + √2)/4 = (1/2) * 2 * (√6 + √2)/4 = (√6 + √2)/4Then, multiply by (2√6 / 3):Wait, no, actually, the order is:(1/2) * (2√6 / 3) * 2 * (√6 + √2)/4Let me compute step by step:First, (1/2) * 2 = 1So, now we have 1 * (√6 / 3) * (√6 + √2)/4Multiply the fractions:(√6 / 3) * (√6 + √2)/4 = [√6*(√6 + √2)] / 12Multiply numerator:√6*√6 = 6√6*√2 = √12 = 2√3So, numerator is 6 + 2√3Thus, Area = (6 + 2√3)/12 = (3 + √3)/6Wait, that's not matching my initial thought. Let me check the calculation again.Wait, perhaps I messed up the order.Wait, Area = (1/2) * a * b * sin Ca = 2√6 / 3, b = 2, sin C = (√6 + √2)/4So,Area = (1/2) * (2√6 / 3) * 2 * (√6 + √2)/4Compute step by step:Multiply constants:(1/2) * 2 * (√6 + √2)/4 = (1/2) * 2 = 1, then 1 * (√6 + √2)/4 = (√6 + √2)/4Then, multiply by (2√6 / 3):Wait, no, actually, the formula is (1/2)*a*b*sin C, so:(1/2) * (2√6 / 3) * 2 * sin CBut sin C is (√6 + √2)/4So, let's compute:(1/2) * (2√6 / 3) * 2 * (√6 + √2)/4Multiply constants:(1/2) * 2 * (√6 + √2)/4 = (1/2) * 2 = 1, then 1 * (√6 + √2)/4 = (√6 + √2)/4Then, multiply by (2√6 / 3):Wait, no, actually, the constants are:(1/2) * (2√6 / 3) * 2 * (√6 + √2)/4Let me rearrange:(1/2) * 2 * (√6 / 3) * (√6 + √2)/4Simplify:(1/2 * 2) = 1So, 1 * (√6 / 3) * (√6 + √2)/4 = (√6*(√6 + √2)) / 12Compute numerator:√6*√6 = 6√6*√2 = √12 = 2√3So, numerator is 6 + 2√3Thus, Area = (6 + 2√3)/12 = (3 + √3)/6Wait, but earlier I thought the answer was (3 + √3)/3. Hmm, seems like I made a mistake.Wait, let me recast the calculation:Area = (1/2) * a * b * sin Ca = 2√6 / 3, b = 2, sin C = (√6 + √2)/4So,Area = (1/2) * (2√6 / 3) * 2 * (√6 + √2)/4Compute step by step:First, multiply (1/2) * 2 = 1Then, multiply 1 * (2√6 / 3) = 2√6 / 3Then, multiply by (√6 + √2)/4:(2√6 / 3) * (√6 + √2)/4 = [2√6*(√6 + √2)] / 12Compute numerator:2√6*√6 = 2*6 = 122√6*√2 = 2*√12 = 2*2√3 = 4√3So, numerator is 12 + 4√3Thus, Area = (12 + 4√3)/12 = (12/12) + (4√3)/12 = 1 + (√3)/3 = (3 + √3)/3Ah, there we go! So, the area is (3 + √3)/3.I think I messed up the multiplication order earlier, but now it makes sense.So, the area is (3 + √3)/3.**Final Answer**(1) The range of ( g(x) ) is (boxed{[0, 3]}).(2) The area of ( triangle ABC ) is (boxed{dfrac{3 + sqrt{3}}{3}}).

answer：Okay, so I have this problem here: I need to find how many distinct ordered pairs of positive integers (m, n) satisfy the equation 1/m + 1/n = 1/5. Hmm, let me think about how to approach this.First, I know that when dealing with equations involving reciprocals, it's often helpful to combine them into a single fraction. So, let me try that. If I add 1/m and 1/n, I get (n + m)/(mn). So, the equation becomes:(n + m)/(mn) = 1/5Now, to get rid of the fractions, I can cross-multiply. That means multiplying both sides by mn and by 5. Let's do that:5(n + m) = mnExpanding the left side gives:5n + 5m = mnHmm, okay, so now I have 5n + 5m = mn. I need to find positive integers m and n that satisfy this equation. It might help to rearrange the equation to make it easier to factor or apply some kind of factoring trick.Let me try moving all terms to one side:mn - 5m - 5n = 0Hmm, this looks a bit like a quadratic in terms of m or n, but it's a bit tricky. I remember something called Simon's Favorite Factoring Trick, where you can add a constant to both sides to make the equation factorable. Let me see if that works here.If I add 25 to both sides, I get:mn - 5m - 5n + 25 = 25Now, let's see if this left side can be factored. It looks like it might factor into (m - 5)(n - 5). Let me check:(m - 5)(n - 5) = mn - 5m - 5n + 25Yes! That's exactly the left side. So, now the equation is:(m - 5)(n - 5) = 25Alright, so now I need to find all pairs of positive integers (m, n) such that when I subtract 5 from each, their product is 25. Since m and n are positive integers, (m - 5) and (n - 5) must also be positive integers. So, I need to find all pairs of positive integers that multiply to 25.Let me list the positive integer factors of 25:1, 5, 25So, the possible pairs (a, b) where a * b = 25 are:(1, 25), (5, 5), (25, 1)Each of these pairs corresponds to (m - 5, n - 5). Therefore, I can find m and n by adding 5 back to each component of these pairs.Let's do that:1. For (1, 25): m = 1 + 5 = 6 n = 25 + 5 = 30 So, one pair is (6, 30)2. For (5, 5): m = 5 + 5 = 10 n = 5 + 5 = 10 So, another pair is (10, 10)3. For (25, 1): m = 25 + 5 = 30 n = 1 + 5 = 6 So, the third pair is (30, 6)Now, I need to check if these are all distinct ordered pairs. An ordered pair means that the order matters, so (6, 30) and (30, 6) are considered different. However, in this case, we have both (6, 30) and (30, 6), as well as (10, 10). So, how many distinct ordered pairs do we have?Looking at the pairs:- (6, 30)- (10, 10)- (30, 6)These are three distinct ordered pairs because (6, 30) and (30, 6) are different, and (10, 10) is the same when reversed.Wait, but I should make sure that these are the only possible pairs. Let me double-check if there are any other factor pairs of 25 that I might have missed. The factors of 25 are 1, 5, and 25, so the pairs are indeed (1,25), (5,5), and (25,1). There are no other positive integer pairs that multiply to 25.Therefore, the distinct ordered pairs of positive integers (m, n) that satisfy the equation are (6, 30), (10, 10), and (30, 6). So, there are three such pairs.I think that's all. I don't see any other possibilities, and I've checked each step to make sure I didn't make a mistake. So, the answer should be 3 distinct ordered pairs.

answer：First, I need to simplify the expression (overrightarrow{AB} + overrightarrow{BD} - overrightarrow{AC} - overrightarrow{CD}).I'll start by grouping the vectors in pairs to make the addition and subtraction easier. So, I'll consider (overrightarrow{AB} + overrightarrow{BD}) and (overrightarrow{AC} + overrightarrow{CD}) separately.For the first pair, (overrightarrow{AB} + overrightarrow{BD}), I can use the triangle rule of vector addition. This rule states that if you have two vectors (overrightarrow{AB}) and (overrightarrow{BD}), their sum is equivalent to the vector from the starting point of the first vector to the endpoint of the second vector. Applying this rule, (overrightarrow{AB} + overrightarrow{BD}) simplifies to (overrightarrow{AD}).Next, I'll simplify the second pair, (overrightarrow{AC} + overrightarrow{CD}). Again, using the triangle rule, the sum of these two vectors is the vector from the starting point of (overrightarrow{AC}) to the endpoint of (overrightarrow{CD}), which is (overrightarrow{AD}).Now, substituting these simplified vectors back into the original expression, I have:[overrightarrow{AB} + overrightarrow{BD} - overrightarrow{AC} - overrightarrow{CD} = overrightarrow{AD} - overrightarrow{AD}]Finally, subtracting (overrightarrow{AD}) from itself results in the zero vector:[overrightarrow{AD} - overrightarrow{AD} = overrightarrow{0}]Therefore, the simplified expression is the zero vector.

answer：First, I need to determine the number of ways to choose the President, considering there are 15 boys and 15 girls in the club.Next, the Vice-President must be of the opposite gender from the President. This means if the President is a boy, there are 15 girls to choose from for Vice-President, and vice versa.Then, the Secretary must be of the same gender as the President but cannot be the same person as the President or Vice-President. This leaves 14 members of the President's gender available for the Secretary role.Finally, I calculate the total number of possible combinations by multiplying the number of choices for each role and then summing the possibilities for both scenarios: President being a boy or a girl.