💡Alright, so I've got this math problem here, and I need to figure out what happens when you take the negative of a rational number. Let's see, the question says: "If m is a rational number, then -m is..." and then it gives four options: A is a negative rational number, B is a non-zero rational number, C is a non-positive rational number, and D is a rational number.Okay, first off, I need to remember what a rational number is. From what I recall, a rational number is any number that can be expressed as the quotient of two integers, where the denominator isn't zero. So, like, if I have a number like 1/2 or 3/4, those are rational numbers. Even integers themselves are rational because you can write them as, say, 5/1 or -3/1.Now, the question is about taking the negative of such a number. So, if m is rational, then -m should be... well, let's think about it. If m is positive, then -m would be negative, right? But what if m is negative to begin with? Then -m would be positive. And if m is zero, then -m is still zero.Hmm, so depending on what m is, -m could be positive, negative, or zero. That makes me think that -m doesn't necessarily have to be negative or non-positive or non-zero. It could be any of those things depending on m.Looking back at the options, A says -m is a negative rational number. But wait, if m is zero, then -m is zero, which isn't negative. So A can't be the right answer because it's not always negative.Option B says -m is a non-zero rational number. But again, if m is zero, then -m is zero, which is a rational number but it's zero. So B isn't always true either because -m can be zero.Option C says -m is a non-positive rational number. Non-positive means it's either negative or zero. Well, if m is positive, then -m is negative, which is non-positive. If m is negative, then -m is positive, which isn't non-positive. And if m is zero, then -m is zero, which is non-positive. So C isn't always true because when m is negative, -m becomes positive, which contradicts C.Finally, option D says -m is a rational number. Well, since m is rational, and taking the negative of a rational number just flips its sign, it's still a rational number. Whether m is positive, negative, or zero, -m will always be a rational number because you're just multiplying by -1, which doesn't change the fact that it's a ratio of two integers.So, putting it all together, the correct answer should be D: rational number.

💡To solve the problem of finding the probability that a randomly selected number from the interval ((0, frac{1}{2})) is less than (frac{1}{3}), we can follow these steps:1. **Determine the Total Interval Length:** The interval from which the number is selected is ((0, frac{1}{2})). The total length of this interval is: [ frac{1}{2} - 0 = frac{1}{2} ]2. **Determine the Favorable Interval Length:** We are interested in numbers less than (frac{1}{3}). The interval of numbers less than (frac{1}{3}) within ((0, frac{1}{2})) is ((0, frac{1}{3})). The length of this interval is: [ frac{1}{3} - 0 = frac{1}{3} ]3. **Calculate the Probability:** Probability is calculated as the ratio of the favorable interval length to the total interval length. Therefore: [ text{Probability} = frac{text{Favorable Length}}{text{Total Length}} = frac{frac{1}{3}}{frac{1}{2}} = frac{1}{3} times frac{2}{1} = frac{2}{3} ]Thus, the probability that the selected number is less than (frac{1}{3}) is (frac{2}{3}).

💡Alright, so I have this problem about a cylindrical silo with a diameter of 30 feet and a height of 80 feet. There's a red stripe that's 3 feet wide going around it twice, and I need to find the area of that stripe. Hmm, okay, let me try to visualize this.First, the silo is a cylinder, right? So it has a circular base with a certain diameter and a certain height. The stripe is wrapped around it, making two complete revolutions. That means if I were to look at the silo from the side, the stripe would spiral around it twice from the bottom to the top.Now, the stripe is 3 feet wide horizontally. I'm assuming that means the width of the stripe as it goes around the cylinder is 3 feet. So, if I were to somehow "unwrap" the cylinder into a flat surface, the stripe would become a slanted rectangle or parallelogram.Wait, unwrapping the cylinder? That might be a useful concept here. If I can imagine cutting the cylinder vertically and laying it flat, it would form a rectangle. The height of this rectangle would be the same as the height of the silo, which is 80 feet. The width of this rectangle would be the circumference of the silo.Let me calculate the circumference. The diameter is 30 feet, so the radius is half of that, which is 15 feet. The circumference of a circle is given by (2pi r), so that would be (2 times pi times 15 = 30pi) feet. So, the flat rectangle would be 80 feet tall and (30pi) feet wide.Now, the stripe makes two complete revolutions around the silo. So, on this flat rectangle, the stripe would be a slanted line that goes from the bottom to the top, but it would also have to cover two times the circumference in width. Wait, no, actually, since it's making two complete revolutions, the horizontal distance it covers would be twice the circumference.But the stripe itself is 3 feet wide. So, on the flat rectangle, the stripe would be a parallelogram with a base of 3 feet and a height equal to the height of the silo, which is 80 feet. But I'm not sure if that's correct.Wait, no, maybe I need to think about it differently. If the stripe makes two complete revolutions, then the horizontal distance it covers is twice the circumference. So, the stripe would be slanting upwards as it goes around the cylinder twice.But the width of the stripe is 3 feet. So, if I unwrap it, the stripe would be a parallelogram with a base of 3 feet and a height equal to the height of the silo, 80 feet. But the slanting part is because it's making two revolutions.Wait, maybe I need to consider the slope of the stripe. If it makes two complete revolutions over the height of 80 feet, then the slope would be related to how much horizontal distance it covers per unit height.The total horizontal distance for two revolutions is (2 times 30pi = 60pi) feet. So, over a vertical distance of 80 feet, the stripe covers 60π feet horizontally. That would give a slope of ( frac{60pi}{80} = frac{3pi}{4} ).But how does that help me find the area of the stripe? Maybe I need to think about the stripe as a parallelogram on the unwrapped rectangle. The area of a parallelogram is base times height. The base would be the width of the stripe, which is 3 feet, and the height would be the vertical height of the silo, which is 80 feet.Wait, but is that correct? Because the stripe is slanting, does that affect the area? Or is it still just base times height regardless of the slant?I think it's still base times height because the area of a parallelogram is indeed base times height, where the height is the perpendicular distance between the bases. In this case, the height is the vertical height of the silo, 80 feet, and the base is the width of the stripe, 3 feet. So, the area should be (3 times 80 = 240) square feet.But wait, I'm not sure if I'm accounting for the fact that the stripe is wrapped around twice. Does that affect the area? Or is it just the same as if it were wrapped around once?I think it doesn't affect the area because whether it's wrapped around once or twice, the stripe's width and the height remain the same. The number of revolutions just changes the slope of the stripe on the unwrapped rectangle, but the area remains the same.So, maybe the area is indeed 240 square feet.But let me double-check. If I consider the stripe as a parallelogram on the unwrapped rectangle, the area is base times height. The base is 3 feet, and the height is 80 feet, so 3*80=240.Alternatively, if I think about the stripe as a helical band around the cylinder, the area can be found by considering the lateral surface area of the cylinder. The lateral surface area is circumference times height, which is (30pi times 80 = 2400pi) square feet. But the stripe is only a part of that area.The stripe is 3 feet wide, so in terms of the circumference, which is (30pi) feet, the fraction of the circumference that the stripe covers is ( frac{3}{30pi} = frac{1}{10pi} ). But wait, that doesn't seem right because the stripe is making two revolutions, so maybe the width is not a fraction of the circumference but rather a fixed width.Wait, perhaps I'm overcomplicating it. If I unwrap the cylinder, the stripe becomes a parallelogram with base 3 feet and height 80 feet, so area is 240 square feet. That seems straightforward.Alternatively, if I think about the stripe as a rectangle when unwrapped, but slanted, the area remains the same as if it were a rectangle because the slant doesn't change the area, only the shape.So, I think the area of the stripe is 240 square feet.

💡Okay, so I have this math problem here: "When a number is divided by 8, the remainder is 3. What is the remainder when four times the number minus 9 is divided by 8?" Hmm, let me try to figure this out step by step.First, I need to understand what it means when a number is divided by 8 and leaves a remainder of 3. That means if I have some number, let's call it 'n', and I divide it by 8, I get a quotient and a remainder of 3. So, mathematically, I can express this as:n = 8k + 3where 'k' is some integer. This equation tells me that 'n' is 3 more than a multiple of 8.Now, the problem asks for the remainder when four times the number minus 9 is divided by 8. So, I need to calculate 4n - 9 and then find the remainder when this result is divided by 8.Let's substitute the expression for 'n' into this equation:4n - 9 = 4*(8k + 3) - 9Now, let's distribute the 4:4*(8k) + 4*3 - 9 = 32k + 12 - 9Simplify the constants:32k + 3So, 4n - 9 simplifies to 32k + 3.Now, I need to find the remainder when 32k + 3 is divided by 8. Let's think about this. 32k is a multiple of 8 because 32 is 8 times 4. So, 32k divided by 8 will give me an integer with no remainder. That means the remainder comes solely from the '+3' part.Therefore, when I divide 32k + 3 by 8, the remainder is 3.Wait, let me double-check that. If I take a specific example, say k = 1, then n = 8*1 + 3 = 11. Then 4n - 9 = 4*11 - 9 = 44 - 9 = 35. Now, dividing 35 by 8 gives 4 with a remainder of 3. Yep, that matches.Let me try another example to be sure. Let's say k = 2, so n = 8*2 + 3 = 19. Then 4n - 9 = 4*19 - 9 = 76 - 9 = 67. Dividing 67 by 8 gives 8 with a remainder of 3. Again, the remainder is 3.Okay, so it seems consistent. The remainder is always 3 when four times the number minus 9 is divided by 8, regardless of the value of 'k'. That makes sense because the term involving 'k' is a multiple of 8, so it doesn't affect the remainder.I think I've got it. The remainder is 3.

💡Alright, so I have this problem here: I need to translate the function ( f(x) = sin 2x + sqrt{3}cos 2x ) to the left by ( varphi ) units, where ( varphi > 0 ). After this translation, the resulting graph should be symmetric about the y-axis. I need to find the minimum value of ( varphi ).Hmm, okay. Let me break this down. First, I remember that translating a function to the left by ( varphi ) units means replacing ( x ) with ( x + varphi ) in the function. So, the translated function would be ( f(x + varphi) ).But before I get into that, maybe I should simplify the original function ( f(x) ). It looks like a combination of sine and cosine functions with the same argument ( 2x ). I think I can express this as a single sine or cosine function using the amplitude-phase form. Let me recall the formula: ( Asin theta + Bcos theta = Csin(theta + phi) ) or ( Ccos(theta + phi) ), where ( C = sqrt{A^2 + B^2} ) and ( phi ) is the phase shift.So, for ( f(x) = sin 2x + sqrt{3}cos 2x ), the coefficients are ( A = 1 ) and ( B = sqrt{3} ). Let me compute the amplitude ( C ):( C = sqrt{1^2 + (sqrt{3})^2} = sqrt{1 + 3} = sqrt{4} = 2 ).Okay, so the amplitude is 2. Now, to find the phase shift ( phi ), I can use the relationships:( sin phi = frac{B}{C} = frac{sqrt{3}}{2} ) and ( cos phi = frac{A}{C} = frac{1}{2} ).Wait, actually, I think it's the other way around. Let me double-check. If I express ( Asin theta + Bcos theta ) as ( Csin(theta + phi) ), then expanding this gives:( Csin(theta + phi) = Csin theta cos phi + Ccos theta sin phi ).Comparing this with ( Asin theta + Bcos theta ), we have:( A = Ccos phi ) and ( B = Csin phi ).So, ( cos phi = frac{A}{C} = frac{1}{2} ) and ( sin phi = frac{B}{C} = frac{sqrt{3}}{2} ).Therefore, ( phi ) is the angle whose cosine is ( frac{1}{2} ) and sine is ( frac{sqrt{3}}{2} ). That's ( phi = frac{pi}{3} ).So, putting it all together, ( f(x) = 2sinleft(2x + frac{pi}{3}right) ).Wait, let me verify that. If I expand ( 2sinleft(2x + frac{pi}{3}right) ), I should get back the original function.Using the sine addition formula:( sin(2x + frac{pi}{3}) = sin 2x cos frac{pi}{3} + cos 2x sin frac{pi}{3} ).We know that ( cos frac{pi}{3} = frac{1}{2} ) and ( sin frac{pi}{3} = frac{sqrt{3}}{2} ).So, multiplying by 2:( 2sinleft(2x + frac{pi}{3}right) = 2left( sin 2x cdot frac{1}{2} + cos 2x cdot frac{sqrt{3}}{2} right) = sin 2x + sqrt{3}cos 2x ).Yes, that's correct. So, ( f(x) = 2sinleft(2x + frac{pi}{3}right) ).Now, translating this function to the left by ( varphi ) units means replacing ( x ) with ( x + varphi ). So, the translated function is:( f(x + varphi) = 2sinleft(2(x + varphi) + frac{pi}{3}right) = 2sinleft(2x + 2varphi + frac{pi}{3}right) ).We need this translated function to be symmetric about the y-axis. A function is symmetric about the y-axis if it's an even function, meaning ( f(-x) = f(x) ).So, let's write the condition for the translated function to be even:( 2sinleft(2(-x) + 2varphi + frac{pi}{3}right) = 2sinleft(2x + 2varphi + frac{pi}{3}right) ).Simplifying the left side:( 2sinleft(-2x + 2varphi + frac{pi}{3}right) = 2sinleft(2x + 2varphi + frac{pi}{3}right) ).We can divide both sides by 2:( sinleft(-2x + 2varphi + frac{pi}{3}right) = sinleft(2x + 2varphi + frac{pi}{3}right) ).Now, I remember that ( sin(-theta) = -sintheta ), so:( sinleft(-2x + 2varphi + frac{pi}{3}right) = sinleft(-(2x - 2varphi - frac{pi}{3})right) = -sinleft(2x - 2varphi - frac{pi}{3}right) ).But the right side is ( sinleft(2x + 2varphi + frac{pi}{3}right) ).So, putting it together:( -sinleft(2x - 2varphi - frac{pi}{3}right) = sinleft(2x + 2varphi + frac{pi}{3}right) ).Hmm, this seems a bit complicated. Maybe there's another approach. Since the function needs to be even, the translated function must satisfy ( f(-x) = f(x) ). So, let's write that condition again:( 2sinleft(-2x + 2varphi + frac{pi}{3}right) = 2sinleft(2x + 2varphi + frac{pi}{3}right) ).Divide both sides by 2:( sinleft(-2x + 2varphi + frac{pi}{3}right) = sinleft(2x + 2varphi + frac{pi}{3}right) ).Now, using the identity ( sin(A) = sin(B) ) implies that either:1. ( A = B + 2pi n ), or2. ( A = pi - B + 2pi n ), for some integer ( n ).But since this must hold for all ( x ), the coefficients of ( x ) must satisfy certain conditions.Let me consider the first case:( -2x + 2varphi + frac{pi}{3} = 2x + 2varphi + frac{pi}{3} + 2pi n ).Simplify:( -2x = 2x + 2pi n ).( -4x = 2pi n ).But this must hold for all ( x ), which is only possible if ( -4 = 0 ), which is not true. So, this case doesn't work.Now, consider the second case:( -2x + 2varphi + frac{pi}{3} = pi - (2x + 2varphi + frac{pi}{3}) + 2pi n ).Simplify the right side:( pi - 2x - 2varphi - frac{pi}{3} + 2pi n ).Combine like terms:( pi - frac{pi}{3} - 2x - 2varphi + 2pi n = frac{2pi}{3} - 2x - 2varphi + 2pi n ).So, the equation becomes:( -2x + 2varphi + frac{pi}{3} = frac{2pi}{3} - 2x - 2varphi + 2pi n ).Let's cancel out the ( -2x ) terms on both sides:( 2varphi + frac{pi}{3} = frac{2pi}{3} - 2varphi + 2pi n ).Now, bring all terms involving ( varphi ) to one side and constants to the other:( 2varphi + 2varphi = frac{2pi}{3} - frac{pi}{3} + 2pi n ).Simplify:( 4varphi = frac{pi}{3} + 2pi n ).Therefore,( varphi = frac{pi}{12} + frac{pi n}{2} ).Since ( varphi > 0 ), we need to find the smallest positive ( varphi ). Let's consider ( n = 0 ):( varphi = frac{pi}{12} ).If ( n = 1 ), ( varphi = frac{pi}{12} + frac{pi}{2} = frac{7pi}{12} ), which is larger. Similarly, higher ( n ) will give larger ( varphi ). Negative ( n ) would give negative ( varphi ), which is not allowed since ( varphi > 0 ).Therefore, the minimum value of ( varphi ) is ( frac{pi}{12} ).Wait, let me double-check. If I translate the function ( f(x) = 2sin(2x + frac{pi}{3}) ) to the left by ( frac{pi}{12} ), the new function becomes:( f(x + frac{pi}{12}) = 2sinleft(2(x + frac{pi}{12}) + frac{pi}{3}right) = 2sinleft(2x + frac{pi}{6} + frac{pi}{3}right) = 2sinleft(2x + frac{pi}{2}right) ).Is ( 2sin(2x + frac{pi}{2}) ) an even function?Let's check ( f(-x) ):( 2sin(-2x + frac{pi}{2}) = 2sinleft(frac{pi}{2} - 2xright) ).Using the identity ( sinleft(frac{pi}{2} - thetaright) = cos theta ):( 2cos(2x) ).On the other hand, ( f(x) = 2sin(2x + frac{pi}{2}) = 2cos(2x) ).So, ( f(-x) = 2cos(2x) = f(x) ). Therefore, yes, it is even. So, the translation by ( frac{pi}{12} ) units to the left makes the function symmetric about the y-axis.Hence, the minimum value of ( varphi ) is indeed ( frac{pi}{12} ).**Final Answer**The minimum value of ( varphi ) is boxed{dfrac{pi}{12}}.

💡Okay, so I have this problem about sequences. It's about a sequence {a_n} where a₁ = 1, and each subsequent term is given by a_{n+1} = (2a_n)/(a_n + 2). Then there's another sequence {b_n} defined as b_n = 1/a_n. Part (1) asks me to prove that {b_n} is an arithmetic sequence. Hmm, arithmetic sequences have a constant difference between consecutive terms, right? So I need to show that b_{n+1} - b_n is constant for all n.Let me start by writing down what I know. Since b_n = 1/a_n, then b_{n+1} = 1/a_{n+1}. And a_{n+1} is given by (2a_n)/(a_n + 2). So substituting that into b_{n+1}, I get:b_{n+1} = 1 / [(2a_n)/(a_n + 2)] = (a_n + 2)/(2a_n)Let me simplify that. (a_n + 2)/(2a_n) can be split into two fractions: a_n/(2a_n) + 2/(2a_n). Simplifying each term, that's 1/2 + 1/a_n. So:b_{n+1} = 1/2 + 1/a_n = 1/2 + b_nSo that means b_{n+1} = b_n + 1/2. Therefore, the difference between consecutive terms is 1/2, which is constant. So {b_n} is indeed an arithmetic sequence with common difference 1/2.Wait, let me double-check that. Starting from b_{n+1} = (a_n + 2)/(2a_n). Let me compute that again:(a_n + 2)/(2a_n) = a_n/(2a_n) + 2/(2a_n) = 1/2 + 1/a_n. Yep, that's correct. So b_{n+1} = b_n + 1/2. So that's an arithmetic sequence.Okay, so part (1) seems solid. Now, part (2) asks for the general term formula for {a_n}. Since we have {b_n} as an arithmetic sequence, and b_n = 1/a_n, maybe I can find b_n first and then invert it to get a_n.Given that {b_n} is arithmetic, with b₁ = 1/a₁ = 1/1 = 1, and common difference d = 1/2. So the general term for an arithmetic sequence is b_n = b₁ + (n - 1)d. Plugging in the values:b_n = 1 + (n - 1)(1/2) = 1 + (n/2 - 1/2) = (2/2) + (n/2 - 1/2) = (2 + n - 1)/2 = (n + 1)/2.So b_n = (n + 1)/2. Therefore, since b_n = 1/a_n, then a_n = 1/b_n = 2/(n + 1).Let me verify this formula with the initial terms. For n=1, a₁ should be 1. Plugging n=1 into 2/(1+1) = 2/2 = 1. Correct.For n=2, a₂ = 2/(2+1) = 2/3. Let's compute a₂ using the recursive formula: a₂ = (2a₁)/(a₁ + 2) = (2*1)/(1 + 2) = 2/3. Correct.For n=3, a₃ = 2/(3+1) = 2/4 = 1/2. Using the recursive formula: a₃ = (2a₂)/(a₂ + 2) = (2*(2/3))/(2/3 + 2) = (4/3)/(8/3) = (4/3)*(3/8) = 12/24 = 1/2. Correct.So the formula seems to hold for the first few terms. Therefore, I think the general term for {a_n} is 2/(n + 1).I should also consider whether this formula makes sense for larger n. Let's try n=4: a₄ = 2/(4+1) = 2/5. Using the recursive formula: a₄ = (2a₃)/(a₃ + 2) = (2*(1/2))/(1/2 + 2) = (1)/(5/2) = 2/5. Correct again.So it seems consistent. Therefore, I'm confident that the general term is a_n = 2/(n + 1).**Final Answer**The general term formula for the sequence {a_n} is boxed{dfrac{2}{n+1}}.