💡Okay, so I have this problem about a train passing an electric pole. Let me try to understand what's being asked here. The train is traveling at 68 kmph, and it's 170 meters long. There's also a 5% incline on the track and a headwind of 20 kmph affecting the train's speed. I need to figure out how long it will take for the train to pass the electric pole.First, I know that when a train passes a pole, the time it takes depends on the length of the train and its speed. The basic formula I remember is Time = Distance / Speed. So, in this case, the distance would be the length of the train, which is 170 meters, and the speed would be the speed of the train.But wait, the problem mentions a headwind and a 5% incline. I guess these factors will affect the train's speed. I need to adjust the train's speed based on these conditions before I can calculate the time.Let's start with the headwind. A headwind is blowing against the direction the train is moving, so it will slow the train down. The headwind speed is given as 20 kmph. So, I think I need to subtract this from the train's speed to get the effective speed. That would be 68 kmph minus 20 kmph, which equals 48 kmph. So, the train's effective speed is now 48 kmph.Now, I need to consider the 5% incline. I'm not entirely sure how a 5% incline affects the train's speed. I think it means that the train has to work harder to go uphill, which might reduce its speed further. But I'm not sure by how much. Maybe I need to look up how inclines affect train speeds, but since I'm just solving this problem, I'll assume that the 5% incline also reduces the speed by 5%. So, I'll take the already reduced speed of 48 kmph and reduce it by 5%. To calculate 5% of 48 kmph, I can do 0.05 times 48, which is 2.4 kmph. So, subtracting that from 48 kmph gives me 45.6 kmph. Now, the train's effective speed is 45.6 kmph.But wait, I'm not sure if the 5% incline reduces the speed by 5% or if it's a different kind of reduction. Maybe it's better to think of it as the train's speed being reduced proportionally to the incline. I think in some contexts, a 5% incline means that for every 100 meters horizontally, the track rises 5 meters. But I'm not sure how that translates to speed reduction. Maybe it's better to stick with the assumption that the speed is reduced by 5%.Okay, so with the effective speed now at 45.6 kmph, I need to convert this speed into meters per second to match the length of the train, which is in meters. I remember that 1 kmph is equal to 1000 meters per hour, and there are 3600 seconds in an hour. So, to convert kmph to m/s, I can multiply by 1000 and divide by 3600, which simplifies to multiplying by 5/18.So, 45.6 kmph times 5/18 equals... let's see, 45.6 times 5 is 228, and 228 divided by 18 is 12.666... m/s. So, approximately 12.67 m/s.Now, I can use the formula Time = Distance / Speed. The distance is 170 meters, and the speed is 12.67 m/s. So, 170 divided by 12.67 equals... let's calculate that. 170 divided by 12.67 is approximately 13.41 seconds.Wait, that seems a bit long. Let me double-check my calculations. Maybe I made a mistake somewhere. Let's go back step by step.First, the train's speed is 68 kmph, and the headwind is 20 kmph. So, the effective speed is 68 minus 20, which is 48 kmph. That seems correct.Then, I assumed a 5% incline reduces the speed by 5%, so 5% of 48 is 2.4, and 48 minus 2.4 is 45.6 kmph. That seems reasonable, but I'm not entirely sure if that's the correct way to apply the incline's effect. Maybe the incline affects the speed differently, like through some other factor, but without more information, I'll stick with this assumption.Next, converting 45.6 kmph to m/s: 45.6 times 5/18. Let's do that again. 45.6 times 5 is 228, and 228 divided by 18 is 12.666... m/s, which is approximately 12.67 m/s. That seems correct.Finally, calculating the time: 170 meters divided by 12.67 m/s equals approximately 13.41 seconds. Hmm, that seems a bit long for a train passing a pole, but considering the speed reduction due to headwind and incline, maybe it's reasonable.Alternatively, maybe I should not have reduced the speed by 5% for the incline. Perhaps the 5% incline doesn't reduce the speed by 5%, but instead affects it in a different way. Maybe it's better to think of the incline as adding resistance, which would require more power from the train, but without knowing the train's power or mass, I can't calculate the exact speed reduction. So, maybe I should ignore the incline's effect on speed since I don't have enough information.If I ignore the incline, then the effective speed is just 48 kmph, which is 48 times 5/18 equals 13.33 m/s. Then, time equals 170 divided by 13.33, which is approximately 12.75 seconds. That seems more reasonable.But the problem did mention a 5% incline, so I think I should include it somehow. Maybe the 5% incline increases the effective grade, which could reduce the speed by a certain percentage. But without specific data on how the incline affects the speed, it's hard to be precise. Maybe I can look up typical speed reductions on a 5% incline, but I don't have access to that information right now.Alternatively, perhaps the 5% incline is just additional information that doesn't affect the calculation, and the main factors are the headwind and the train's speed. In that case, I should proceed with the headwind adjustment only.Given that, I think the more accurate answer would be approximately 12.75 seconds, considering only the headwind's effect on the train's speed. The 5% incline might complicate things further, but without specific data on how it affects the speed, it's safer to exclude it from the calculation.So, to summarize, the train's speed is reduced by the headwind to 48 kmph, which converts to approximately 13.33 m/s. The time to pass the pole is the length of the train divided by this speed, which is about 12.75 seconds.

💡First, I need to determine the original dimensions of the rectangle. Let’s denote the original length as ( L ) and the original width as ( W ).When the length is doubled, the new length becomes ( 2L ). Similarly, when the width is tripled, the new width becomes ( 3W ).The area of the new rectangle is given by multiplying the new length and the new width:[text{New Area} = 2L times 3W = 6LW]We know that the new area is 1800 square meters:[6LW = 1800]To find the original area ( LW ), I divide both sides of the equation by 6:[LW = frac{1800}{6} = 300]Therefore, the area of the original rectangle is 300 square meters.

💡Okay, so I have this problem here where I'm told that the surface area of a sphere circumscribed around a cube is 16π. I need to find the edge length of the cube. Hmm, okay, let's break this down step by step.First, I remember that a sphere circumscribed around a cube means that all the vertices of the cube touch the sphere. So, the sphere is perfectly surrounding the cube, and the distance from the center of the sphere to any vertex of the cube is the radius of the sphere.Now, the surface area of a sphere is given by the formula 4πr², where r is the radius. The problem says this surface area is 16π. So, I can set up the equation:4πr² = 16πTo find r, I can divide both sides by π:4r² = 16Then, divide both sides by 4:r² = 4Taking the square root of both sides gives:r = 2So, the radius of the sphere is 2 units.Now, since the sphere is circumscribed around the cube, the radius of the sphere is related to the edge length of the cube. I recall that the space diagonal of the cube is equal to the diameter of the sphere. The space diagonal of a cube with edge length a is given by a√3. This comes from the Pythagorean theorem extended to three dimensions.So, the space diagonal is a√3, and this is equal to the diameter of the sphere, which is 2r. Since we found r = 2, the diameter is 4.Therefore, we have:a√3 = 4To solve for a, divide both sides by √3:a = 4 / √3But it's usually not good practice to have a square root in the denominator, so I can rationalize it by multiplying numerator and denominator by √3:a = (4√3) / 3So, the edge length of the cube is 4√3 divided by 3.Let me double-check my steps to make sure I didn't make a mistake. Starting from the surface area, I correctly found the radius. Then, I related the space diagonal of the cube to the diameter of the sphere, which is correct. The formula for the space diagonal is indeed a√3, so that seems right. Solving for a, I got 4√3 over 3, which is one of the answer choices. Looking back at the options, D is 4√3/3, so that must be the correct answer.I think that's solid. I don't see any errors in my reasoning. It all flows logically from the surface area to the radius, then to the space diagonal, and finally to the edge length of the cube.

💡Okay, so I have this problem about a geometric sequence. Let me try to understand it step by step. The sequence given is {2^{a_n}} with a common ratio of 2. Hmm, so that means each term is 2 times the previous term. So, if I denote the first term as 2^{a_1}, then the next term is 2^{a_2} = 2 * 2^{a_1} = 2^{a_1 + 1}. Similarly, the third term would be 2^{a_3} = 2 * 2^{a_2} = 2^{a_2 + 1} = 2^{a_1 + 2}, and so on. Wait, so does that mean that the exponent a_n is increasing by 1 each time? Let me check. If 2^{a_n} is a geometric sequence with ratio 2, then 2^{a_n} = 2^{a_1} * 2^{n-1} = 2^{a_1 + n - 1}. Therefore, 2^{a_n} = 2^{a_1 + n - 1}, which implies that a_n = a_1 + n - 1. So, the sequence {a_n} is an arithmetic sequence with common difference 1. That makes sense because the exponents are increasing linearly.So, part (1) is asking for the general term formula for {a_n}. Since it's an arithmetic sequence, the general term is a_n = a_1 + (n - 1)d, where d is the common difference. In this case, d = 1, so a_n = a_1 + n - 1. But I don't know the value of a_1 yet. The problem also gives me another equation: a_4 + a_3^2 = 21. Let me write that down. Since a_n is an arithmetic sequence, a_4 = a_1 + 3 and a_3 = a_1 + 2. So, substituting into the equation: (a_1 + 3) + (a_1 + 2)^2 = 21. Let me expand that:(a_1 + 3) + (a_1^2 + 4a_1 + 4) = 21Simplify:a_1 + 3 + a_1^2 + 4a_1 + 4 = 21Combine like terms:a_1^2 + 5a_1 + 7 = 21Subtract 21 from both sides:a_1^2 + 5a_1 - 14 = 0Now, I need to solve this quadratic equation for a_1. Let's use the quadratic formula: a = 1, b = 5, c = -14.Discriminant D = b^2 - 4ac = 25 + 56 = 81So, a_1 = [-5 ± sqrt(81)] / 2 = [-5 ± 9]/2That gives two solutions:a_1 = (-5 + 9)/2 = 4/2 = 2a_1 = (-5 - 9)/2 = -14/2 = -7So, a_1 can be either 2 or -7. Therefore, the general term formula for {a_n} is:If a_1 = 2, then a_n = 2 + (n - 1)*1 = n + 1If a_1 = -7, then a_n = -7 + (n - 1)*1 = n - 8So, that should be the answer for part (1). Now, moving on to part (2). It says if a_1 > 0, find the sum of the first n terms, S_n, for the sequence {1 / [(2a_n - 1)(2n - 1)]}.Since a_1 > 0, from part (1), a_1 must be 2, so a_n = n + 1. Therefore, let's substitute a_n into the expression:1 / [(2a_n - 1)(2n - 1)] = 1 / [(2(n + 1) - 1)(2n - 1)] = 1 / [(2n + 2 - 1)(2n - 1)] = 1 / [(2n + 1)(2n - 1)]So, the sequence becomes 1 / [(2n + 1)(2n - 1)]. I need to find the sum S_n of the first n terms of this sequence.This looks like a telescoping series. Maybe I can use partial fractions to decompose it. Let me try that.Let me write 1 / [(2n + 1)(2n - 1)] as A / (2n - 1) + B / (2n + 1). So, 1 = A(2n + 1) + B(2n - 1)Let me solve for A and B. Expanding the right side:1 = (2A + 2B)n + (A - B)Since this must hold for all n, the coefficients of n and the constant term must be equal on both sides. Coefficient of n: 2A + 2B = 0Constant term: A - B = 1From the first equation: 2A + 2B = 0 => A + B = 0 => A = -BSubstitute into the second equation: (-B) - B = 1 => -2B = 1 => B = -1/2Then, A = -B = 1/2So, the partial fraction decomposition is:1 / [(2n + 1)(2n - 1)] = (1/2) / (2n - 1) - (1/2) / (2n + 1)Therefore, each term in the sequence can be written as:1 / [(2n + 1)(2n - 1)] = (1/2)[1/(2n - 1) - 1/(2n + 1)]So, the sum S_n is the sum from k = 1 to n of (1/2)[1/(2k - 1) - 1/(2k + 1)]Let me write out the terms to see the telescoping effect:For k = 1: (1/2)[1/1 - 1/3]For k = 2: (1/2)[1/3 - 1/5]For k = 3: (1/2)[1/5 - 1/7]...For k = n: (1/2)[1/(2n - 1) - 1/(2n + 1)]When we add all these up, most of the intermediate terms cancel out:S_n = (1/2)[1/1 - 1/3 + 1/3 - 1/5 + 1/5 - 1/7 + ... + 1/(2n - 1) - 1/(2n + 1)]All the -1/3 and +1/3 cancel, -1/5 and +1/5 cancel, and so on, until the last term is -1/(2n + 1). So, we're left with:S_n = (1/2)[1 - 1/(2n + 1)]Simplify this expression:1 - 1/(2n + 1) = (2n + 1 - 1)/(2n + 1) = 2n/(2n + 1)Therefore, S_n = (1/2)(2n/(2n + 1)) = n/(2n + 1)So, the sum of the first n terms is n/(2n + 1). Let me just double-check my steps to make sure I didn't make any mistakes. Starting from the partial fractions, I found A and B correctly. Then, when I wrote out the terms, the telescoping worked as expected, leaving only the first and the last terms. The algebra seems correct, so I think this is the right answer.**Final Answer**(1) The general term formula for {a_n} is boxed{a_n = n + 1} or boxed{a_n = n - 8}.(2) The sum of the first n terms is boxed{dfrac{n}{2n + 1}}.

💡First, I need to determine the number of legs required for each type of furniture. Typically, a chair has 4 legs and a table also has 4 legs.Next, I'll calculate the total number of legs for the chairs. There are 6 chairs, and each chair has 4 legs. So, 6 chairs multiplied by 4 legs per chair equals 24 legs.Then, I'll calculate the total number of legs for the tables. There are 4 tables, and each table has 4 legs. So, 4 tables multiplied by 4 legs per table equals 16 legs.Finally, I'll add the legs from the chairs and the legs from the tables to find the total number of furniture legs. 24 legs from chairs plus 16 legs from tables equals 40 legs in total.

💡Okay, so I have this problem here: I need to find a unique polynomial Q(x) of degree 4 with rational coefficients and leading coefficient 1, which has √3 + √7 as a root. Then, I have to find Q(2). Hmm, okay, let me think about how to approach this.First, I remember that if a polynomial with rational coefficients has an irrational root like √3 + √7, then its conjugate roots must also be included. So, the conjugate of √3 + √7 would be √3 - √7, right? Because we're changing the sign of the square root part. Similarly, maybe also -√3 + √7 and -√3 - √7? Wait, no, actually, I think for polynomials with rational coefficients, the minimal polynomial would include all combinations of the square roots with both positive and negative signs. So, if √3 + √7 is a root, then √3 - √7, -√3 + √7, and -√3 - √7 should all be roots as well. That makes sense because the polynomial needs to have rational coefficients, so it can't just have one irrational root without its conjugates.So, if that's the case, then the polynomial Q(x) should have these four roots: √3 + √7, √3 - √7, -√3 + √7, and -√3 - √7. Since it's a degree 4 polynomial, that should cover all the roots. Now, how do I construct this polynomial?I think I can start by considering the minimal polynomial for √3 + √7. To find that, I can let x = √3 + √7 and then try to eliminate the square roots to get a polynomial equation with rational coefficients.Let me set x = √3 + √7. Then, if I square both sides, I get x² = (√3 + √7)². Expanding that, it's (√3)² + 2*√3*√7 + (√7)², which is 3 + 2√21 + 7. So, x² = 10 + 2√21.Hmm, okay, so x² = 10 + 2√21. Now, I still have a square root in there, √21. To eliminate that, I can isolate the square root term and then square again. Let's subtract 10 from both sides: x² - 10 = 2√21. Then, divide both sides by 2: (x² - 10)/2 = √21.Now, square both sides again to eliminate the square root: [(x² - 10)/2]^2 = (√21)^2. That simplifies to (x² - 10)^2 / 4 = 21. Multiply both sides by 4: (x² - 10)^2 = 84.Expanding the left side: (x² - 10)^2 = x^4 - 20x² + 100. So, x^4 - 20x² + 100 = 84. Subtract 84 from both sides: x^4 - 20x² + 16 = 0.So, the minimal polynomial is x^4 - 20x² + 16. Wait, but the problem says the polynomial has rational coefficients and leading coefficient 1, which this does. So, is this the Q(x) we're looking for? Let me check.Yes, it's a degree 4 polynomial with rational coefficients, leading coefficient 1, and √3 + √7 is a root. So, that must be Q(x). Therefore, Q(x) = x^4 - 20x² + 16.Now, the question asks for Q(2). So, I need to substitute x = 2 into this polynomial. Let's compute that step by step.First, compute x^4 when x = 2: 2^4 = 16.Next, compute 20x²: 20*(2)^2 = 20*4 = 80.Then, the constant term is +16.So, putting it all together: Q(2) = 16 - 80 + 16.Wait, 16 - 80 is -64, and then -64 + 16 is -48. Hmm, that seems negative. Is that correct?Wait, let me double-check my calculations because that seems a bit unexpected. Let me go through each term again.x^4 when x=2: 2*2*2*2 = 16. Correct.20x² when x=2: 20*(2*2) = 20*4 = 80. Correct.Constant term: +16. Correct.So, Q(2) = 16 - 80 + 16 = (16 + 16) - 80 = 32 - 80 = -48. Hmm, so is it -48? But wait, in the initial problem statement, the user had a different approach and got 16. Did I make a mistake somewhere?Wait, let me go back to how I constructed Q(x). I started with x = √3 + √7, squared it, got x² = 10 + 2√21, then squared again to get rid of the square root, leading to x^4 - 20x² + 16 = 0. So, Q(x) = x^4 - 20x² + 16.But in the initial problem, the user had a different polynomial: x^4 - 4x² + 16. Wait, that's different from what I have here. So, maybe I made a mistake in my calculation.Let me check the squaring steps again.Starting with x = √3 + √7.x² = (√3 + √7)^2 = 3 + 2√21 + 7 = 10 + 2√21. That seems correct.Then, x² - 10 = 2√21.Divide both sides by 2: (x² - 10)/2 = √21.Square both sides: [(x² - 10)/2]^2 = 21.Expand the left side: (x² - 10)^2 / 4 = 21.Multiply both sides by 4: (x² - 10)^2 = 84.Expand (x² - 10)^2: x^4 - 20x² + 100 = 84.Subtract 84: x^4 - 20x² + 16 = 0. Hmm, that seems correct.Wait, so why does the initial problem have a different polynomial? Maybe I misunderstood the initial problem.Wait, let me check the initial problem again. It says: "There is a unique polynomial Q(x) of degree 4 with rational coefficients and leading coefficient 1 which has √3 + √7 as a root. What is Q(2)?" So, the user's initial thought process had a different polynomial, but I think my calculation is correct.Wait, let me see. The user started by considering (x - (√3 + √7))(x - (√3 - √7)) = x² - 2√3 x - 4. Then multiplied by (x² + 2√3 x - 4) to get x^4 - 4x² + 16. But that seems different from my result.Wait, so which one is correct? Let me see.If I take (√3 + √7) as a root, then the minimal polynomial should have degree 4 because Q(√3 + √7) is degree 4 over Q. So, both approaches are trying to find the minimal polynomial, but perhaps the user made a mistake in their initial steps.Wait, let me check the user's steps. They said:(x - (√3 + √7))(x - (√3 - √7)) = x² - 2√3 x + ( (√3)^2 - (√7)^2 ) = x² - 2√3 x - 4.Wait, hold on, (√3)^2 is 3, (√7)^2 is 7, so 3 - 7 is -4. So, that part is correct. So, the quadratic factor is x² - 2√3 x - 4.Then, the user multiplied this by its conjugate, which would be (x² + 2√3 x - 4). Let me compute that:(x² - 2√3 x - 4)(x² + 2√3 x - 4) = ?Let me multiply these two quadratics:First, x² * x² = x^4.Then, x² * 2√3 x = 2√3 x³.x² * (-4) = -4x².Then, (-2√3 x) * x² = -2√3 x³.(-2√3 x) * 2√3 x = (-2√3)(2√3) x² = (-4*3) x² = -12x².(-2√3 x) * (-4) = 8√3 x.Then, (-4) * x² = -4x².(-4) * 2√3 x = -8√3 x.(-4) * (-4) = 16.Now, let's combine all these terms:x^4 + 2√3 x³ - 4x² - 2√3 x³ - 12x² + 8√3 x - 4x² - 8√3 x + 16.Now, let's combine like terms:x^4.2√3 x³ - 2√3 x³ = 0.-4x² -12x² -4x² = (-4 -12 -4)x² = -20x².8√3 x -8√3 x = 0.And the constant term is +16.So, putting it all together: x^4 - 20x² + 16. So, that's the same as what I got earlier. So, the user initially thought they had x^4 - 4x² + 16, but actually, when multiplying out, it's x^4 - 20x² + 16.Wait, so the user made a mistake in their initial calculation when they thought the product was x^4 - 4x² + 16. It's actually x^4 - 20x² + 16. So, that explains the discrepancy.Therefore, the correct polynomial is x^4 - 20x² + 16, and Q(2) is 16 - 80 + 16 = -48.But wait, the user's initial answer was 16, but that was based on an incorrect polynomial. So, I think my calculation is correct, and the correct answer should be -48.Wait, but let me double-check by plugging in x=2 into x^4 - 20x² + 16.2^4 = 16.20*(2)^2 = 20*4 = 80.So, 16 - 80 + 16 = (16 + 16) - 80 = 32 - 80 = -48. Yes, that seems correct.Alternatively, maybe I can verify by another method. Let me compute Q(2) using another approach.Since Q(x) = (x² - 2√3 x - 4)(x² + 2√3 x - 4), as the user initially thought, but then when expanded, it's x^4 - 20x² + 16.So, Q(2) = (2² - 2√3*2 - 4)(2² + 2√3*2 - 4).Compute each quadratic at x=2:First quadratic: 4 - 4√3 - 4 = (-4√3).Second quadratic: 4 + 4√3 - 4 = 4√3.So, Q(2) = (-4√3)(4√3) = (-4√3)(4√3) = (-4*4)*(√3*√3) = (-16)*(3) = -48. Yep, same result.So, that confirms that Q(2) is indeed -48.Wait, but the user initially thought it was 16, but that was because they incorrectly expanded the product. So, the correct answer is -48.But just to be thorough, let me check if Q(x) is indeed the minimal polynomial. Let me see if √3 + √7 satisfies x^4 - 20x² + 16 = 0.Compute (√3 + √7)^4 - 20*(√3 + √7)^2 + 16.First, compute (√3 + √7)^2 = 3 + 2√21 + 7 = 10 + 2√21.Then, (√3 + √7)^4 = (10 + 2√21)^2 = 100 + 40√21 + 4*21 = 100 + 40√21 + 84 = 184 + 40√21.Now, compute 20*(√3 + √7)^2 = 20*(10 + 2√21) = 200 + 40√21.So, Q(√3 + √7) = (√3 + √7)^4 - 20*(√3 + √7)^2 + 16 = (184 + 40√21) - (200 + 40√21) + 16.Simplify: 184 + 40√21 - 200 - 40√21 + 16 = (184 - 200 + 16) + (40√21 - 40√21) = 0 + 0 = 0. So, yes, it satisfies the equation. Therefore, Q(x) is correct.So, Q(2) = -48.Wait, but the user initially thought it was 16, but that was based on an incorrect polynomial. So, I think my answer is correct.Alternatively, maybe I can use another method to find Q(2). Let me think about it.Since Q(x) is a polynomial with rational coefficients and leading coefficient 1, and it has √3 + √7 as a root, then Q(2) is just the value of the polynomial at x=2. Since we've already derived Q(x) as x^4 - 20x² + 16, plugging in x=2 gives us -48.Alternatively, maybe I can compute Q(2) without expanding the polynomial. Let me see.We know that Q(x) = (x² - 2√3 x - 4)(x² + 2√3 x - 4). So, Q(2) = (4 - 4√3 - 4)(4 + 4√3 - 4) = (-4√3)(4√3) = (-4√3)(4√3) = -16*3 = -48. Yep, same result.So, I think I'm confident that Q(2) is -48.Wait, but just to make sure, let me compute it step by step again.Compute x^4 when x=2: 2^4 = 16.Compute 20x² when x=2: 20*(2)^2 = 20*4 = 80.So, Q(2) = 16 - 80 + 16 = (16 + 16) - 80 = 32 - 80 = -48.Yes, that's correct.So, the final answer is -48.Wait, but in the initial problem, the user had a different polynomial and got 16. So, maybe they made a mistake in their calculation. But in any case, based on my calculations, the correct answer is -48.I think I've double-checked enough times, so I'm pretty sure that's the correct answer.