💡First, let's denote the total number of workers in the company as ( W ).Given that a third of the workers do not have a retirement plan, the number of workers without a retirement plan is:[frac{1}{3}W]20% of these workers are women, so the number of women without a retirement plan is:[0.20 times frac{1}{3}W = frac{1}{15}W]Since there are approximately 140 women workers in total, we set up the equation:[frac{1}{15}W = 140]Solving for ( W ):[W = 140 times 15 = 2100]So, there are 2100 workers in total.Next, 40% of the workers who do have a retirement plan are men. Since two-thirds of the workers have a retirement plan:[frac{2}{3}W]The number of men with a retirement plan is:[0.40 times frac{2}{3}W = frac{8}{30}W]Substituting ( W = 2100 ):[frac{8}{30} times 2100 = 560]Now, 80% of the workers without a retirement plan are men:[0.80 times frac{1}{3}W = frac{8}{30}W]Substituting ( W = 2100 ):[frac{8}{30} times 2100 = 560]Finally, the total number of men workers is:[560 + 560 = 1120]

💡Okay, so I have this problem here: I need to find the sum of the arithmetic sequence starting from 3, then 5, then 7, and so on, up to the term (2n + 7). Hmm, arithmetic sequences, right? They have a common difference between consecutive terms. Let me think about how to approach this.First, let me recall what an arithmetic sequence is. It's a sequence where each term after the first is obtained by adding a constant difference. In this case, starting from 3, the next term is 5, which is 3 + 2, then 7, which is 5 + 2, and so on. So the common difference, d, is 2. That seems straightforward.Now, the problem is asking for the sum of this sequence up to the term (2n + 7). So, I need to figure out how many terms are in this sequence and then use the formula for the sum of an arithmetic series.Wait, let me make sure I understand the sequence correctly. The first term is 3, the second is 5, the third is 7, and it goes on until (2n + 7). So, each term can be represented as 2k + 1, where k starts from 1. Let me check: when k=1, 2(1)+1=3; k=2, 2(2)+1=5; k=3, 2(3)+1=7. Yeah, that works. So, the general term is 2k + 1.But wait, the last term is given as (2n + 7). Hmm, so if the general term is 2k + 1, then setting 2k + 1 equal to 2n + 7 should give me the value of k for the last term. Let me solve for k:2k + 1 = 2n + 7 Subtract 1 from both sides: 2k = 2n + 6 Divide both sides by 2: k = n + 3So, the last term is the (n + 3)th term of the sequence. That means there are (n + 3) terms in total.Now, the formula for the sum of an arithmetic series is S = (number of terms)/2 * (first term + last term). So, applying that here, the sum S should be:S = [(n + 3)/2] * [3 + (2n + 7)]Let me compute that step by step. First, compute the sum inside the brackets:3 + (2n + 7) = 2n + 10So, now the formula becomes:S = [(n + 3)/2] * (2n + 10)Hmm, let me simplify this expression. First, notice that (2n + 10) can be factored as 2(n + 5). So, substituting back:S = [(n + 3)/2] * 2(n + 5)The 2 in the numerator and the denominator cancel out, so we're left with:S = (n + 3)(n + 5)Now, let's multiply these two binomials:(n + 3)(n + 5) = n(n + 5) + 3(n + 5) = n^2 + 5n + 3n + 15 = n^2 + 8n + 15So, the sum of the sequence is n squared plus 8n plus 15. Let me just double-check my steps to make sure I didn't make a mistake.1. Identified the common difference as 2. Correct.2. Expressed the general term as 2k + 1. Correct.3. Set 2k + 1 equal to the last term 2n + 7 and solved for k, getting k = n + 3. That seems right.4. Number of terms is n + 3. Makes sense.5. Applied the sum formula: (number of terms)/2 * (first term + last term). Plugged in the values correctly.6. Simplified step by step, factoring out the 2, canceling it, then multiplying the binomials. All steps look good.I think that's solid. So, the final answer should be n squared plus 8n plus 15.**Final Answer**The sum of the sequence is boxed{n^2 + 8n + 15}.

💡First, I'll define the time it takes for each pipe to drain the tank. Let ( t_P ) be the time for pipe P, ( t_Q ) for pipe Q, and ( t_R ) for pipe R.According to the problem, pipe P drains the tank in some fraction of the time it takes pipe Q, which I'll denote as ( frac{t_P}{t_Q} ). Additionally, pipe P drains the tank in ( frac{3}{3} ) the time it takes pipe R, simplifying to ( t_P = t_R ).When all three pipes work together, pipe Q drains ( 0.2727272727272727 ) of the tank, which is equivalent to ( frac{3}{11} ). Therefore, the remaining portion drained by pipes P and R together is ( 1 - frac{3}{11} = frac{8}{11} ).Since ( t_P = t_R ), pipes P and R drain the tank at the same rate. Thus, each drains half of the remaining ( frac{8}{11} ), which is ( frac{4}{11} ) each.To find the fraction of time it takes for pipe P compared to pipe Q, I'll set up the ratio of the work done by pipes P and Q:[frac{t_P}{t_Q} = frac{text{Work done by Q}}{text{Work done by P}} = frac{frac{3}{11}}{frac{4}{11}} = frac{3}{4}]Therefore, pipe P can drain the tank in ( frac{3}{4} ) the time it takes pipe Q.

💡Okay, so I have this problem here: If the point (n, 3) is on the graph of the function y = 3^x, then what is the value of cos(π/(3n))? Hmm, let me try to figure this out step by step.First, I know that if a point (n, 3) is on the graph of y = 3^x, that means when x is n, y is 3. So, substituting n into the function should give me 3. So, mathematically, that would be 3^n = 3. Okay, that makes sense.Now, I need to solve for n. So, 3^n = 3. Hmm, since 3 is the same as 3^1, I can rewrite the equation as 3^n = 3^1. Since the bases are the same, the exponents must be equal. So, n must be 1. That seems straightforward.Alright, so now I know that n is 1. The next part of the problem asks for the value of cos(π/(3n)). Since n is 1, I can substitute that into the expression. So, it becomes cos(π/(3*1)) which simplifies to cos(π/3).Now, I need to remember the value of cos(π/3). I recall that π/3 radians is 60 degrees. From the unit circle, I know that cos(60°) is 0.5 or 1/2. So, cos(π/3) is 1/2.Wait, let me double-check that. I remember that cos(0) is 1, cos(π/2) is 0, and cos(π) is -1. So, at π/3, which is between 0 and π/2, the cosine value should be positive and less than 1. Yes, 1/2 is correct because in a 30-60-90 triangle, the sides are in the ratio 1:√3:2, and the cosine of 60 degrees is adjacent over hypotenuse, which is 1/2.So, putting it all together, since n is 1, the expression cos(π/(3n)) becomes cos(π/3), which is 1/2. Therefore, the value is 1/2.I think that's it. I don't see any mistakes in my reasoning. I solved for n correctly by equating the exponents, substituted it into the cosine function, and recalled the correct value from the unit circle. Yeah, I'm confident that the answer is 1/2.**Final Answer**The value of cos dfrac {pi}{3n} is boxed{dfrac{1}{2}}.

💡First, I need to determine the total distance of the trip, which is 950 km.Next, I'll calculate the time taken for each segment of the journey by dividing the distance of each segment by the speed during that segment.1. For the first 200 km at 20 km/h:Time = 200 km / 20 km/h = 10 hours2. For the next 350 km at 30 km/h:Time = 350 km / 30 km/h ≈ 11.67 hours3. For the following 200 km at 25 km/h:Time = 200 km / 25 km/h = 8 hours4. For the remaining 200 km at 15 km/h:Time = 200 km / 15 km/h ≈ 13.33 hoursNow, I'll add up all the times to find the total time taken for the trip:Total time = 10 hours + 11.67 hours + 8 hours + 13.33 hours = 43 hoursFinally, I'll calculate the average speed by dividing the total distance by the total time:Average speed = 950 km / 43 hours ≈ 22.09 km/h

💡Alright, so I have this problem here: there's a line passing through the point M(1,1), and it intersects an ellipse given by the equation x²/4 + y²/3 = 1 at points A and B. The point M is the midpoint of the chord AB. I need to find the equation of the line AB, and there are four options provided.Hmm, okay. Let me think about how to approach this. I remember that when dealing with ellipses and midpoints of chords, there's something called the equation of the chord with a given midpoint. Maybe that's what I need to use here.First, let me recall the standard form of an ellipse: x²/a² + y²/b² = 1. In this case, a² is 4 and b² is 3, so a is 2 and b is sqrt(3). That might come in handy later.Now, the midpoint of the chord AB is given as M(1,1). I think the formula for the equation of the chord of an ellipse with a given midpoint (h,k) is T = S₁, where T is the tangent at the midpoint, but wait, no, that's for the tangent. Maybe it's different for the chord.Wait, I think it's actually similar to the tangent equation but adjusted for the midpoint. Let me try to recall. For the ellipse x²/a² + y²/b² = 1, the equation of the chord with midpoint (h,k) is given by (xx₁)/a² + (yy₁)/b² = (h²)/a² + (k²)/b². But actually, that might be the equation of the tangent at (x₁,y₁). Hmm, I'm getting confused.Wait, no. Maybe it's better to use the concept of the equation of the chord bisected at a point. I think the formula is (x h)/a² + (y k)/b² = (h²)/a² + (k²)/b². Let me check that.So, substituting h = 1 and k = 1, the equation becomes (x * 1)/4 + (y * 1)/3 = (1²)/4 + (1²)/3. Simplifying the right side: 1/4 + 1/3 = 3/12 + 4/12 = 7/12.So, the left side is x/4 + y/3, and the right side is 7/12. Therefore, the equation is x/4 + y/3 = 7/12.To make it look like the options given, let's multiply both sides by 12 to eliminate denominators: 3x + 4y = 7. So, the equation is 3x + 4y - 7 = 0.Looking at the options, that's option B: 3x + 4y - 7 = 0.Wait, but let me double-check my steps to make sure I didn't make a mistake. I used the formula for the chord bisected at (h,k), which I think is correct. Plugging in h=1 and k=1, I got x/4 + y/3 = 7/12, which simplifies to 3x + 4y = 7. That seems right.Alternatively, maybe I can approach this using the point-slope form of a line. Let's assume the line AB has a slope m. Since it passes through M(1,1), its equation would be y - 1 = m(x - 1). Then, I can find the points where this line intersects the ellipse and ensure that M is the midpoint.So, substituting y = m(x - 1) + 1 into the ellipse equation:x²/4 + [m(x - 1) + 1]²/3 = 1.Expanding that:x²/4 + [m²(x - 1)² + 2m(x - 1) + 1]/3 = 1.Let me expand the numerator:x²/4 + [m²(x² - 2x + 1) + 2m(x - 1) + 1]/3 = 1.Multiplying through by 12 to eliminate denominators:3x² + 4[m²(x² - 2x + 1) + 2m(x - 1) + 1] = 12.Expanding the terms inside the brackets:3x² + 4[m²x² - 2m²x + m² + 2mx - 2m + 1] = 12.Distribute the 4:3x² + 4m²x² - 8m²x + 4m² + 8mx - 8m + 4 = 12.Combine like terms:(3 + 4m²)x² + (-8m² + 8m)x + (4m² - 8m + 4) = 12.Bring the 12 to the left side:(3 + 4m²)x² + (-8m² + 8m)x + (4m² - 8m + 4 - 12) = 0.Simplify the constants:(3 + 4m²)x² + (-8m² + 8m)x + (4m² - 8m - 8) = 0.Now, since M(1,1) is the midpoint of AB, the sum of the roots x₁ + x₂ should be equal to 2 (since the midpoint x-coordinate is 1). For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a.So, in our case:x₁ + x₂ = [8m² - 8m] / [3 + 4m²] = 2.Setting that equal to 2:[8m² - 8m] / [3 + 4m²] = 2.Multiply both sides by [3 + 4m²]:8m² - 8m = 2(3 + 4m²).Expand the right side:8m² - 8m = 6 + 8m².Subtract 8m² from both sides:-8m = 6.Divide both sides by -8:m = -6/8 = -3/4.So, the slope m is -3/4. Therefore, the equation of the line is:y - 1 = (-3/4)(x - 1).Multiplying both sides by 4 to eliminate the fraction:4(y - 1) = -3(x - 1).Expanding:4y - 4 = -3x + 3.Bring all terms to one side:3x + 4y - 7 = 0.Which is the same as option B. So, both methods lead me to the same answer, which is reassuring.I think I'm confident now that the correct equation is 3x + 4y - 7 = 0, which is option B.