answer：First, I need to determine the man's rowing speed in still water. I know that when rowing with the stream, his effective speed is the sum of his rowing speed and the stream's speed. Conversely, when rowing against the stream, his effective speed is the difference between his rowing speed and the stream's speed.Let’s denote the man's rowing speed in still water as ( R ) and the stream's speed as ( S ).Given:- Speed with the stream (( R + S )) = 20 km/h- Speed against the stream (( R - S )) = 4 km/hI can set up two equations based on the given information:1. ( R + S = 20 )2. ( R - S = 4 )To find ( R ), I can add these two equations together:( (R + S) + (R - S) = 20 + 4 )( 2R = 24 )( R = 12 ) km/hTherefore, the man's rowing speed in still water is 12 km/h.

answer：Okay, so I have this integral to solve: the integral of (x³ - 6x² + 10x - 10) divided by (x + 1)(x - 2)³ dx. Hmm, that looks a bit complicated, but I think I can handle it by using partial fractions. Let me try to remember how partial fractions work.First, I know that when you have a rational function like this, you can break it down into simpler fractions. The denominator here is (x + 1)(x - 2)³, which means we have a linear factor (x + 1) and a repeated linear factor (x - 2) raised to the third power. So, for partial fractions, each distinct linear factor gets its own term, and for repeated factors, each power up to the highest gets its own term.So, I should set up the partial fraction decomposition as follows:(x³ - 6x² + 10x - 10) / [(x + 1)(x - 2)³] = A/(x + 1) + B/(x - 2) + C/(x - 2)² + D/(x - 2)³Okay, so I have four constants to find: A, B, C, and D. To find these, I need to combine the right-hand side over a common denominator and then equate the numerators.Let me write that out:A(x - 2)³ + B(x + 1)(x - 2)² + C(x + 1)(x - 2) + D(x + 1) = x³ - 6x² + 10x - 10Now, I need to expand each term on the left-hand side and then collect like terms.Starting with A(x - 2)³:A(x³ - 6x² + 12x - 8)Then B(x + 1)(x - 2)²:First, expand (x - 2)²: x² - 4x + 4Then multiply by (x + 1):(x + 1)(x² - 4x + 4) = x³ - 4x² + 4x + x² - 4x + 4 = x³ - 3x² + 0x + 4So, B times that is Bx³ - 3Bx² + 4BNext, C(x + 1)(x - 2):Expand (x + 1)(x - 2): x² - 2x + x - 2 = x² - x - 2So, C times that is Cx² - Cx - 2CFinally, D(x + 1):That's Dx + DNow, let's combine all these terms:A(x³ - 6x² + 12x - 8) + B(x³ - 3x² + 4x + 4) + C(x² - x - 2) + D(x + 1)Let me distribute each constant:A x³ - 6A x² + 12A x - 8A + B x³ - 3B x² + 4B x + 4B + C x² - C x - 2C + D x + DNow, let's collect like terms:x³ terms: A x³ + B x³ = (A + B) x³x² terms: -6A x² - 3B x² + C x² = (-6A - 3B + C) x²x terms: 12A x + 4B x - C x + D x = (12A + 4B - C + D) xconstant terms: -8A + 4B - 2C + DSo, putting it all together:(A + B) x³ + (-6A - 3B + C) x² + (12A + 4B - C + D) x + (-8A + 4B - 2C + D) = x³ - 6x² + 10x - 10Now, we can set up equations by equating the coefficients of corresponding powers of x.For x³: A + B = 1For x²: -6A - 3B + C = -6For x: 12A + 4B - C + D = 10For constants: -8A + 4B - 2C + D = -10So, we have a system of four equations:1. A + B = 12. -6A - 3B + C = -63. 12A + 4B - C + D = 104. -8A + 4B - 2C + D = -10Now, let's solve this system step by step.From equation 1: A = 1 - BLet's substitute A = 1 - B into the other equations.Equation 2: -6(1 - B) - 3B + C = -6Simplify:-6 + 6B - 3B + C = -6Combine like terms:-6 + 3B + C = -6Add 6 to both sides:3B + C = 0So, equation 2 becomes: 3B + C = 0Equation 3: 12(1 - B) + 4B - C + D = 10Simplify:12 - 12B + 4B - C + D = 10Combine like terms:12 - 8B - C + D = 10Subtract 12 from both sides:-8B - C + D = -2Equation 3 becomes: -8B - C + D = -2Equation 4: -8(1 - B) + 4B - 2C + D = -10Simplify:-8 + 8B + 4B - 2C + D = -10Combine like terms:-8 + 12B - 2C + D = -10Add 8 to both sides:12B - 2C + D = -2So, equation 4 becomes: 12B - 2C + D = -2Now, let's write the updated system:1. A = 1 - B2. 3B + C = 03. -8B - C + D = -24. 12B - 2C + D = -2Now, let's express C from equation 2: C = -3BSubstitute C = -3B into equations 3 and 4.Equation 3: -8B - (-3B) + D = -2Simplify:-8B + 3B + D = -2-5B + D = -2Equation 3 becomes: -5B + D = -2Equation 4: 12B - 2(-3B) + D = -2Simplify:12B + 6B + D = -218B + D = -2Equation 4 becomes: 18B + D = -2Now, we have:Equation 3: -5B + D = -2Equation 4: 18B + D = -2Subtract equation 3 from equation 4:(18B + D) - (-5B + D) = -2 - (-2)18B + D + 5B - D = 023B = 0So, B = 0Now, substitute B = 0 into equation 2: 3(0) + C = 0 => C = 0Then, from equation 3: -5(0) + D = -2 => D = -2From equation 1: A = 1 - B = 1 - 0 = 1So, we have A = 1, B = 0, C = 0, D = -2Therefore, the partial fraction decomposition is:1/(x + 1) + 0/(x - 2) + 0/(x - 2)² - 2/(x - 2)³Simplifying, we get:1/(x + 1) - 2/(x - 2)³Now, the integral becomes:∫ [1/(x + 1) - 2/(x - 2)³] dxLet's integrate term by term.First term: ∫ 1/(x + 1) dx = ln|x + 1| + CSecond term: ∫ -2/(x - 2)³ dxLet me rewrite this as -2 ∫ (x - 2)^(-3) dxIntegrate:-2 * [ (x - 2)^(-2) / (-2) ) ] + CSimplify:-2 * [ -1/(2(x - 2)²) ] + C = (1)/(x - 2)² + CSo, combining both integrals:ln|x + 1| + 1/(x - 2)² + CWait, let me double-check the integration of the second term.∫ -2/(x - 2)^3 dxLet u = x - 2, then du = dxSo, integral becomes -2 ∫ u^(-3) du = -2 [ u^(-2)/(-2) ] + C = -2 [ -1/(2u²) ] + C = (1)/u² + C = 1/(x - 2)² + CYes, that's correct.So, the final integral is:ln|x + 1| + 1/(x - 2)² + CI think that's the answer. Let me just make sure I didn't make any mistakes in the partial fractions part.We had A = 1, B = 0, C = 0, D = -2. Plugging back into the original expression:1/(x + 1) - 2/(x - 2)^3If I combine these, do I get the original numerator?Let me check:1/(x + 1) - 2/(x - 2)^3 = [ (x - 2)^3 - 2(x + 1) ] / [(x + 1)(x - 2)^3]Compute numerator:(x - 2)^3 - 2(x + 1) = (x³ - 6x² + 12x - 8) - 2x - 2 = x³ - 6x² + 10x - 10Yes, that's exactly the original numerator. So, the partial fractions were correct.Therefore, the integral is indeed ln|x + 1| + 1/(x - 2)² + C.**Final Answer**The indefinite integral is boxed{ln |x+1| + frac{1}{(x-2)^2} + C}.

answer：First, determine the total amount of work required to complete the job. This can be calculated by multiplying the number of workers by the number of days they are supposed to work.Next, calculate the amount of work completed by the initial 6 workers in the first 3 days.Subtract the work completed from the total work to find the remaining work.After 3 days, 4 additional workers join, making the total number of workers 10. Calculate the new work rate with 10 workers.Finally, determine how many additional days are needed to complete the remaining work with the increased number of workers.

answer：Okay, so I have this problem here: There's a point P(-4, a) on the terminal side of an angle α, and it's given that sinα multiplied by cosα equals √3 over 4. I need to find the value of a from the given options. Hmm, let me think about how to approach this.First, I remember that for any point (x, y) on the terminal side of an angle in standard position, the sine and cosine of that angle can be found using the coordinates. Specifically, sinα is y over the distance from the origin to the point, and cosα is x over that same distance. The distance itself is calculated using the Pythagorean theorem, right?So, for point P(-4, a), the x-coordinate is -4, and the y-coordinate is a. The distance from the origin to P, which I'll call r, should be sqrt((-4)^2 + a^2). Let me write that down:r = sqrt((-4)^2 + a^2) = sqrt(16 + a^2)Okay, so sinα is a/r, which is a over sqrt(16 + a^2), and cosα is -4/r, which is -4 over sqrt(16 + a^2). Got that.Now, the problem says that sinα multiplied by cosα equals √3 over 4. So, I can set up that equation:sinα * cosα = (a / sqrt(16 + a^2)) * (-4 / sqrt(16 + a^2)) = √3 / 4Let me simplify the left side of the equation. Multiplying those two fractions together gives:(a * -4) / (sqrt(16 + a^2) * sqrt(16 + a^2)) = (-4a) / (16 + a^2)So, now I have:(-4a) / (16 + a^2) = √3 / 4Hmm, okay. Let me write that as:(-4a) / (16 + a^2) = √3 / 4To solve for a, I can cross-multiply to get rid of the fractions. So, multiplying both sides by (16 + a^2) and by 4 gives:-4a * 4 = √3 * (16 + a^2)Simplifying both sides:-16a = 16√3 + √3 a^2Let me rearrange this equation to bring all terms to one side. I'll add 16a to both sides and subtract √3 a^2 from both sides:0 = √3 a^2 + 16√3 + 16aWait, that seems a bit messy. Let me write it as:√3 a^2 + 16a + 16√3 = 0Hmm, this is a quadratic equation in terms of a. It might be easier if I factor out √3, but let me see:√3 a^2 + 16a + 16√3 = 0Alternatively, maybe I can divide both sides by √3 to simplify:a^2 + (16 / √3) a + 16 = 0But that still looks complicated. Maybe I should instead multiply both sides by √3 to eliminate the radical in the denominator. Let me try that.Wait, actually, let me step back. Maybe I made a mistake in the cross-multiplication step. Let me double-check.Original equation after cross-multiplying:-4a * 4 = √3 * (16 + a^2)Which is:-16a = 16√3 + √3 a^2Yes, that seems correct. So, moving all terms to one side:√3 a^2 + 16a + 16√3 = 0Hmm, maybe I can factor this quadratic equation. Let me see if it's factorable.Looking for two numbers that multiply to (√3)(16√3) = 16*3 = 48 and add up to 16. Hmm, 12 and 4? 12*4=48 and 12+4=16. Wait, but the coefficients involve √3, so maybe I need to adjust.Alternatively, maybe I can use the quadratic formula. Let me set it up.The quadratic equation is:√3 a^2 + 16a + 16√3 = 0So, in the standard form ax^2 + bx + c = 0, the coefficients are:A = √3B = 16C = 16√3Applying the quadratic formula:a = [-B ± sqrt(B^2 - 4AC)] / (2A)Plugging in the values:a = [-16 ± sqrt(16^2 - 4 * √3 * 16√3)] / (2 * √3)Let me compute the discriminant first:D = 16^2 - 4 * √3 * 16√316^2 is 256.4 * √3 * 16√3 = 4 * 16 * (√3 * √3) = 64 * 3 = 192So, D = 256 - 192 = 64That's a perfect square, so sqrt(64) is 8.So, plugging back into the formula:a = [-16 ± 8] / (2 * √3)So, two possible solutions:a = (-16 + 8) / (2√3) = (-8) / (2√3) = -4 / √3Or,a = (-16 - 8) / (2√3) = (-24) / (2√3) = -12 / √3Simplify both solutions:-4 / √3 can be rationalized as (-4√3)/3Similarly, -12 / √3 is (-12√3)/3 = -4√3So, the two possible values for a are -4√3 and -4√3/3.Wait, but looking back at the options, option C is "-4√3 or -4/3√3", which matches my solutions.But let me double-check my steps to make sure I didn't make a mistake.Starting from sinα * cosα = √3 / 4Computed sinα as a / sqrt(16 + a^2), cosα as -4 / sqrt(16 + a^2)Multiplying them gives (-4a) / (16 + a^2) = √3 / 4Cross-multiplied: -16a = √3(16 + a^2)Which leads to √3 a^2 + 16a + 16√3 = 0Used quadratic formula, got a = -4√3 and -4√3/3Yes, that seems correct.But wait, let me think about the point P(-4, a). The x-coordinate is negative, so the point is in either the second or third quadrant. Since sinα is a / r, which is a over a positive number, so the sign of sinα depends on a. Similarly, cosα is negative because x is -4.Given that sinα * cosα is positive (√3 / 4), which is positive. So, sinα and cosα must have the same sign.But cosα is negative because x is negative. Therefore, sinα must also be negative, meaning a must be negative.So, a is negative, which matches our solutions of -4√3 and -4√3/3.Therefore, the correct answer should be option C: -4√3 or -4/3√3.I think that's it. I don't see any mistakes in my calculations, so I'm confident that option C is correct.

answer：Alright, so I've got this problem here about Sharon driving from her house to her mom's house. Normally, it takes her 200 minutes, but one day she hits a snowstorm after driving one-third of the way and has to reduce her speed by 30 miles per hour. Because of that, her total trip takes 310 minutes. I need to figure out how many miles Sharon's trip is. The answer choices are A) 216, B) 220, C) 225, D) 230, and E) 235. Okay, let's break this down step by step. First, I know that distance equals speed multiplied by time, so maybe I can use that formula somewhere here. Sharon usually takes 200 minutes, which is a little over 3 hours. But on the day with the snowstorm, her trip takes 310 minutes, which is about 5 hours and 10 minutes. That's a significant increase in time, so the snowstorm must have really slowed her down.Let me denote the total distance of the trip as ( x ) miles. Since she usually takes 200 minutes, her usual speed would be ( frac{x}{200} ) miles per minute. But wait, that's in miles per minute. Maybe it's easier to convert that to miles per hour because the speed reduction is given in miles per hour. There are 60 minutes in an hour, so her usual speed in miles per hour would be ( frac{x}{200} times 60 = frac{3x}{10} ) mph. That seems right because if she drives ( x ) miles in 200 minutes, which is ( frac{200}{60} = frac{10}{3} ) hours, so speed is ( frac{x}{frac{10}{3}} = frac{3x}{10} ) mph. Okay, that makes sense.Now, on the day with the snowstorm, she drives one-third of the way at her usual speed and then reduces her speed by 30 mph for the remaining two-thirds. So, let's figure out how long each part of the trip takes.First, the distance for the first part is ( frac{x}{3} ) miles. Her speed for this part is ( frac{3x}{10} ) mph, so the time taken for this part would be ( frac{frac{x}{3}}{frac{3x}{10}} ). Let me compute that:( frac{frac{x}{3}}{frac{3x}{10}} = frac{x}{3} times frac{10}{3x} = frac{10}{9} ) hours. Hmm, that's approximately 1.111 hours, which is about 66.666 minutes. Okay, so the first third of the trip takes her roughly 66.666 minutes.Now, for the remaining two-thirds of the trip, which is ( frac{2x}{3} ) miles, she reduces her speed by 30 mph. So her new speed is ( frac{3x}{10} - 30 ) mph. Let me write that down: new speed = ( frac{3x}{10} - 30 ) mph. Now, the time taken for the second part of the trip would be ( frac{frac{2x}{3}}{frac{3x}{10} - 30} ). Let me compute that:( frac{frac{2x}{3}}{frac{3x}{10} - 30} = frac{2x}{3} div left( frac{3x}{10} - 30 right) ).To simplify this, I can write it as:( frac{2x}{3} times frac{1}{frac{3x}{10} - 30} ).But this seems a bit complicated. Maybe I should convert everything into minutes to make it consistent with the total time given in minutes. Let's see.Wait, her usual speed is ( frac{3x}{10} ) mph, which is ( frac{3x}{10} times frac{1}{60} = frac{x}{200} ) miles per minute, which matches the initial speed given in miles per minute. So, maybe it's better to keep everything in minutes.Wait, no, because the speed reduction is given in mph, so it's probably better to work in hours for the speed and then convert the time accordingly.Alternatively, I can convert her usual speed to miles per minute, which is ( frac{x}{200} ) miles per minute, as I initially thought. Then, her speed reduction is 30 mph, which is 0.5 miles per minute because 30 miles per hour is 0.5 miles per minute (since 30 divided by 60 is 0.5). So, her reduced speed is ( frac{x}{200} - 0.5 ) miles per minute.Wait, that might be a better approach because then I can keep everything in minutes. Let me try that.So, her usual speed is ( frac{x}{200} ) miles per minute. After the snowstorm, her speed is ( frac{x}{200} - 0.5 ) miles per minute.Now, the first part of the trip is ( frac{x}{3} ) miles at ( frac{x}{200} ) miles per minute, so the time taken is ( frac{frac{x}{3}}{frac{x}{200}} = frac{x}{3} times frac{200}{x} = frac{200}{3} ) minutes, which is approximately 66.666 minutes, as I calculated earlier.The second part of the trip is ( frac{2x}{3} ) miles at ( frac{x}{200} - 0.5 ) miles per minute. So, the time taken for this part is ( frac{frac{2x}{3}}{frac{x}{200} - 0.5} ) minutes.The total time for the trip is the sum of the time for the first part and the time for the second part, which is given as 310 minutes. So, I can set up the equation:( frac{200}{3} + frac{frac{2x}{3}}{frac{x}{200} - 0.5} = 310 ).Now, I need to solve this equation for ( x ). Let me write it out:( frac{200}{3} + frac{frac{2x}{3}}{frac{x}{200} - frac{1}{2}} = 310 ).To simplify this, I can first subtract ( frac{200}{3} ) from both sides:( frac{frac{2x}{3}}{frac{x}{200} - frac{1}{2}} = 310 - frac{200}{3} ).Let me compute ( 310 - frac{200}{3} ). First, 310 is ( frac{930}{3} ), so:( frac{930}{3} - frac{200}{3} = frac{730}{3} ).So, the equation becomes:( frac{frac{2x}{3}}{frac{x}{200} - frac{1}{2}} = frac{730}{3} ).Now, let's simplify the left side. The numerator is ( frac{2x}{3} ) and the denominator is ( frac{x}{200} - frac{1}{2} ). Let me write the denominator with a common denominator:( frac{x}{200} - frac{1}{2} = frac{x - 100}{200} ).So, the left side becomes:( frac{frac{2x}{3}}{frac{x - 100}{200}} = frac{2x}{3} times frac{200}{x - 100} = frac{400x}{3(x - 100)} ).So, now the equation is:( frac{400x}{3(x - 100)} = frac{730}{3} ).I can multiply both sides by 3 to eliminate the denominators:( frac{400x}{x - 100} = 730 ).Now, I can write this as:( 400x = 730(x - 100) ).Expanding the right side:( 400x = 730x - 73000 ).Now, subtract 730x from both sides:( 400x - 730x = -73000 ).Which simplifies to:( -330x = -73000 ).Dividing both sides by -330:( x = frac{73000}{330} ).Let me compute that:Dividing 73000 by 330. Let's see, 330 times 221 is 73, 330*200=66,000, 330*21=6,930, so 66,000 + 6,930 = 72,930. That's close to 73,000. So, 330*221=72,930. The difference is 73,000 - 72,930 = 70. So, 70/330 is approximately 0.212. So, x ≈ 221.212 miles.Looking at the answer choices, 220 is the closest, so I think the answer is 220 miles.Wait, but let me double-check my calculations because sometimes when dealing with fractions, it's easy to make a mistake.Starting from the equation:( frac{400x}{3(x - 100)} = frac{730}{3} ).Multiplying both sides by 3:( frac{400x}{x - 100} = 730 ).Then, cross-multiplying:400x = 730(x - 100).Expanding:400x = 730x - 73,000.Subtracting 730x:-330x = -73,000.Dividing:x = (-73,000)/(-330) = 73,000/330.Simplify:Divide numerator and denominator by 10: 7,300/33.Now, 33*221 = 7,293, which is 7,300 - 7,293 = 7. So, 7,300/33 = 221 + 7/33 ≈ 221.212.So, x ≈ 221.212 miles.But the answer choices are 216, 220, 225, 230, 235. So, 221.212 is closest to 220, so I think the answer is 220 miles.Wait, but let me check if 220 miles works.If x = 220 miles, her usual speed is 220/200 = 1.1 miles per minute, which is 66 mph (since 1.1*60=66). On the day with the snowstorm, she drives the first third, which is 220/3 ≈ 73.333 miles, at 66 mph. The time taken for that is 73.333/66 ≈ 1.111 hours ≈ 66.666 minutes.Then, her speed reduces by 30 mph, so her new speed is 66 - 30 = 36 mph. The remaining distance is 220 - 73.333 ≈ 146.666 miles. Time taken at 36 mph is 146.666/36 ≈ 4.074 hours ≈ 244.444 minutes.Total time is 66.666 + 244.444 ≈ 311.11 minutes, which is close to 310 minutes, but not exact. Hmm, that's a bit off. Maybe I made a rounding error.Wait, let's do it more precisely.If x = 220, then:First part: 220/3 ≈ 73.333 miles at 66 mph. Time = 73.333/66 ≈ 1.111 hours ≈ 66.666 minutes.Second part: 220 - 73.333 ≈ 146.666 miles at 36 mph. Time = 146.666/36 ≈ 4.074 hours ≈ 244.444 minutes.Total time ≈ 66.666 + 244.444 ≈ 311.11 minutes.But the problem says the total time is 310 minutes. So, 311.11 is a bit more than 310. Maybe the exact value is 220, but due to rounding, it's a bit off. Alternatively, maybe I made a mistake in the calculations.Wait, let's try x = 220 exactly.First part: 220/3 miles at 66 mph.Time = (220/3)/66 = (220)/(3*66) = 220/198 ≈ 1.1111 hours ≈ 66.6667 minutes.Second part: (2*220)/3 = 440/3 miles at 36 mph.Time = (440/3)/36 = 440/(3*36) = 440/108 ≈ 4.0741 hours ≈ 244.4444 minutes.Total time ≈ 66.6667 + 244.4444 ≈ 311.1111 minutes.But the problem says 310 minutes, so it's off by about 1.1111 minutes. That's about 66.666 seconds. Hmm, that's a bit much. Maybe I need to check my initial setup.Wait, perhaps I made a mistake in converting the speed reduction. Let me go back.The problem says she reduces her speed by 30 mph. So, her new speed is (3x/10 - 30) mph, not (x/200 - 0.5) miles per minute. Wait, I think I confused the units here.Let me re-express everything in hours.Her usual speed is 3x/10 mph. After the snowstorm, her speed is 3x/10 - 30 mph.Time for the first part: (x/3) miles at 3x/10 mph.Time = distance/speed = (x/3)/(3x/10) = (x/3)*(10/(3x)) = 10/9 hours ≈ 1.1111 hours ≈ 66.6667 minutes.Time for the second part: (2x/3) miles at (3x/10 - 30) mph.Time = (2x/3)/(3x/10 - 30).Total time is 10/9 + (2x/3)/(3x/10 - 30) = 310 minutes.But 310 minutes is 310/60 ≈ 5.1667 hours.So, the equation is:10/9 + (2x/3)/(3x/10 - 30) = 5.1667.Let me write this in fractions to avoid decimal approximations.10/9 + (2x/3)/(3x/10 - 30) = 310/60 = 31/6.So, 10/9 + (2x/3)/(3x/10 - 30) = 31/6.Subtract 10/9 from both sides:(2x/3)/(3x/10 - 30) = 31/6 - 10/9.Find a common denominator for 31/6 and 10/9, which is 18.31/6 = 93/18, 10/9 = 20/18.So, 93/18 - 20/18 = 73/18.Thus, (2x/3)/(3x/10 - 30) = 73/18.Now, let's solve for x.Multiply both sides by (3x/10 - 30):2x/3 = (73/18)(3x/10 - 30).Simplify the right side:(73/18)*(3x/10) - (73/18)*30.Compute each term:First term: (73/18)*(3x/10) = (73*3x)/(18*10) = (219x)/180 = 73x/60.Second term: (73/18)*30 = (73*30)/18 = (2190)/18 = 121.6667.So, the equation becomes:2x/3 = 73x/60 - 121.6667.Multiply both sides by 60 to eliminate denominators:60*(2x/3) = 60*(73x/60) - 60*121.6667.Simplify:40x = 73x - 7300.Subtract 73x from both sides:40x - 73x = -7300.-33x = -7300.Divide both sides by -33:x = 7300/33 ≈ 221.2121 miles.Again, this is approximately 221.21 miles, which is closest to 220 miles. But when I plug in 220, the total time is about 311.11 minutes, which is a bit more than 310. Maybe the exact answer is 220, considering the options given, or perhaps I need to check if I made a mistake in the setup.Wait, perhaps I should express everything in minutes from the start.Her usual speed is x/200 miles per minute.After the snowstorm, her speed is x/200 - 0.5 miles per minute (since 30 mph is 0.5 miles per minute).Time for first part: (x/3)/(x/200) = (x/3)*(200/x) = 200/3 ≈ 66.6667 minutes.Time for second part: (2x/3)/(x/200 - 0.5).Total time: 200/3 + (2x/3)/(x/200 - 0.5) = 310.Let me write this equation:200/3 + (2x/3)/(x/200 - 1/2) = 310.Multiply both sides by 3 to eliminate denominators:200 + 2x/(x/200 - 1/2) = 930.Subtract 200 from both sides:2x/(x/200 - 1/2) = 730.Now, let me simplify the denominator:x/200 - 1/2 = (x - 100)/200.So, the equation becomes:2x / ((x - 100)/200) = 730.Which is:2x * (200)/(x - 100) = 730.Simplify:400x/(x - 100) = 730.Multiply both sides by (x - 100):400x = 730(x - 100).Expand:400x = 730x - 73000.Subtract 730x:-330x = -73000.Divide:x = 73000/330 ≈ 221.2121.Again, same result. So, x ≈ 221.21 miles. Since the answer choices are in whole numbers, 220 is the closest. But when I plug in 220, the total time is about 311.11 minutes, which is 1.11 minutes more than 310. Maybe the exact answer is 220, considering the options, or perhaps I need to check if I made a mistake in the setup.Wait, maybe I should consider that the speed reduction is 30 mph, so her new speed is (3x/10 - 30) mph, and I need to ensure that this speed is positive. So, 3x/10 - 30 > 0 => x > 100 miles. Since x is around 220, that's fine.Alternatively, maybe I should express the time in minutes directly.Let me try that.Her usual speed is x/200 miles per minute.After the snowstorm, her speed is x/200 - 0.5 miles per minute.Time for first part: (x/3)/(x/200) = 200/3 ≈ 66.6667 minutes.Time for second part: (2x/3)/(x/200 - 0.5).Total time: 200/3 + (2x/3)/(x/200 - 0.5) = 310.Let me write this equation:200/3 + (2x/3)/(x/200 - 1/2) = 310.Multiply both sides by 3:200 + 2x/(x/200 - 1/2) = 930.Subtract 200:2x/(x/200 - 1/2) = 730.Simplify denominator:x/200 - 1/2 = (x - 100)/200.So, equation becomes:2x / ((x - 100)/200) = 730.Which is:2x * (200)/(x - 100) = 730.Simplify:400x/(x - 100) = 730.Multiply both sides by (x - 100):400x = 730(x - 100).Expand:400x = 730x - 73000.Subtract 730x:-330x = -73000.Divide:x = 73000/330 ≈ 221.2121.Same result again. So, x ≈ 221.21 miles.Given the answer choices, 220 is the closest. Maybe the problem expects us to round to the nearest whole number, so 220 is the answer.Alternatively, perhaps I made a mistake in the setup. Let me check.Wait, when I converted 30 mph to miles per minute, I did 30/60 = 0.5 miles per minute. That's correct.Her reduced speed is x/200 - 0.5 miles per minute.Yes, that seems right.So, I think the answer is 220 miles, which is option B.

answer：Okay, so I have this problem in polar coordinates. It says that the circle ρ = 2 cos θ is tangent to the line 3ρ cos θ + 4ρ sin θ + a = 0, and I need to find the value of the real number a. Hmm, okay, let me try to break this down step by step.First, I remember that in polar coordinates, ρ is the radius and θ is the angle. The equation ρ = 2 cos θ represents a circle. I think this is a circle with a diameter along the x-axis, right? Because when θ is 0, ρ is 2, and when θ is π/2, ρ is 0. So it's a circle that touches the origin and has a radius of 1, centered at (1, 0) in Cartesian coordinates. Wait, is that right? Let me recall: the general form of a circle in polar coordinates is ρ = 2a cos θ, which is a circle with radius a centered at (a, 0). So in this case, a is 1, so the circle has a radius of 1 and is centered at (1, 0). Got it.Now, the line is given by 3ρ cos θ + 4ρ sin θ + a = 0. I need to find the value of a such that this line is tangent to the circle ρ = 2 cos θ. Tangent means they touch at exactly one point. So, I need to find the condition where this line touches the circle at only one point.I think the best approach is to convert both equations to Cartesian coordinates because I'm more familiar with working with lines and circles in Cartesian form. Let me recall that in polar coordinates, ρ cos θ is x and ρ sin θ is y. So, substituting these into the line equation:3ρ cos θ + 4ρ sin θ + a = 0 becomes 3x + 4y + a = 0.Okay, so the line in Cartesian coordinates is 3x + 4y + a = 0. Now, the circle is ρ = 2 cos θ, which I can also convert to Cartesian. Since ρ = 2 cos θ, multiplying both sides by ρ gives ρ² = 2ρ cos θ. But ρ² is x² + y², and ρ cos θ is x. So, substituting, we get x² + y² = 2x. Rearranging, x² - 2x + y² = 0. Completing the square for x, we get (x - 1)² + y² = 1. So, this is a circle with center at (1, 0) and radius 1. Perfect, that matches what I thought earlier.Now, I need to find the value of a such that the line 3x + 4y + a = 0 is tangent to the circle (x - 1)² + y² = 1. I remember that the condition for a line to be tangent to a circle is that the distance from the center of the circle to the line is equal to the radius of the circle. So, if I can compute the distance from the center (1, 0) to the line 3x + 4y + a = 0 and set it equal to the radius, which is 1, I can solve for a.The formula for the distance from a point (x₀, y₀) to the line Ax + By + C = 0 is |Ax₀ + By₀ + C| / sqrt(A² + B²). So, applying this formula, the distance from (1, 0) to the line 3x + 4y + a = 0 is |3*1 + 4*0 + a| / sqrt(3² + 4²) = |3 + a| / 5.Since this distance must equal the radius, which is 1, we have |3 + a| / 5 = 1. Multiplying both sides by 5, we get |3 + a| = 5. This absolute value equation has two solutions: 3 + a = 5 and 3 + a = -5.Solving the first equation: 3 + a = 5 ⇒ a = 5 - 3 ⇒ a = 2.Solving the second equation: 3 + a = -5 ⇒ a = -5 - 3 ⇒ a = -8.So, the possible values for a are 2 and -8. Therefore, the real number a can be either 2 or -8 for the line to be tangent to the circle.Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from the beginning, converting the polar equations to Cartesian seems correct. The circle equation becomes (x - 1)² + y² = 1, which is a circle with center (1, 0) and radius 1. The line equation becomes 3x + 4y + a = 0. The distance from the center (1, 0) to the line is |3*1 + 4*0 + a| / sqrt(3² + 4²) = |3 + a| / 5. Setting this equal to the radius 1 gives |3 + a| = 5, leading to a = 2 or a = -8. That seems correct.Alternatively, I could have approached this problem without converting to Cartesian coordinates. Let me try that method as a check. Starting with the polar equations:Circle: ρ = 2 cos θ.Line: 3ρ cos θ + 4ρ sin θ + a = 0.If I substitute ρ from the circle's equation into the line's equation, I get:3*(2 cos θ)*cos θ + 4*(2 cos θ)*sin θ + a = 0.Simplifying, that's 6 cos² θ + 8 cos θ sin θ + a = 0.Now, I can write this as 6 cos² θ + 8 cos θ sin θ + a = 0.I need this equation to have exactly one solution for θ, meaning the discriminant should be zero. But this is a quadratic in terms of cos θ and sin θ, which complicates things. Maybe I can express everything in terms of sin θ or cos θ.Alternatively, I can use the identity cos² θ = 1 - sin² θ to rewrite the equation:6(1 - sin² θ) + 8 cos θ sin θ + a = 0.Expanding, that's 6 - 6 sin² θ + 8 cos θ sin θ + a = 0.Rearranging, -6 sin² θ + 8 cos θ sin θ + (6 + a) = 0.This is a quadratic in sin θ, but it also has a cos θ term, which complicates things. Maybe I can square both sides or use another identity. Alternatively, I can express cos θ in terms of sin θ using cos θ = sqrt(1 - sin² θ), but that introduces square roots, which might complicate things further.Alternatively, perhaps I can use the fact that for the line to be tangent to the circle, the system of equations has exactly one solution. So, substituting ρ = 2 cos θ into the line equation gives 6 cos² θ + 8 cos θ sin θ + a = 0. To have exactly one solution, the discriminant of this equation must be zero. But since it's not a standard quadratic, maybe I need to manipulate it differently.Wait, perhaps I can write this equation as 6 cos² θ + 8 cos θ sin θ + a = 0 and then use the identity cos² θ = (1 + cos 2θ)/2 and sin θ cos θ = (sin 2θ)/2 to rewrite the equation in terms of double angles.Let's try that:6*(1 + cos 2θ)/2 + 8*(sin 2θ)/2 + a = 0.Simplifying, that's 3(1 + cos 2θ) + 4 sin 2θ + a = 0.Which becomes 3 + 3 cos 2θ + 4 sin 2θ + a = 0.Rearranging, 3 cos 2θ + 4 sin 2θ = - (3 + a).Now, this is of the form A cos 2θ + B sin 2θ = C, where A = 3, B = 4, and C = -(3 + a).I remember that the maximum value of A cos x + B sin x is sqrt(A² + B²), and the minimum is -sqrt(A² + B²). So, for the equation 3 cos 2θ + 4 sin 2θ = - (3 + a) to have a solution, the absolute value of the right-hand side must be less than or equal to sqrt(3² + 4²) = 5.So, | - (3 + a) | ≤ 5 ⇒ |3 + a| ≤ 5.This gives -5 ≤ 3 + a ≤ 5 ⇒ -8 ≤ a ≤ 2.But since the line is tangent to the circle, the equation must have exactly one solution, meaning that the right-hand side must be equal to the maximum or minimum value of the left-hand side. Therefore, | - (3 + a) | = 5 ⇒ |3 + a| = 5 ⇒ 3 + a = ±5 ⇒ a = 2 or a = -8.So, this method also gives the same result: a = 2 or a = -8.Therefore, both methods confirm that the possible values of a are 2 and -8. I think this solidifies the answer.Just to be thorough, let me visualize this. The circle is centered at (1, 0) with radius 1. The line 3x + 4y + a = 0 is a straight line. The distance from the center to the line must be equal to the radius for tangency. Since the radius is 1, the distance from (1, 0) to the line is |3*1 + 4*0 + a| / 5 = |3 + a| / 5 = 1. Solving this gives |3 + a| = 5, so a = 2 or a = -8. Yep, that makes sense.I don't think I made any mistakes in my reasoning. Both approaches—converting to Cartesian coordinates and using trigonometric identities—led me to the same conclusion. So, I'm confident that the values of a are 2 and -8.