💡First, I need to determine the relative speed between the train and the man since they are moving in opposite directions. The train is moving at 25 km/h, and the man is moving at 2 km/h. Adding these speeds gives a relative speed of 27 km/h.Next, I should convert this relative speed from kilometers per hour to meters per second to match the time unit provided in seconds. Using the conversion factor, 27 km/h is equivalent to 7.5 m/s.Finally, with the relative speed in meters per second and the time it takes for the train to pass the man, I can calculate the length of the train by multiplying the relative speed by the time. This gives me the length of the train as 330 meters.

💡Alright, so I have this problem about coordinate systems and parametric equations. It's divided into two parts. Let me try to tackle each part step by step.Starting with part (1): I need to find the standard equation of circle C and the Cartesian equation of line l.First, the parametric equations for circle C are given as:[begin{cases}x = -5 + sqrt{2}cos t y = 3 + sqrt{2}sin tend{cases}]where ( t ) is the parameter. I remember that parametric equations for a circle usually have the form ( x = h + rcos t ) and ( y = k + rsin t ), where ( (h, k) ) is the center and ( r ) is the radius. So, comparing, it looks like the center of the circle is at ( (-5, 3) ) and the radius is ( sqrt{2} ). To write the standard equation, I just need to square both ( x + 5 ) and ( y - 3 ) and set their sum equal to ( (sqrt{2})^2 = 2 ). So, the standard equation should be:[(x + 5)^2 + (y - 3)^2 = 2]That seems straightforward.Next, the polar equation of line ( l ) is given as:[rho cosleft(theta + frac{pi}{4}right) = -sqrt{2}]I need to convert this into Cartesian coordinates. I recall that in polar coordinates, ( rho cos(theta + phi) ) can be expanded using the cosine addition formula:[cos(theta + phi) = costheta cosphi - sintheta sinphi]So, substituting ( phi = frac{pi}{4} ), we have:[rho left( costheta cosfrac{pi}{4} - sintheta sinfrac{pi}{4} right) = -sqrt{2}]Since ( cosfrac{pi}{4} = sinfrac{pi}{4} = frac{sqrt{2}}{2} ), this simplifies to:[rho left( frac{sqrt{2}}{2} costheta - frac{sqrt{2}}{2} sintheta right) = -sqrt{2}]Multiplying through by ( frac{2}{sqrt{2}} ) to simplify, we get:[rho costheta - rho sintheta = -2]But in Cartesian coordinates, ( rho costheta = x ) and ( rho sintheta = y ). So substituting these in, the equation becomes:[x - y = -2]Or rearranged:[x - y + 2 = 0]So, the Cartesian equation of line ( l ) is ( x - y + 2 = 0 ).Moving on to part (2): I need to find the minimum value of the area of triangle ( PAB ) where ( P ) is any point on circle ( C ).First, the points ( A ) and ( B ) are given in polar coordinates: ( A(2, frac{pi}{2}) ) and ( B(2, pi) ). I should convert these to Cartesian coordinates to work with them more easily.For point ( A ):[x = 2 cosleft( frac{pi}{2} right) = 0 y = 2 sinleft( frac{pi}{2} right) = 2]So, ( A ) is at ( (0, 2) ).For point ( B ):[x = 2 cos(pi) = -2 y = 2 sin(pi) = 0]So, ( B ) is at ( (-2, 0) ).Now, I need to find the area of triangle ( PAB ) where ( P ) is any point on circle ( C ). The area of a triangle given three points can be found using the formula:[text{Area} = frac{1}{2} | vec{AB} times vec{AP} |]But maybe it's easier to use the formula involving base and height. Alternatively, since I know the coordinates of all three points, I can use the shoelace formula.But perhaps a more efficient method is to realize that the area of triangle ( PAB ) can be expressed as ( frac{1}{2} times text{base} times text{height} ). Here, I can take ( AB ) as the base and the distance from point ( P ) to line ( AB ) as the height.First, let's compute the length of ( AB ). Points ( A(0, 2) ) and ( B(-2, 0) ).The distance between ( A ) and ( B ) is:[|AB| = sqrt{(0 - (-2))^2 + (2 - 0)^2} = sqrt{(2)^2 + (2)^2} = sqrt{4 + 4} = sqrt{8} = 2sqrt{2}]So, the base length is ( 2sqrt{2} ).Now, I need to find the distance from point ( P ) to the line ( AB ). Wait, but line ( AB ) is the same as line ( l ) from part (1), right? Because line ( l ) has equation ( x - y + 2 = 0 ). Let me check if points ( A ) and ( B ) lie on this line.For point ( A(0, 2) ):[0 - 2 + 2 = 0 quad text{True}]For point ( B(-2, 0) ):[-2 - 0 + 2 = 0 quad text{True}]Yes, both points lie on line ( l ), so line ( AB ) is indeed line ( l ). Therefore, the distance from ( P ) to line ( AB ) is the same as the distance from ( P ) to line ( l ).So, the area of triangle ( PAB ) is:[text{Area} = frac{1}{2} times |AB| times d = frac{1}{2} times 2sqrt{2} times d = sqrt{2} times d]Where ( d ) is the distance from ( P ) to line ( l ).Therefore, to minimize the area, I need to minimize ( d ), the distance from ( P ) to line ( l ).Given that ( P ) is any point on circle ( C ), which has equation ( (x + 5)^2 + (y - 3)^2 = 2 ). So, I need to find the minimum distance from any point on this circle to the line ( x - y + 2 = 0 ).I remember that the distance from a point ( (x_0, y_0) ) to the line ( ax + by + c = 0 ) is given by:[d = frac{|ax_0 + by_0 + c|}{sqrt{a^2 + b^2}}]In our case, the line is ( x - y + 2 = 0 ), so ( a = 1 ), ( b = -1 ), ( c = 2 ).So, the distance from ( P(x, y) ) to line ( l ) is:[d = frac{|x - y + 2|}{sqrt{1 + 1}} = frac{|x - y + 2|}{sqrt{2}}]But ( P ) lies on circle ( C ), so ( x = -5 + sqrt{2}cos t ) and ( y = 3 + sqrt{2}sin t ). Let me substitute these into the distance formula.Substituting ( x ) and ( y ):[d = frac{|(-5 + sqrt{2}cos t) - (3 + sqrt{2}sin t) + 2|}{sqrt{2}} = frac{|-5 + sqrt{2}cos t - 3 - sqrt{2}sin t + 2|}{sqrt{2}}]Simplify the numerator:[-5 - 3 + 2 = -6 sqrt{2}cos t - sqrt{2}sin t = sqrt{2}(cos t - sin t)]So, the numerator becomes:[|-6 + sqrt{2}(cos t - sin t)|]Therefore, the distance ( d ) is:[d = frac{|-6 + sqrt{2}(cos t - sin t)|}{sqrt{2}} = frac{|-6 + sqrt{2}(cos t - sin t)|}{sqrt{2}}]I can factor out ( sqrt{2} ) from the numerator:[d = frac{|-6 + sqrt{2}(cos t - sin t)|}{sqrt{2}} = frac{|-6 + sqrt{2} cdot sqrt{2} left( frac{cos t - sin t}{sqrt{2}} right)|}{sqrt{2}} ]Wait, that might complicate things. Alternatively, I can write ( cos t - sin t ) as ( sqrt{2} cos(t + frac{pi}{4}) ) using the identity ( cos t - sin t = sqrt{2} cos(t + frac{pi}{4}) ). Let me verify that:Using the identity:[cos t - sin t = sqrt{2} cosleft(t + frac{pi}{4}right)]Yes, that's correct because:[sqrt{2} cosleft(t + frac{pi}{4}right) = sqrt{2} left( cos t cosfrac{pi}{4} - sin t sinfrac{pi}{4} right) = sqrt{2} left( frac{sqrt{2}}{2} cos t - frac{sqrt{2}}{2} sin t right) = cos t - sin t]Perfect. So, substituting back into ( d ):[d = frac{|-6 + sqrt{2} cdot sqrt{2} cosleft(t + frac{pi}{4}right)|}{sqrt{2}} = frac{|-6 + 2 cosleft(t + frac{pi}{4}right)|}{sqrt{2}}]Simplify:[d = frac{|-6 + 2 cosleft(t + frac{pi}{4}right)|}{sqrt{2}} = frac{|2 cosleft(t + frac{pi}{4}right) - 6|}{sqrt{2}} = frac{|2(cosleft(t + frac{pi}{4}right) - 3)|}{sqrt{2}}]Wait, that doesn't seem helpful. Alternatively, perhaps I can factor out the 2:[d = frac{2 |cosleft(t + frac{pi}{4}right) - 3|}{sqrt{2}} = sqrt{2} |cosleft(t + frac{pi}{4}right) - 3|]But ( cos ) function ranges between -1 and 1, so ( cosleft(t + frac{pi}{4}right) - 3 ) ranges between -4 and -2. Therefore, the absolute value becomes:[|cosleft(t + frac{pi}{4}right) - 3| = 3 - cosleft(t + frac{pi}{4}right)]So, substituting back:[d = sqrt{2} left( 3 - cosleft(t + frac{pi}{4}right) right)]Therefore, the distance ( d ) is:[d = sqrt{2} left( 3 - cosleft(t + frac{pi}{4}right) right)]To find the minimum distance, we need to minimize ( d ). Since ( cosleft(t + frac{pi}{4}right) ) has a maximum value of 1 and a minimum value of -1, the expression ( 3 - cosleft(t + frac{pi}{4}right) ) will be minimized when ( cosleft(t + frac{pi}{4}right) ) is maximized, i.e., when ( cosleft(t + frac{pi}{4}right) = 1 ).So, the minimum value of ( d ) is:[d_{text{min}} = sqrt{2} (3 - 1) = sqrt{2} times 2 = 2sqrt{2}]Therefore, the minimum distance from ( P ) to line ( l ) is ( 2sqrt{2} ).Now, going back to the area of triangle ( PAB ):[text{Area} = sqrt{2} times d]Substituting ( d_{text{min}} = 2sqrt{2} ):[text{Area}_{text{min}} = sqrt{2} times 2sqrt{2} = 2 times 2 = 4]So, the minimum area of triangle ( PAB ) is 4.Wait, let me double-check my steps to make sure I didn't make a mistake.1. Converted parametric equations to standard circle equation: correct.2. Converted polar equation of line to Cartesian: correct, got ( x - y + 2 = 0 ).3. Converted points ( A ) and ( B ) to Cartesian: correct, ( A(0,2) ) and ( B(-2,0) ).4. Calculated distance ( AB ): correct, ( 2sqrt{2} ).5. Expressed area in terms of distance from ( P ) to line ( AB ): correct.6. Expressed distance from ( P ) to line ( l ) using parametric equations: correct.7. Used trigonometric identity to simplify: correct, ( cos t - sin t = sqrt{2} cos(t + pi/4) ).8. Expressed distance ( d ) in terms of ( cos(t + pi/4) ): correct.9. Found the minimum distance by considering the range of cosine: correct, minimum when cosine is 1, giving ( 2sqrt{2} ).10. Calculated minimum area: correct, ( sqrt{2} times 2sqrt{2} = 4 ).Everything seems to check out. So, I think the minimum area is indeed 4.**Final Answer**The minimum value of the area of triangle ( Delta PAB ) is boxed{4}.

💡Okay, so I have this problem here: "What quantity of water should be taken out to concentrate 24 liters of a certain percentage acidic liquid to 60% acidic liquid if 8 liters of water need to be removed?" Hmm, that's a bit confusing at first glance. Let me try to break it down.First, I need to understand what's being asked. It seems like we have 24 liters of an acidic liquid, and we want to make it more concentrated—specifically, to 60% acidity. To do this, we need to remove some water. The problem mentions that 8 liters of water need to be removed, but it's asking for the quantity of water to be taken out. Wait, that seems a bit contradictory. If it's already stating that 8 liters need to be removed, why is it asking for the quantity? Maybe I'm misinterpreting it.Let me read it again: "What quantity of water should be taken out to concentrate 24 liters of a certain percentage acidic liquid to 60% acidic liquid if 8 liters of water need to be removed?" Okay, perhaps it's saying that in order to achieve the concentration of 60%, 8 liters of water need to be removed, and it's asking for the quantity of water that should be taken out. That still seems a bit circular. Maybe it's a translation issue or a typo.Assuming that the problem is asking: "What quantity of water should be taken out to concentrate 24 liters of a certain percentage acidic liquid to 60% acidic liquid?" and it's given that 8 liters of water need to be removed. Wait, that still doesn't make sense. If it's given that 8 liters need to be removed, then why is it asking for the quantity? Perhaps the original problem was different, and there was a mistake in the translation or wording.Alternatively, maybe the problem is stating that 8 liters of water need to be removed to achieve a certain concentration, and now it's asking for the quantity of water to be removed to reach 60% acidity. That would make more sense. So, perhaps the initial concentration is different, and removing 8 liters brings it to a certain concentration, and now we need to find out how much more water needs to be removed to get it to 60%.But the problem as stated is a bit unclear. Let me try to rephrase it: We have 24 liters of an acidic liquid with an unknown concentration. We need to remove some water to make it 60% acidic. It's given that 8 liters of water need to be removed. So, perhaps the question is confirming that 8 liters is the correct amount to remove to reach 60% acidity, and it's asking for the quantity, which is 8 liters. But that seems too straightforward.Alternatively, maybe the problem is saying that 8 liters of water need to be removed to achieve a certain concentration, and now we need to find out how much water needs to be removed to reach 60% acidity. In that case, we would need to know the initial concentration or the concentration after removing 8 liters.Wait, the problem doesn't mention the initial concentration or the concentration after removing 8 liters. It only mentions that 8 liters need to be removed to reach 60% acidity. So, perhaps the answer is simply 8 liters. But that seems too simple, and the problem is phrased in a way that suggests it's asking for a calculation.Maybe I'm overcomplicating it. Let's try to approach it step by step. We have 24 liters of an acidic liquid. Let's assume that the acidic liquid is a mixture of acid and water. The total volume is 24 liters. We need to remove some water to increase the concentration of acid to 60%. The problem states that 8 liters of water need to be removed. So, perhaps it's asking us to confirm that removing 8 liters of water will result in a 60% concentration.But to confirm that, we need to know the initial concentration of the acid. Without knowing the initial concentration, we can't verify if removing 8 liters of water will indeed result in a 60% concentration. Therefore, the problem might be incomplete or missing some information.Alternatively, maybe the problem is asking us to find out how much water needs to be removed to reach 60% acidity, and it's given that 8 liters need to be removed. But that still doesn't make sense because it's both asking and providing the answer.Perhaps there's a misunderstanding in the translation or wording. Maybe it's supposed to say that 8 liters of acid need to be added, or 8 liters of the solution need to be removed. That would make the problem more solvable.Assuming that the problem is asking: "What quantity of water should be taken out to concentrate 24 liters of a certain percentage acidic liquid to 60% acidic liquid?" and it's given that 8 liters of water need to be removed, then the answer would simply be 8 liters. But that seems too straightforward, and the problem is likely expecting a calculation.Alternatively, if the problem is saying that 8 liters of water need to be removed to achieve a certain concentration, and now we need to find out how much more water needs to be removed to reach 60%, then we would need to know the initial concentration or the concentration after removing 8 liters.Given the ambiguity, perhaps the best approach is to assume that the problem is asking for the quantity of water to be removed to reach 60% acidity, and it's given that 8 liters need to be removed. Therefore, the answer is 8 liters.But to be thorough, let's consider the possibility that the problem is missing some information. If we assume that the initial concentration is, say, 50%, then we can calculate how much water needs to be removed to reach 60%. Let's try that.Let's denote:- V = total volume = 24 liters- C_initial = initial concentration (unknown)- C_final = final concentration = 60% = 0.6- W_initial = initial amount of water- W_final = final amount of water after removing 8 litersThe amount of acid remains constant, so:C_initial * V = C_final * (V - W_removed)But we don't know C_initial or W_removed.Alternatively, if we assume that the initial concentration is such that removing 8 liters of water results in 60% acidity, then we can set up the equation:C_initial * 24 = 0.6 * (24 - 8)C_initial * 24 = 0.6 * 16C_initial = (0.6 * 16) / 24C_initial = 9.6 / 24C_initial = 0.4 or 40%So, if the initial concentration was 40%, then removing 8 liters of water would result in a 60% concentration. Therefore, the quantity of water to be taken out is 8 liters.But since the problem doesn't provide the initial concentration, we can't perform this calculation. Therefore, the problem is incomplete or there's a misunderstanding in the wording.In conclusion, the problem as stated is unclear and likely missing some information. If we assume that the initial concentration is 40%, then removing 8 liters of water would result in a 60% concentration. Therefore, the quantity of water to be taken out is 8 liters.

💡First, I need to determine which of the given points lie above the line ( y = 3x + 4 ). To do this, I'll compare the ( y )-coordinate of each point to the value of ( 3x + 4 ) for the corresponding ( x )-coordinate.1. **For the point (4, 15):** [ 3(4) + 4 = 12 + 4 = 16 ] Since ( 15 < 16 ), this point does not lie above the line.2. **For the point (8, 25):** [ 3(8) + 4 = 24 + 4 = 28 ] Since ( 25 < 28 ), this point does not lie above the line.3. **For the point (14, 42):** [ 3(14) + 4 = 42 + 4 = 46 ] Since ( 42 < 46 ), this point does not lie above the line.4. **For the point (19, 48):** [ 3(19) + 4 = 57 + 4 = 61 ] Since ( 48 < 61 ), this point does not lie above the line.5. **For the point (22, 60):** [ 3(22) + 4 = 66 + 4 = 70 ] Since ( 60 < 70 ), this point does not lie above the line.After evaluating all points, none of them lie above the line ( y = 3x + 4 ). Therefore, the sum of the ( x )-coordinates of the points above the line is ( 0 ).

💡First, determine the number of female officers on duty by calculating half of the total officers on duty.Next, recognize that this number represents 17% of the total female officers on the police force.Set up an equation where 85 equals 17% of the total female officers.Solve for the total number of female officers by dividing 85 by 0.17.The final result is 500 female officers on the police force.

💡First, I need to identify the two numbers that differ by 2 and add up to 20. Let's call the smaller number ( x ) and the larger number ( y ).Since the numbers differ by 2, I can write the equation:[y = x + 2]I also know that their sum is 20:[x + y = 20]Substituting the expression for ( y ) from the first equation into the second equation:[x + (x + 2) = 20]Simplifying this:[2x + 2 = 20]Subtracting 2 from both sides:[2x = 18]Dividing both sides by 2:[x = 9]Now, using the expression for ( y ):[y = 9 + 2 = 11]Therefore, the larger number is 11.