answer：Okay, so I have this problem here about a courier traveling from point A to point B. Let me try to understand what it's saying. The courier starts at A and goes towards B. After some time, the distance he has already covered is in the ratio of 2:3 compared to the remaining distance. Hmm, okay. So if I think about it, the total distance must be the sum of the distance covered and the remaining distance. So if the covered distance is 2 parts and the remaining is 3 parts, the total is 5 parts. Let me denote the total distance as D. So, if the ratio is 2:3, that means the courier has covered (2/5)D and has (3/5)D left to go. That makes sense. Now, the problem says that if the courier travels another 60 km, the ratio of the distance already covered to the remaining distance will be 6:5. So, after covering an additional 60 km, the new ratio becomes 6:5. Let me write this down step by step. Initially, the distance covered is (2/5)D, and the remaining is (3/5)D. Then, the courier travels another 60 km. So, the new distance covered becomes (2/5)D + 60 km, and the remaining distance becomes (3/5)D - 60 km. According to the problem, after this additional 60 km, the ratio of the distance covered to the remaining distance is 6:5. So, I can set up a proportion:[(2/5)D + 60] / [(3/5)D - 60] = 6/5Okay, so that's the equation I need to solve for D. Let me write that equation again:( (2/5)D + 60 ) / ( (3/5)D - 60 ) = 6/5To solve this, I can cross-multiply to eliminate the fractions. So, 5*( (2/5)D + 60 ) = 6*( (3/5)D - 60 )Let me compute each side step by step. First, the left side:5*( (2/5)D + 60 ) = 5*(2/5 D) + 5*60 = 2D + 300Right side:6*( (3/5)D - 60 ) = 6*(3/5 D) - 6*60 = (18/5)D - 360So, now the equation is:2D + 300 = (18/5)D - 360Hmm, okay. Let me get all the D terms on one side and constants on the other. First, subtract (18/5)D from both sides:2D - (18/5)D + 300 = -360And subtract 300 from both sides:2D - (18/5)D = -360 - 300Simplify the left side. 2D is the same as (10/5)D, so:(10/5)D - (18/5)D = (-8/5)DAnd the right side is -660.So, (-8/5)D = -660Multiply both sides by (-5/8) to solve for D:D = (-660)*(-5/8) = (660*5)/8Calculate that:660 divided by 8 is 82.5, and 82.5 multiplied by 5 is 412.5Wait, 660*5 is 3300, and 3300 divided by 8 is 412.5So, D = 412.5 kmHmm, that seems a bit odd because it's a decimal. Is that correct? Let me check my steps again.Starting from the beginning:After some time, the distance covered is 2:3 of the remaining distance. So, covered is 2x, remaining is 3x, total is 5x.Then, after traveling another 60 km, the ratio becomes 6:5.So, covered becomes 2x + 60, remaining becomes 3x - 60.So, (2x + 60)/(3x - 60) = 6/5Cross-multiplying: 5*(2x + 60) = 6*(3x - 60)10x + 300 = 18x - 360Subtract 10x: 300 = 8x - 360Add 360: 660 = 8xSo, x = 660 / 8 = 82.5Therefore, total distance D = 5x = 5*82.5 = 412.5 kmOkay, so that seems consistent. So, the total distance is 412.5 km.But 412.5 km is equal to 412.5 km, which is 412 km and 500 meters. It's a decimal, but it's a valid answer. So, maybe that's correct.Alternatively, maybe I can represent it as a fraction. 412.5 is equal to 825/2, so 825/2 km.But let me think again if I did everything correctly.Let me denote the initial covered distance as 2k and remaining as 3k, so total is 5k.After traveling another 60 km, covered is 2k + 60, remaining is 3k - 60.So, (2k + 60)/(3k - 60) = 6/5Cross-multiplying: 5*(2k + 60) = 6*(3k - 60)10k + 300 = 18k - 360Subtract 10k: 300 = 8k - 360Add 360: 660 = 8kSo, k = 660 / 8 = 82.5Therefore, total distance D = 5k = 412.5 kmYes, that seems consistent.Alternatively, maybe I can represent 412.5 as 412 1/2 km, which is 412.5 km.So, unless I made a mistake in the ratio somewhere.Wait, let me think about the initial ratio. It says the distance already covered is in the ratio of 2:3 to the remaining distance. So, covered:remaining = 2:3.So, covered is 2x, remaining is 3x, total is 5x.Then, after 60 km, covered is 2x + 60, remaining is 3x - 60, and the ratio is 6:5.So, (2x + 60)/(3x - 60) = 6/5.Yes, that's correct.So, solving, 5*(2x + 60) = 6*(3x - 60)10x + 300 = 18x - 36010x - 18x = -360 - 300-8x = -660x = (-660)/(-8) = 82.5So, x = 82.5, so total distance is 5x = 412.5 km.So, that seems correct.Alternatively, maybe I can represent it as a fraction: 82.5 is 165/2, so 5*(165/2) = 825/2 = 412.5.Yes, that's correct.So, the total distance is 412.5 km.But let me check if 412.5 km satisfies the original conditions.So, initially, covered is 2/5 of 412.5, which is (2/5)*412.5 = 165 km.Remaining is 3/5 of 412.5 = 247.5 km.So, ratio is 165:247.5, which simplifies to 2:3, since 165*3=495 and 247.5*2=495. So, yes, 165:247.5 is 2:3.Then, after traveling another 60 km, covered becomes 165 + 60 = 225 km.Remaining becomes 247.5 - 60 = 187.5 km.So, ratio is 225:187.5.Divide both by 22.5: 225/22.5 = 10, 187.5/22.5 = 8.333... Wait, that's not 6:5.Wait, hold on, that can't be right. 225:187.5.Wait, 225 divided by 187.5 is 1.2, which is 6/5.Yes, 225:187.5 is 6:5 because 225/187.5 = 1.2 = 6/5.So, 225:187.5 is 6:5.So, that's correct.Wait, but when I tried to simplify 225:187.5 by dividing both by 22.5, I get 10:8.333..., which is not 6:5. So, maybe I shouldn't have divided by 22.5.Alternatively, let's divide both by 37.5: 225/37.5 = 6, 187.5/37.5 = 5. So, 6:5.Yes, that's correct.So, 225:187.5 is 6:5.So, the calculations are correct.Therefore, the total distance is 412.5 km.But in the problem statement, they might expect an integer, but since 412.5 is a valid number, it's okay.Alternatively, maybe I made a mistake in the initial ratio.Wait, let me read the problem again:"A courier departs from A towards B. After a certain time, the distance already covered is in the ratio of 2:3 to the remaining distance."So, covered:remaining = 2:3.Yes, that's correct."If the courier travels another 60 km, the ratio of the distance already covered to the remaining distance will be 6:5."Yes, so covered becomes 2x + 60, remaining becomes 3x - 60, ratio 6:5.Yes, so that's correct.So, with x = 82.5, total distance is 412.5 km.So, unless I made a calculation error, that's correct.Wait, let me compute 2x + 60 and 3x - 60 with x = 82.5.2x = 165, so 165 + 60 = 225.3x = 247.5, so 247.5 - 60 = 187.5.225:187.5 is 6:5, as 225/187.5 = 1.2 = 6/5.Yes, that's correct.So, all the steps check out.Therefore, the total distance is 412.5 km.But in the problem statement, it's written as "60 km", so 60 is an integer, but the total distance is 412.5 km, which is a decimal. So, that's acceptable.Alternatively, if we want to write it as a fraction, 412.5 is equal to 825/2, so 825/2 km.But 412.5 is also correct.So, I think that's the answer.**Final Answer**The total distance from A to B is boxed{412.5} kilometers.

answer：Okay, so I have this problem here: I need to find the smallest positive integer ( n ) such that ( sqrt{n} - sqrt{n - 1} < 0.01 ). The options are A. 2499, B. 2500, C. 2501, and D. No such integer. Hmm, let me think about how to approach this.First, I remember that the difference between consecutive square roots can be approximated using calculus, maybe derivatives? Or perhaps there's an algebraic way to manipulate the inequality. Let me try both approaches and see which one works better.Starting with the inequality:( sqrt{n} - sqrt{n - 1} < 0.01 )I think I can rationalize the left side to make it easier to handle. To rationalize, I'll multiply and divide by the conjugate, which is ( sqrt{n} + sqrt{n - 1} ). Let's see:( sqrt{n} - sqrt{n - 1} = frac{(sqrt{n} - sqrt{n - 1})(sqrt{n} + sqrt{n - 1})}{sqrt{n} + sqrt{n - 1}} )Multiplying out the numerator:( (sqrt{n})^2 - (sqrt{n - 1})^2 = n - (n - 1) = 1 )So, the expression simplifies to:( frac{1}{sqrt{n} + sqrt{n - 1}} < 0.01 )That's much simpler! So now, the inequality becomes:( frac{1}{sqrt{n} + sqrt{n - 1}} < 0.01 )To solve for ( n ), I can take reciprocals on both sides, remembering that this will reverse the inequality sign because both sides are positive:( sqrt{n} + sqrt{n - 1} > 100 )Hmm, okay. So the sum of these two square roots needs to be greater than 100. Since ( sqrt{n} ) and ( sqrt{n - 1} ) are consecutive square roots, their values are close to each other. Maybe I can approximate ( sqrt{n - 1} ) as ( sqrt{n} - delta ) where ( delta ) is a small number. But perhaps a better approach is to consider that ( sqrt{n - 1} ) is approximately ( sqrt{n} - frac{1}{2sqrt{n}} ) using the linear approximation from calculus.Wait, let me recall that for small ( h ), ( sqrt{n + h} approx sqrt{n} + frac{h}{2sqrt{n}} ). So, if ( h = -1 ), then ( sqrt{n - 1} approx sqrt{n} - frac{1}{2sqrt{n}} ). So, substituting this into the sum:( sqrt{n} + sqrt{n - 1} approx sqrt{n} + sqrt{n} - frac{1}{2sqrt{n}} = 2sqrt{n} - frac{1}{2sqrt{n}} )So, the inequality becomes approximately:( 2sqrt{n} - frac{1}{2sqrt{n}} > 100 )Hmm, since ( frac{1}{2sqrt{n}} ) is a small term, especially for large ( n ), maybe I can ignore it for an initial approximation. So, approximately:( 2sqrt{n} > 100 )Which simplifies to:( sqrt{n} > 50 )So,( n > 2500 )Therefore, ( n ) must be greater than 2500. So, the smallest integer ( n ) would be 2501. Let me check if 2501 satisfies the original inequality.Wait, but let me verify this because approximations can sometimes be misleading. Let me compute ( sqrt{2501} - sqrt{2500} ). Since ( sqrt{2500} = 50 ), and ( sqrt{2501} ) is just a bit more than 50. Let me compute it precisely.I know that ( (50.01)^2 = 2501.0001 ), because ( (50 + 0.01)^2 = 50^2 + 2*50*0.01 + (0.01)^2 = 2500 + 1 + 0.0001 = 2501.0001 ). So, ( sqrt{2501} ) is approximately 50.01. Therefore, ( sqrt{2501} - sqrt{2500} approx 50.01 - 50 = 0.01 ). But the inequality is strict: less than 0.01. So, is 0.01 less than 0.01? No, it's equal. So, does that mean 2501 doesn't satisfy the inequality?Wait, that's a problem. So, according to this, ( sqrt{2501} - sqrt{2500} = 0.01 ), which is not less than 0.01. So, 2501 doesn't satisfy the inequality. Hmm, so maybe I need to go to the next integer, 2502?Wait, but let's compute more accurately. Maybe my approximation is too rough. Let me compute ( sqrt{2501} ) more precisely.Let me use the Newton-Raphson method to approximate ( sqrt{2501} ). Let me start with an initial guess. Since ( 50^2 = 2500 ), let me take ( x_0 = 50.01 ). Then, the next iteration is:( x_{1} = frac{1}{2}(x_0 + frac{2501}{x_0}) )Compute ( x_0 = 50.01 )Compute ( frac{2501}{50.01} ). Let's compute 2501 divided by 50.01.First, 50.01 * 50 = 2500.5, which is 2500.5. So, 2501 - 2500.5 = 0.5. So, 2501 / 50.01 = 50 + 0.5 / 50.01 ≈ 50 + 0.009998 ≈ 50.009998.So, ( x_1 = frac{1}{2}(50.01 + 50.009998) = frac{1}{2}(100.019998) = 50.009999 ).So, ( sqrt{2501} approx 50.009999 ). Therefore, ( sqrt{2501} - sqrt{2500} approx 50.009999 - 50 = 0.009999 ), which is approximately 0.01, but slightly less. So, 0.009999 is less than 0.01. Therefore, 2501 does satisfy the inequality.Wait, so my initial approximation was too rough, but with a better approximation, ( sqrt{2501} - sqrt{2500} ) is just under 0.01. So, 2501 is the smallest integer where the difference is less than 0.01.But let me check 2500 as well, just to be thorough. ( sqrt{2500} = 50 ), ( sqrt{2499} ) is approximately 49.99. Let me compute ( sqrt{2499} ).Again, using Newton-Raphson. Let me take ( x_0 = 49.99 ). Then, ( x_1 = frac{1}{2}(49.99 + 2499 / 49.99) ).Compute ( 2499 / 49.99 ). Let's see, 49.99 * 50 = 2499.5, which is 2499.5. So, 2499 / 49.99 = 50 - (0.5 / 49.99) ≈ 50 - 0.010002 ≈ 49.989998.So, ( x_1 = frac{1}{2}(49.99 + 49.989998) = frac{1}{2}(99.979998) = 49.989999 ).So, ( sqrt{2499} approx 49.989999 ). Therefore, ( sqrt{2500} - sqrt{2499} approx 50 - 49.989999 = 0.010001 ), which is just over 0.01. So, 2500 doesn't satisfy the inequality because the difference is slightly more than 0.01.Therefore, 2501 is the smallest integer where the difference is less than 0.01. So, the answer should be C. 2501.But wait, let me make sure I didn't make any calculation errors. Let me compute ( sqrt{2501} ) more accurately.Alternatively, I can use the formula for the difference of square roots:( sqrt{n} - sqrt{n - 1} = frac{1}{sqrt{n} + sqrt{n - 1}} )We can write this as:( frac{1}{sqrt{n} + sqrt{n - 1}} < 0.01 )Which implies:( sqrt{n} + sqrt{n - 1} > 100 )Let me denote ( sqrt{n} = x ), so ( sqrt{n - 1} = sqrt{x^2 - 1} ). So, the inequality becomes:( x + sqrt{x^2 - 1} > 100 )Let me solve for ( x ). Let me denote ( y = x + sqrt{x^2 - 1} ). Then, I can write:( y = x + sqrt{x^2 - 1} )Let me solve for ( x ) in terms of ( y ). Let me subtract ( x ) from both sides:( y - x = sqrt{x^2 - 1} )Square both sides:( (y - x)^2 = x^2 - 1 )Expand the left side:( y^2 - 2xy + x^2 = x^2 - 1 )Simplify:( y^2 - 2xy = -1 )Bring all terms to one side:( y^2 - 2xy + 1 = 0 )This is a quadratic equation in terms of ( x ):( -2y x + (y^2 + 1) = 0 )Solving for ( x ):( x = frac{y^2 + 1}{2y} )So, ( x = frac{y}{2} + frac{1}{2y} )Since ( y > 100 ), let's plug ( y = 100 ):( x = frac{100}{2} + frac{1}{2*100} = 50 + 0.005 = 50.005 )So, ( x = 50.005 ), which means ( sqrt{n} = 50.005 ), so ( n = (50.005)^2 ).Compute ( (50.005)^2 ):( 50^2 = 2500 )( 2*50*0.005 = 0.5 )( (0.005)^2 = 0.000025 )So, ( (50.005)^2 = 2500 + 0.5 + 0.000025 = 2500.500025 )So, ( n ) must be greater than 2500.500025. Since ( n ) must be an integer, the smallest integer greater than 2500.500025 is 2501. Therefore, ( n = 2501 ) is the smallest integer satisfying the inequality.Wait, so that confirms my earlier conclusion. So, 2501 is indeed the smallest integer where the difference is less than 0.01. Therefore, the correct answer is C. 2501.But just to be thorough, let me compute ( sqrt{2501} ) and ( sqrt{2500} ) precisely.We know that ( 50^2 = 2500 ), so ( sqrt{2500} = 50 ).For ( sqrt{2501} ), let's compute it more accurately.Let me use the binomial expansion for ( sqrt{2500 + 1} ). So,( sqrt{2500 + 1} = sqrt{2500(1 + 1/2500)} = 50sqrt{1 + 0.0004} )Using the Taylor series expansion for ( sqrt{1 + x} ) around ( x = 0 ):( sqrt{1 + x} approx 1 + frac{x}{2} - frac{x^2}{8} + frac{x^3}{16} - dots )So, plugging ( x = 0.0004 ):( sqrt{1 + 0.0004} approx 1 + 0.0002 - frac{(0.0004)^2}{8} + frac{(0.0004)^3}{16} )Compute each term:First term: 1Second term: 0.0002Third term: ( frac{0.00000016}{8} = 0.00000002 )Fourth term: ( frac{0.000000000064}{16} = 0.000000000004 )So, adding them up:1 + 0.0002 = 1.0002Subtract 0.00000002: 1.00019998Add 0.000000000004: approximately 1.000199980004So, ( sqrt{2501} approx 50 * 1.000199980004 approx 50.009999 )Therefore, ( sqrt{2501} - sqrt{2500} approx 50.009999 - 50 = 0.009999 ), which is approximately 0.01, but just slightly less. So, indeed, 2501 satisfies the inequality.On the other hand, for ( n = 2500 ), ( sqrt{2500} - sqrt{2499} approx 0.010001 ), which is just over 0.01, so 2500 doesn't satisfy the inequality.Therefore, the smallest integer ( n ) is 2501, which is option C.I think that's thorough enough. I considered multiple methods: rationalizing, linear approximation, Newton-Raphson, and binomial expansion, all leading me to the same conclusion. So, I'm confident that 2501 is the correct answer.**Final Answer**boxed{C}

answer：First, I need to determine how many daisies are in each daisy bouquet. I know that the flower shop sold a total of 20 bouquets, with 10 being rose bouquets and 10 being daisy bouquets.Each rose bouquet contains 12 roses, so the total number of roses sold is 10 bouquets multiplied by 12 roses per bouquet, which equals 120 roses.The total number of flowers sold today is 190. Since 120 of these are roses, the remaining flowers must be daisies. Subtracting the roses from the total gives 190 minus 120, which equals 70 daisies.There are 10 daisy bouquets sold, so to find out how many daisies are in each bouquet, I divide the total number of daisies by the number of daisy bouquets. That is 70 daisies divided by 10 bouquets, which equals 7 daisies per bouquet.

answer：Alright, so I have this equation: tan 20° + m sin 20° = √3. I need to find the value of m. Hmm, okay. Let me think about how to approach this.First, I know that tan θ is equal to sin θ over cos θ. So maybe I can rewrite the equation in terms of sine and cosine to make it easier to solve for m. Let me try that.So, tan 20° is sin 20° / cos 20°. Therefore, the equation becomes:(sin 20° / cos 20°) + m sin 20° = √3Hmm, okay. Now, I can factor out sin 20° from both terms on the left side. Let me see:sin 20° (1 / cos 20° + m) = √3So, that simplifies to:sin 20° (1 / cos 20° + m) = √3Now, I can write 1 / cos 20° as sec 20°, but I'm not sure if that helps. Maybe I should just keep it as 1 / cos 20° for now.So, let me denote sin 20° as S and cos 20° as C for simplicity. Then the equation becomes:S (1/C + m) = √3So, expanding that, it's S/C + mS = √3But S/C is tan 20°, which is the original term. Hmm, maybe this substitution isn't helping much. Let me think differently.Alternatively, I can write the equation as:tan 20° + m sin 20° = √3I need to solve for m, so maybe I can isolate m. Let me subtract tan 20° from both sides:m sin 20° = √3 - tan 20°Then, divide both sides by sin 20°:m = (√3 - tan 20°) / sin 20°Okay, that seems straightforward. Let me compute the value step by step.First, compute tan 20°. I know that tan 20° is approximately 0.3640. Let me verify that with a calculator. Yes, tan 20° ≈ 0.3640.Next, compute sin 20°. Sin 20° is approximately 0.3420.So, plugging these approximate values into the equation:m ≈ (√3 - 0.3640) / 0.3420Compute √3 first. √3 is approximately 1.7320.So, √3 - 0.3640 ≈ 1.7320 - 0.3640 = 1.3680Then, divide that by 0.3420:1.3680 / 0.3420 ≈ 4.0Wait, that's exactly 4.0? Let me check the calculation again.1.3680 divided by 0.3420. Let me compute 1.3680 / 0.3420.First, 0.3420 * 4 = 1.3680. Yes, exactly. So, m ≈ 4.0Hmm, so m is 4. That seems clean. But let me verify if this is exact or just an approximation.Because sometimes, with trigonometric identities, exact values can be found without resorting to decimal approximations.Let me try to express everything in terms of exact trigonometric functions.We have:m = (√3 - tan 20°) / sin 20°Let me write tan 20° as sin 20° / cos 20°, so:m = (√3 - (sin 20° / cos 20°)) / sin 20°Let me combine the terms in the numerator:= (√3 cos 20° - sin 20°) / cos 20° / sin 20°Wait, no. Let me do it step by step.First, write the numerator as √3 - (sin 20° / cos 20°). To combine these, we can write √3 as √3 * (cos 20° / cos 20°):= (√3 cos 20° / cos 20° - sin 20° / cos 20°) / sin 20°Now, combine the terms:= (√3 cos 20° - sin 20°) / cos 20° / sin 20°Which is equal to:(√3 cos 20° - sin 20°) / (cos 20° sin 20°)Hmm, okay. So, m = (√3 cos 20° - sin 20°) / (cos 20° sin 20°)I wonder if this can be simplified further. Maybe using some trigonometric identities.I recall that expressions of the form a cos θ + b sin θ can be written as R cos(θ - φ), where R = √(a² + b²) and tan φ = b/a.But in our case, it's √3 cos 20° - sin 20°, which is similar to a cos θ + b sin θ with a = √3 and b = -1.So, let's compute R and φ.R = √( (√3)² + (-1)² ) = √(3 + 1) = √4 = 2And tan φ = |b/a| = |(-1)/√3| = 1/√3, so φ = 30°, because tan 30° = 1/√3.But since b is negative, the angle is in the fourth quadrant, so φ = -30° or 330°, but since we're dealing with positive angles, we can write it as 30° below the x-axis.Therefore, √3 cos 20° - sin 20° can be written as 2 cos(20° + 30°) because of the negative sign in front of sin.Wait, let me think. The identity is a cos θ + b sin θ = R cos(θ - φ). But in our case, it's a cos θ - b sin θ, so it's a cos θ + (-b) sin θ.So, that would be R cos(θ + φ), where φ is such that tan φ = b/a.Wait, maybe I should write it as:√3 cos 20° - sin 20° = 2 cos(20° + 30°)Because:cos(A + B) = cos A cos B - sin A sin BSo, 2 cos(20° + 30°) = 2 [cos 20° cos 30° - sin 20° sin 30°]Compute cos 30° and sin 30°:cos 30° = √3 / 2 ≈ 0.8660sin 30° = 1/2 = 0.5So, 2 [cos 20° * (√3 / 2) - sin 20° * (1/2)] = 2*(√3 / 2 cos 20° - 1/2 sin 20°) = √3 cos 20° - sin 20°, which matches.Therefore, √3 cos 20° - sin 20° = 2 cos(50°)Because 20° + 30° = 50°, so yes, that's correct.Therefore, m = 2 cos 50° / (cos 20° sin 20°)Hmm, okay. So, m = 2 cos 50° / (cos 20° sin 20°)I wonder if I can simplify this further.I know that sin 2θ = 2 sin θ cos θ, so 2 sin θ cos θ = sin 2θ.In the denominator, we have cos 20° sin 20°, which is (1/2) sin 40°, because 2 sin 20° cos 20° = sin 40°, so sin 20° cos 20° = (1/2) sin 40°.Therefore, the denominator becomes (1/2) sin 40°, so:m = 2 cos 50° / ( (1/2) sin 40° ) = (2 cos 50°) / ( (1/2) sin 40° ) = (2 / (1/2)) * (cos 50° / sin 40° ) = 4 * (cos 50° / sin 40° )But cos 50° is equal to sin 40°, because cos(90° - θ) = sin θ. So, cos 50° = sin 40°.Therefore, cos 50° / sin 40° = sin 40° / sin 40° = 1So, m = 4 * 1 = 4Wow, that's neat! So, m is exactly 4. That matches the approximate calculation I did earlier.So, after going through the trigonometric identities and simplifying step by step, I found that m is exactly 4.I think that's the answer. Let me recap:Starting with tan 20° + m sin 20° = √3Expressed tan 20° as sin 20° / cos 20°, then factored out sin 20°, leading to an expression for m.Then, by expressing the numerator as a single cosine function using the identity for a cos θ + b sin θ, I was able to simplify the expression.Recognizing that cos 50° is equal to sin 40°, and using the double-angle identity for sine, everything simplified down to m = 4.So, yeah, m is 4.**Final Answer**The value of ( m ) is boxed{4}.

answer：Okay, so I have this problem about triangle ABC with coordinates given for points A, B, and C. I need to find the equation of the height AH from A to side BC and then find the area of triangle ABC. Hmm, let me think about how to approach this step by step.First, let me write down the coordinates to make sure I have them right. Point A is at (0, 2), point B is at (2, 0), and point C is at (-2, -1). So, triangle ABC has these three vertices.Starting with part (1): finding the equation of the line on which the height AH from A to side BC lies. I remember that a height (or altitude) of a triangle is a perpendicular segment from a vertex to the opposite side. So, in this case, AH is perpendicular to BC and starts at point A.To find the equation of line AH, I think I need two things: the slope of BC and then the slope of AH since they should be negative reciprocals (because they are perpendicular). Once I have the slope of AH and a point it passes through (which is point A), I can write the equation of the line.Alright, let me find the slope of BC first. The coordinates of B are (2, 0) and C are (-2, -1). The slope formula is (y2 - y1)/(x2 - x1). So, plugging in the values:Slope of BC = (-1 - 0)/(-2 - 2) = (-1)/(-4) = 1/4.Wait, so the slope of BC is 1/4. That means the slope of AH, which is perpendicular to BC, should be the negative reciprocal. The reciprocal of 1/4 is 4, so the negative reciprocal is -4. So, the slope of AH is -4.Now that I have the slope of AH and a point it passes through (A(0,2)), I can use the point-slope form to write the equation. Point-slope is y - y1 = m(x - x1), where m is the slope and (x1, y1) is the point.Plugging in the values:y - 2 = -4(x - 0)Simplifying that:y - 2 = -4xSo, adding 2 to both sides:y = -4x + 2Okay, so the equation of the height AH is y = -4x + 2. That seems straightforward. Let me just double-check my calculations.Slope of BC: (y2 - y1)/(x2 - x1) = (-1 - 0)/(-2 - 2) = (-1)/(-4) = 1/4. Correct.Slope of AH: negative reciprocal of 1/4 is -4. Correct.Point-slope form: y - 2 = -4x. So, y = -4x + 2. Yep, that looks right.Moving on to part (2): finding the area of triangle ABC. I remember there are a few ways to find the area of a triangle given coordinates of its vertices. One common method is using the shoelace formula, which is a formula that uses the coordinates to calculate the area.The shoelace formula is given by:Area = |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|Let me assign the points:Point A: (x1, y1) = (0, 2)Point B: (x2, y2) = (2, 0)Point C: (x3, y3) = (-2, -1)Plugging these into the formula:Area = |(0*(0 - (-1)) + 2*(-1 - 2) + (-2)*(2 - 0))/2|Let me compute each term step by step.First term: 0*(0 - (-1)) = 0*(1) = 0Second term: 2*(-1 - 2) = 2*(-3) = -6Third term: -2*(2 - 0) = -2*(2) = -4Now, adding these together:0 + (-6) + (-4) = -10Take the absolute value and divide by 2:| -10 | / 2 = 10 / 2 = 5So, the area is 5 square units.Wait, that seems a bit small. Let me verify using another method to make sure I didn't make a mistake.Another way to find the area is by using the formula:Area = (base * height)/2In this case, if I can find the length of side BC and then find the height from A to BC, which we've already found in part (1), I can compute the area.First, let me find the length of BC. Using the distance formula between points B(2,0) and C(-2,-1):Distance BC = sqrt[(x2 - x1)^2 + (y2 - y1)^2] = sqrt[(-2 - 2)^2 + (-1 - 0)^2] = sqrt[(-4)^2 + (-1)^2] = sqrt[16 + 1] = sqrt[17]So, the length of BC is sqrt(17).Now, the height from A to BC is the length of AH. Since we have the equation of AH, which is y = -4x + 2, and we know that H lies on BC. So, to find H, we need to find the intersection point of AH and BC.First, let me find the equation of BC. We already found the slope of BC earlier, which was 1/4. So, using point B(2,0) to write the equation.Point-slope form: y - 0 = (1/4)(x - 2)Simplifying:y = (1/4)x - (1/2)So, equation of BC is y = (1/4)x - 1/2.Now, to find point H, which is the intersection of AH and BC.Equation of AH: y = -4x + 2Equation of BC: y = (1/4)x - 1/2Set them equal:-4x + 2 = (1/4)x - 1/2Let me solve for x.Multiply both sides by 4 to eliminate fractions:-16x + 8 = x - 2Bring all terms to one side:-16x - x + 8 + 2 = 0-17x + 10 = 0-17x = -10x = (-10)/(-17) = 10/17Now, plug x back into one of the equations to find y. Let's use equation of AH:y = -4*(10/17) + 2 = -40/17 + 34/17 = (-40 + 34)/17 = (-6)/17So, point H is at (10/17, -6/17)Now, let's find the length of AH. Point A is at (0,2) and point H is at (10/17, -6/17).Using the distance formula:AH = sqrt[(10/17 - 0)^2 + (-6/17 - 2)^2] = sqrt[(100/289) + (-6/17 - 34/17)^2] = sqrt[(100/289) + (-40/17)^2]Wait, let me compute each part step by step.First, x-coordinate difference: 10/17 - 0 = 10/17Y-coordinate difference: -6/17 - 2 = -6/17 - 34/17 = (-40)/17So, AH = sqrt[(10/17)^2 + (-40/17)^2] = sqrt[(100/289) + (1600/289)] = sqrt[(1700)/289] = sqrt[1700]/sqrt[289] = sqrt[1700]/17Simplify sqrt[1700]. 1700 = 100 * 17, so sqrt[100*17] = 10*sqrt(17). Therefore, AH = (10*sqrt(17))/17.So, the height AH is (10*sqrt(17))/17.Earlier, we found the length of BC is sqrt(17). So, plugging into the area formula:Area = (base * height)/2 = (sqrt(17) * (10*sqrt(17))/17)/2Simplify:First, sqrt(17) * sqrt(17) = 17So, numerator becomes 17 * 10 /17 = 10Then, divide by 2: 10 / 2 = 5So, the area is 5 square units. That matches the result from the shoelace formula. So, that's reassuring.Alternatively, another way to find the area is by using vectors or determinants, but I think the shoelace formula and the base-height method both gave me the same answer, so I feel confident that the area is indeed 5.Wait, just to make sure, let me visualize the triangle. Point A is at (0,2), which is on the y-axis, point B is at (2,0), which is on the x-axis, and point C is at (-2,-1), which is in the third quadrant. So, the triangle is spread out across different quadrants. The area being 5 seems reasonable.I think I've covered all the steps and double-checked using two different methods. So, I'm pretty confident about both answers.**Final Answer**(1) The equation of the height AH is boxed{y = -4x + 2}.(2) The area of triangle ABC is boxed{5}.

answer：To determine the sum of the different prime factors of 150,280, I will start by performing prime factorization.First, I notice that 150,280 is an even number, so it is divisible by 2. Dividing 150,280 by 2 gives 75,140. I continue dividing by 2 until I can no longer do so evenly, resulting in 18,785.Next, I check if 18,785 is divisible by 5, since it ends with a 5. Dividing by 5 gives 3,757.I then test 3,757 for divisibility by smaller prime numbers. It is not divisible by 2, 3, or 5. Checking divisibility by 7, I find that 3,757 divided by 7 is 537 with a remainder, so it is not divisible by 7. Continuing this process, I find that 3,757 is a prime number.Finally, I compile the unique prime factors: 2, 5, and 3,757. Adding these together gives the total sum of the different prime factors.