💡Okay, so I have this problem where I need to compare three logarithms: a, b, and c. They are defined as follows:- ( a = log_{4}3 )- ( b = log_{3}4 )- ( c = log_{5}3 )And I need to figure out the order of these three values from largest to smallest. The options are:A: ( a > b > c )B: ( b > a > c )C: ( b > c > a )D: ( a > c > b )Hmm, let me think about how to approach this. I remember that logarithms can be tricky, especially when the bases and the arguments are different. Maybe I can convert them to a common base or use some properties of logarithms to compare them.First, let me recall that ( log_{b}a ) is the exponent to which we need to raise b to get a. So, ( a = log_{4}3 ) means 4 raised to what power gives 3? Similarly for the others.I also remember that if the base is greater than 1, the logarithm function is increasing. So, if I have two logarithms with the same base, the one with the larger argument is larger. But here, the bases are different, so that complicates things.Maybe I can use the change of base formula to express all of them in terms of natural logarithms or base 10 logarithms. The change of base formula is:( log_{b}a = frac{ln a}{ln b} )So, let me apply that to each of them.For ( a = log_{4}3 ):( a = frac{ln 3}{ln 4} )For ( b = log_{3}4 ):( b = frac{ln 4}{ln 3} )For ( c = log_{5}3 ):( c = frac{ln 3}{ln 5} )Okay, so now I have all three expressed in terms of natural logarithms. Maybe I can compare these fractions.Let me note that ( ln 3 ), ( ln 4 ), and ( ln 5 ) are all positive numbers because the natural logarithm of a number greater than 1 is positive.So, ( a = frac{ln 3}{ln 4} ), ( b = frac{ln 4}{ln 3} ), and ( c = frac{ln 3}{ln 5} ).Hmm, interesting. So, ( a ) and ( c ) both have ( ln 3 ) in the numerator, but different denominators. ( b ) is the reciprocal of ( a ).Wait, that's a key point. ( b = frac{ln 4}{ln 3} ) is the reciprocal of ( a = frac{ln 3}{ln 4} ). So, ( b = frac{1}{a} ). That means if I can figure out whether ( a ) is greater than 1 or less than 1, I can know the relationship between ( a ) and ( b ).Similarly, ( c = frac{ln 3}{ln 5} ). Since ( ln 5 > ln 4 > ln 3 ), the denominator in ( c ) is larger than in ( a ), so ( c ) should be smaller than ( a ).Let me elaborate.First, let's consider ( a = log_{4}3 ). Since 4 is greater than 3, and the logarithm base is greater than the argument, the result should be less than 1. Because ( log_{4}4 = 1 ), and since 3 is less than 4, ( log_{4}3 ) is less than 1. So, ( a < 1 ).Similarly, ( b = log_{3}4 ). Here, the base is 3 and the argument is 4, which is greater than 3. So, since the argument is greater than the base, the logarithm should be greater than 1. Because ( log_{3}3 = 1 ), and 4 is greater than 3, ( log_{3}4 ) is greater than 1. So, ( b > 1 ).Now, ( c = log_{5}3 ). Similar to ( a ), since 5 is greater than 3, ( log_{5}3 ) is less than 1. So, ( c < 1 ).So, so far, I know that ( b > 1 ), while both ( a ) and ( c ) are less than 1. So, ( b ) is definitely the largest among the three.Now, I need to compare ( a ) and ( c ) to see which one is larger.We have ( a = frac{ln 3}{ln 4} ) and ( c = frac{ln 3}{ln 5} ).Since ( ln 5 > ln 4 ), the denominator in ( c ) is larger than in ( a ). Therefore, ( c ) is smaller than ( a ) because both have the same numerator, but ( c ) is divided by a larger number.So, ( a > c ).Putting it all together:( b > 1 ), ( a < 1 ), ( c < 1 ), and ( a > c ).Therefore, the order is ( b > a > c ).Looking back at the options:A: ( a > b > c ) → Incorrect because ( b ) is the largest.B: ( b > a > c ) → Correct.C: ( b > c > a ) → Incorrect because ( a > c ).D: ( a > c > b ) → Incorrect because ( b ) is the largest.So, the correct answer should be option B.Wait, let me double-check my reasoning to make sure I didn't make any mistakes.First, I converted all logarithms to natural logs using the change of base formula. That seems correct.Then, I noted that ( b = 1/a ). Since ( a < 1 ), ( b = 1/a > 1 ). That makes sense.For ( c ), since the base is larger (5 vs. 4), the logarithm of 3 will be smaller because the function ( log_{b}3 ) decreases as ( b ) increases, given that ( b > 3 ). So, ( c < a ). That also makes sense.Therefore, I think my reasoning is solid. The order is indeed ( b > a > c ), which is option B.**Final Answer**boxed{B}

💡First, I need to evaluate the expression inside the parentheses: 105 divided by 21.0.Calculating 105 ÷ 21.0 gives me 5.Next, I subtract this result from 6000: 6000 - 5 equals 5995.Therefore, the final answer is 5995.

💡Okay, so I have this problem here: In the line 3x + 5y + c = 0, the sum of the x- and y-intercepts is 55/4. I need to find c. Hmm, let me think about how to approach this.First, I remember that to find the x-intercept of a line, I set y to 0 and solve for x. Similarly, to find the y-intercept, I set x to 0 and solve for y. That makes sense because the x-intercept is where the line crosses the x-axis, so y is 0 there, and the y-intercept is where it crosses the y-axis, so x is 0 there.Alright, so let me start by finding the x-intercept. If I set y = 0 in the equation 3x + 5y + c = 0, it becomes:3x + 5(0) + c = 0 Simplifying that, I get: 3x + c = 0 So, solving for x, I subtract c from both sides: 3x = -c Then, divide both sides by 3: x = -c/3Okay, so the x-intercept is at the point (-c/3, 0). Got that.Now, let's find the y-intercept. I'll set x = 0 in the equation 3x + 5y + c = 0, which gives:3(0) + 5y + c = 0 Simplifying: 5y + c = 0 Solving for y, subtract c from both sides: 5y = -c Divide both sides by 5: y = -c/5So, the y-intercept is at the point (0, -c/5). Got that as well.The problem says that the sum of the x-intercept and y-intercept is 55/4. Hmm, wait a second. The x-intercept is a point on the x-axis, which is (-c/3, 0), and the y-intercept is a point on the y-axis, which is (0, -c/5). But when they say the sum of the intercepts, do they mean the sum of the x-intercept value and the y-intercept value? I think so, because intercepts are often referred to by their coordinates, but in this context, since they're talking about the sum, it's likely the sum of the x-value and the y-value.So, the x-intercept value is -c/3, and the y-intercept value is -c/5. So, adding those together:(-c/3) + (-c/5) = 55/4Wait, that would be negative c over 3 plus negative c over 5 equals 55 over 4. Hmm, let me write that out:(-c/3) + (-c/5) = 55/4Combine the terms on the left side. To add these fractions, I need a common denominator. The denominators are 3 and 5, so the least common denominator is 15. So, let's convert both fractions:(-c/3) is equivalent to (-5c/15) and (-c/5) is equivalent to (-3c/15). So, adding them together:(-5c/15) + (-3c/15) = (-8c)/15So, the equation becomes:(-8c)/15 = 55/4Now, I need to solve for c. Let's rewrite the equation:(-8c)/15 = 55/4To isolate c, I can multiply both sides by 15:-8c = (55/4) * 15Let me compute (55/4) * 15. 55 times 15 is... let's see, 55*10 is 550, 55*5 is 275, so 550 + 275 is 825. So, 55*15 is 825, and then divided by 4 is 825/4.So, now we have:-8c = 825/4To solve for c, divide both sides by -8:c = (825/4) / (-8)Dividing by -8 is the same as multiplying by -1/8:c = (825/4) * (-1/8) = -825/32Wait, but hold on. The problem says the sum of the intercepts is 55/4, which is positive. But when I calculated the sum, I got (-8c)/15 = 55/4, which led to c being negative. Is that correct?Let me double-check my steps. So, x-intercept is -c/3, y-intercept is -c/5. Sum is (-c/3) + (-c/5) = 55/4. So, that's (-8c)/15 = 55/4. So, c is negative because 55/4 is positive, and the left side is negative. So, c is negative. Hmm, that seems correct.But let me verify by plugging the value back into the intercepts. If c is -825/32, then:x-intercept is -c/3 = -(-825/32)/3 = (825/32)/3 = 825/96Similarly, y-intercept is -c/5 = -(-825/32)/5 = (825/32)/5 = 825/160Now, let's compute the sum:825/96 + 825/160To add these, find a common denominator. 96 and 160. Let's see, 96 factors into 16*6, and 160 is 16*10. So, the least common multiple is 16*6*10 = 960.Convert 825/96 to 960 denominator: 96 goes into 960 ten times, so multiply numerator and denominator by 10: 825*10=8250, so 8250/960Convert 825/160 to 960 denominator: 160 goes into 960 six times, so multiply numerator and denominator by 6: 825*6=4950, so 4950/960Now, add them together: 8250/960 + 4950/960 = (8250 + 4950)/960 = 13200/960Simplify 13200/960. Let's divide numerator and denominator by 10: 1320/96Divide numerator and denominator by 12: 110/8Simplify 110/8: divide numerator and denominator by 2: 55/4Yes, that's correct. So, the sum is indeed 55/4, which matches the problem statement. So, even though c is negative, the sum comes out positive because both intercepts are negative, and their sum is positive. So, that makes sense.Therefore, c is -825/32. But let me write that as a positive fraction with the negative sign: c = -825/32.Wait, but in the problem, the equation is 3x + 5y + c = 0. So, if c is negative, that would mean the equation is 3x + 5y - 825/32 = 0. That seems correct.Alternatively, if I factor out the negative sign, it's 3x + 5y + (-825/32) = 0, which is the same as 3x + 5y - 825/32 = 0.So, yeah, that seems consistent.I think I've got it. So, c is -825/32.**Final Answer**The value of ( c ) is boxed{-dfrac{825}{32}}.

💡Okay, so I have this problem where I need to find the points where two curves intersect. The equations given are:First curve: ( y = 3x^2 - 4x + 2 )Second curve: ( y = -x^3 + 9x^2 - 4x + 2 )Hmm, to find the points of intersection, I remember that I need to set the two equations equal to each other because at the points of intersection, the y-values will be the same for the same x-value. So, let me write that down:( 3x^2 - 4x + 2 = -x^3 + 9x^2 - 4x + 2 )Now, I need to solve this equation for x. Let me try to simplify it step by step. First, I'll subtract the left side from both sides to get everything on one side:( 0 = -x^3 + 9x^2 - 4x + 2 - 3x^2 + 4x - 2 )Wait, let me make sure I did that correctly. Subtracting each term:- ( -x^3 ) stays as is.- ( 9x^2 - 3x^2 = 6x^2 )- ( -4x + 4x = 0x ) (they cancel out)- ( 2 - 2 = 0 )So, simplifying, I get:( 0 = -x^3 + 6x^2 )Hmm, that seems right. Let me write it as:( -x^3 + 6x^2 = 0 )I can factor out a common term here. Both terms have an ( x^2 ) in them, so let's factor that out:( x^2(-x + 6) = 0 )Wait, is that correct? Let me check:( x^2 times (-x) = -x^3 )( x^2 times 6 = 6x^2 )Yes, that looks correct. So, the equation becomes:( x^2(-x + 6) = 0 )This gives us two factors: ( x^2 = 0 ) and ( -x + 6 = 0 ).Solving ( x^2 = 0 ) gives ( x = 0 ).Solving ( -x + 6 = 0 ) gives ( x = 6 ).So, the solutions are ( x = 0 ) and ( x = 6 ).Now, I need to find the corresponding y-values for each x to get the points of intersection.Starting with ( x = 0 ):I can plug this into either of the original equations. Let's use the first one:( y = 3(0)^2 - 4(0) + 2 = 0 - 0 + 2 = 2 )So, one point is ( (0, 2) ).Next, for ( x = 6 ):Again, I can plug this into either equation. Let me choose the second one this time:( y = -(6)^3 + 9(6)^2 - 4(6) + 2 )Calculating each term:- ( -(6)^3 = -216 )- ( 9(6)^2 = 9 times 36 = 324 )- ( -4(6) = -24 )- ( +2 ) remains as is.Now, adding them all together:( -216 + 324 - 24 + 2 )Let me compute step by step:- ( -216 + 324 = 108 )- ( 108 - 24 = 84 )- ( 84 + 2 = 86 )So, the y-value is 86. Therefore, the other point is ( (6, 86) ).Wait a second, let me verify this by plugging ( x = 6 ) into the first equation as well to make sure I didn't make a mistake.Using the first equation:( y = 3(6)^2 - 4(6) + 2 )Calculating each term:- ( 3(36) = 108 )- ( -4(6) = -24 )- ( +2 )Adding them up:( 108 - 24 + 2 = 86 )Okay, that matches. So, both points are correct.So, the curves intersect at ( (0, 2) ) and ( (6, 86) ). Since we're asked to list them in order of increasing x-coordinate, it's already in order because 0 comes before 6.I think that's all. Let me just recap to make sure I didn't miss anything.1. Set the equations equal to each other.2. Simplified the equation to ( -x^3 + 6x^2 = 0 ).3. Factored out ( x^2 ) to get ( x^2(-x + 6) = 0 ).4. Solved for x, getting x = 0 and x = 6.5. Plugged these x-values back into one of the original equations to find y.6. Verified the y-values by plugging into both equations to ensure consistency.Everything seems to check out. I don't think I made any calculation errors, but let me just double-check the calculations for ( x = 6 ) in the second equation:( y = -(6)^3 + 9(6)^2 - 4(6) + 2 )Calculating each term:- ( 6^3 = 216 ), so ( -216 )- ( 9 times 6^2 = 9 times 36 = 324 )- ( -4 times 6 = -24 )- ( +2 )Adding them up:( -216 + 324 = 108 )( 108 - 24 = 84 )( 84 + 2 = 86 )Yes, that's correct.And for ( x = 0 ):Both equations give y = 2, so that's consistent.I think I'm confident with my answers. The curves intersect at (0, 2) and (6, 86).**Final Answer**The points of intersection are boxed{(0, 2)} and boxed{(6, 86)}.

💡First, I need to convert the train's speed from kilometers per hour (km/h) to meters per second (m/s) to match the units of the train's length.To convert km/h to m/s, I multiply the speed by 1000 meters per kilometer and divide by 3600 seconds per hour:[ text{Speed in m/s} = 90 , text{km/h} times frac{1000 , text{m}}{1 , text{km}} times frac{1 , text{h}}{3600 , text{s}} = 25 , text{m/s} ]Next, I use the formula for time, which is:[ text{Time} = frac{text{Distance}}{text{Speed}} ]Here, the distance is the length of the train, which is 100.008 meters, and the speed is 25 m/s:[ text{Time} = frac{100.008 , text{m}}{25 , text{m/s}} = 4.00032 , text{seconds} ]Therefore, it takes approximately 4.00032 seconds for the train to cross the pole.

💡Alright, so I have this problem here: If ( a, b in mathbb{R} ), then a sufficient but not necessary condition for ( |a| + |b| > 1 ) to hold is (    ). The options are A, B, C, and D.First, I need to understand what a sufficient but not necessary condition means. A sufficient condition means that if the condition is met, then the statement ( |a| + |b| > 1 ) must be true. However, it's not necessary, meaning that the statement can still be true even if the condition isn't met.So, I need to go through each option and check if it's a sufficient condition for ( |a| + |b| > 1 ). Also, I need to make sure that it's not a necessary condition, meaning there are cases where ( |a| + |b| > 1 ) is true, but the condition isn't satisfied.Let's start with option A: ( |a + b| > 1 ).Hmm, if ( |a + b| > 1 ), does that necessarily mean ( |a| + |b| > 1 )? I know that ( |a + b| leq |a| + |b| ) by the triangle inequality. So, if ( |a + b| > 1 ), then ( |a| + |b| ) must be greater than or equal to ( |a + b| ), which is greater than 1. Therefore, ( |a| + |b| > 1 ) must hold. So, A is a sufficient condition.But is it necessary? That is, can ( |a| + |b| > 1 ) be true without ( |a + b| > 1 ) being true? Let's see. Suppose ( a = 1 ) and ( b = 1 ). Then ( |a| + |b| = 2 > 1 ), and ( |a + b| = 2 > 1 ). But what if ( a = 1 ) and ( b = -0.5 )? Then ( |a| + |b| = 1 + 0.5 = 1.5 > 1 ), but ( |a + b| = |0.5| = 0.5 ), which is not greater than 1. So, in this case, ( |a| + |b| > 1 ) is true, but ( |a + b| > 1 ) is false. Therefore, A is not a necessary condition. So, A is a sufficient but not necessary condition.Okay, so A seems to be a candidate. Let me check the other options just to be thorough.Option B: ( |a| geq frac{1}{2} ) and ( |b| geq frac{1}{2} ).If both ( |a| ) and ( |b| ) are at least ( frac{1}{2} ), then ( |a| + |b| geq frac{1}{2} + frac{1}{2} = 1 ). But the question is about ( |a| + |b| > 1 ). So, if ( |a| = frac{1}{2} ) and ( |b| = frac{1}{2} ), then ( |a| + |b| = 1 ), which is not greater than 1. Therefore, B is not a sufficient condition because it doesn't guarantee that ( |a| + |b| > 1 ). It only guarantees ( |a| + |b| geq 1 ). So, B is incorrect.Option C: ( |a| geq 1 ).If ( |a| geq 1 ), then regardless of the value of ( b ), ( |a| + |b| geq 1 + 0 = 1 ). But again, we need ( |a| + |b| > 1 ). If ( |a| = 1 ) and ( |b| = 0 ), then ( |a| + |b| = 1 ), which is not greater than 1. So, C is not a sufficient condition either because it doesn't guarantee ( |a| + |b| > 1 ). So, C is incorrect.Option D: ( b > -1 ).If ( b > -1 ), does that ensure ( |a| + |b| > 1 )? Let's see. Suppose ( a = 0 ) and ( b = 0 ). Then ( |a| + |b| = 0 + 0 = 0 ), which is not greater than 1. So, even though ( b = 0 > -1 ), ( |a| + |b| ) is not greater than 1. Therefore, D is not a sufficient condition. So, D is incorrect.So, after evaluating all options, only option A is a sufficient but not necessary condition for ( |a| + |b| > 1 ).**Final Answer**boxed{A}