answer：First, I need to understand the relationship between the capacities of Bucket A and Bucket B. Since Bucket B has half the capacity of Bucket A, I can express this as ( C_B = frac{1}{2} C_A ).Next, I'll determine the initial amount of sand in Bucket A, which is given as ( frac{1}{4} ) of its capacity. So, the initial sand in Bucket A is ( frac{1}{4} C_A ).After pouring the contents of Bucket B into Bucket A, the total sand in Bucket A becomes ( 0.4375 C_A ). To find out how much sand was in Bucket B, I'll subtract the initial amount of sand in Bucket A from the total amount after pouring:[0.4375 C_A - frac{1}{4} C_A = frac{7}{16} C_A - frac{4}{16} C_A = frac{3}{16} C_A]Now, I need to express this amount of sand (( frac{3}{16} C_A )) as a fraction of Bucket B's capacity (( C_B )). Since ( C_B = frac{1}{2} C_A ), I can rewrite the sand in Bucket B as:[frac{3}{16} C_A = frac{3}{16} times 2 C_B = frac{3}{8} C_B]Therefore, the initial fraction of Bucket B's capacity that was filled with sand is ( frac{3}{8} ).

answer：First, I need to determine how many gallons of diesel fuel Mr. Alvarez uses each week. He spends 36 on diesel fuel weekly, and the cost per gallon is 3.To find the weekly consumption, I divide the total amount spent by the cost per gallon:36 ÷ 3 per gallon = 12 gallons per week.Next, to find the consumption over two weeks, I multiply the weekly usage by 2:12 gallons per week × 2 weeks = 24 gallons.Therefore, Mr. Alvarez uses 24 gallons of diesel fuel in two weeks.

answer：Okay, so I have this multiple-choice question with four statements labeled A, B, C, and D. I need to figure out which one is correct. Let me go through each statement one by one and think about whether they're true or not.Starting with statement A: "The sum of the exterior angles of a polygon is 360 degrees." Hmm, I remember learning something about exterior angles in geometry. I think for any polygon, if you add up all the exterior angles, it always equals 360 degrees. Is that right? Let me visualize a polygon, like a triangle or a square. For a triangle, each exterior angle would be 120 degrees, and there are three of them, so 120 times 3 is 360. For a square, each exterior angle is 90 degrees, and there are four, so 90 times 4 is also 360. Okay, that seems consistent. So statement A seems correct.Moving on to statement B: "6a²b - 2ab² = 2ab(3a - 2b)." Alright, this looks like an algebraic expression. Let me try to factor the left side. The terms are 6a²b and -2ab². I can factor out a common factor of 2ab from both terms. So, factoring 2ab out of 6a²b gives 3a, and factoring 2ab out of -2ab² gives -b. So, the factored form should be 2ab(3a - b). But the statement says 2ab(3a - 2b). That means they have an extra factor of 2 in the second term. That doesn't seem right. So statement B is incorrect.Next is statement C: "525000 = 5.25 × 10³." Let me check this. 5.25 times 10³ is 5.25 times 1000, which equals 5250. But 525000 is much larger than 5250. To express 525000 in scientific notation, I need to move the decimal point five places to the left, which would give me 5.25 × 10⁵. So the correct scientific notation is 5.25 × 10⁵, not 10³. Therefore, statement C is incorrect.Finally, statement D: "Things with very low possibility are impossible to happen." Hmm, this sounds like it's talking about probability. I know that in probability, an event with a very low probability isn't impossible; it's just unlikely. For example, winning the lottery has a very low probability, but it's not impossible—it can still happen. So saying that things with very low possibility are impossible is not true. Therefore, statement D is incorrect.After going through all four statements, only statement A is correct. The sum of the exterior angles of any polygon is indeed 360 degrees.

answer：Alright, so I've got this problem here where I need to figure out the relationship between m and n, and then determine which of the options A to D is correct. Let me try to break it down step by step.First, let's understand what m and n are.Given:- ( m = log_{0.3}{0.6} )- ( n = frac{1}{2}log_{2}{0.6} )Okay, so m is a logarithm with base 0.3 of 0.6, and n is half of the logarithm with base 2 of 0.6. Hmm, logarithms with bases less than 1 can be tricky because they're decreasing functions, unlike the usual logarithms with bases greater than 1 which are increasing. That might affect the sign of the logarithm.Let me recall that ( log_b{a} ) is positive if a and b are both greater than 1 or both less than 1. Since 0.3 and 0.6 are both less than 1, ( log_{0.3}{0.6} ) should be positive. Similarly, for ( log_{2}{0.6} ), since 2 is greater than 1 and 0.6 is less than 1, this logarithm should be negative. Therefore, n is half of a negative number, so n is negative.So, m is positive and n is negative. That means when I multiply m and n, the product mn will be negative. That's useful information because it tells me that mn is negative, which might help in comparing m + n and mn.Now, let's try to compute or estimate the values of m and n to get a better sense.Starting with m:( m = log_{0.3}{0.6} )I know that ( log_b{a} = frac{ln{a}}{ln{b}} ), so let's use natural logarithms to compute this.So, ( m = frac{ln{0.6}}{ln{0.3}} )Calculating the natural logs:- ( ln{0.6} ) is approximately -0.5108- ( ln{0.3} ) is approximately -1.2039So, ( m = frac{-0.5108}{-1.2039} approx 0.424 )Alright, so m is approximately 0.424.Now, let's compute n:( n = frac{1}{2}log_{2}{0.6} )Again, using natural logs:( log_{2}{0.6} = frac{ln{0.6}}{ln{2}} )We already have ( ln{0.6} approx -0.5108 ) and ( ln{2} approx 0.6931 )So, ( log_{2}{0.6} approx frac{-0.5108}{0.6931} approx -0.7369 )Therefore, ( n = frac{1}{2} times (-0.7369) approx -0.3685 )So, n is approximately -0.3685.Now, let's summarize:- m ≈ 0.424- n ≈ -0.3685Now, let's compute m + n and mn to compare them.First, m + n:0.424 + (-0.3685) = 0.424 - 0.3685 = 0.0555So, m + n ≈ 0.0555Next, mn:0.424 * (-0.3685) ≈ -0.156So, mn ≈ -0.156Now, let's look at the options:A) ( m + n > mn )B) ( m + n < mn )C) ( n - m > mn )D) ( m - n < mn )Let's evaluate each option.Option A: 0.0555 > -0.156Yes, because 0.0555 is positive and -0.156 is negative. So, A is true.Option B: 0.0555 < -0.156No, because 0.0555 is greater than -0.156. So, B is false.Option C: ( n - m > mn )Compute n - m: -0.3685 - 0.424 = -0.7925Compare to mn: -0.7925 > -0.156Is -0.7925 > -0.156? No, because -0.7925 is less than -0.156. So, C is false.Option D: ( m - n < mn )Compute m - n: 0.424 - (-0.3685) = 0.424 + 0.3685 = 0.7925Compare to mn: 0.7925 < -0.156Is 0.7925 < -0.156? No, because 0.7925 is positive and greater than -0.156. So, D is false.Therefore, only option A is correct.Wait, but let me double-check my calculations because sometimes approximations can be misleading.Let me compute m and n more accurately.Calculating m:( ln{0.6} approx -0.510825623766 )( ln{0.3} approx -1.203972804326 )So, ( m = frac{-0.510825623766}{-1.203972804326} approx 0.424 ) (as before)Calculating n:( ln{0.6} approx -0.510825623766 )( ln{2} approx 0.69314718056 )So, ( log_{2}{0.6} = frac{-0.510825623766}{0.69314718056} approx -0.736965594 )Thus, ( n = frac{1}{2} times (-0.736965594) approx -0.368482797 )So, m ≈ 0.424 and n ≈ -0.3685 are accurate.Then, m + n ≈ 0.424 - 0.3685 ≈ 0.0555mn ≈ 0.424 * (-0.3685) ≈ -0.156So, m + n ≈ 0.0555, which is greater than mn ≈ -0.156. So, A is correct.Just to make sure, let's consider if there's another way to approach this without approximating, perhaps using logarithmic identities or inequalities.I remember that for positive numbers a and b, ( log_{a}{b} = frac{1}{log_{b}{a}} ). Maybe that can help.Given that m = ( log_{0.3}{0.6} ), which can be rewritten as ( frac{1}{log_{0.6}{0.3}} ). Similarly, n is ( frac{1}{2}log_{2}{0.6} ), which is ( frac{1}{2} times frac{ln{0.6}}{ln{2}} ).But I'm not sure if that helps directly. Alternatively, maybe expressing both logarithms in terms of the same base could help compare them.Alternatively, perhaps using the change of base formula to express both m and n in terms of natural logs or base 10 logs.But since I already have approximate values, and they seem consistent, I think my initial conclusion is correct.Another thought: since m is positive and n is negative, m + n is a small positive number, and mn is negative. So, m + n is positive and mn is negative, so m + n > mn must be true.Therefore, option A is correct.**Final Answer**boxed{A}

answer：First, I need to determine the distance from marker B to marker C. Since this distance is half of the distance from marker A to marker B, and the distance from A to B is 120 miles, the distance from B to C is 60 miles.Next, I calculate the total distance of the trip from marker A to marker C by adding the distances from A to B and B to C, which gives 180 miles.Given that the average speed for the entire trip is 50 miles per hour, I can find the total time taken for the trip by dividing the total distance by the average speed. This results in a total time of 3.6 hours.The ride from marker A to marker B lasted 3 times as long as the ride from marker B to marker C. Let’s denote the time taken to travel from B to C as ( t ) hours. Therefore, the time taken to travel from A to B is ( 3t ) hours. Adding these times together gives the total time:[3t + t = 3.6 text{ hours}]Solving for ( t ):[4t = 3.6][t = frac{3.6}{4} = 0.9 text{ hours}]Finally, I calculate the average speed from marker B to marker C by dividing the distance from B to C by the time taken for that segment:[text{Average Speed} = frac{60 text{ miles}}{0.9 text{ hours}} = 66.overline{6} text{ mph}]

answer：Alright, so I have this problem here about mathematical induction, and I need to figure out what expression should be added to the left side of the equation when moving from n = k to n = k + 1. The formula given is:{1^{2}} + {2}^{2} + dots + {(n-1)}^{2} + {n}^{2} + {(n-1)}^{2} + dots + {2}^{2} + {1}^{2} = frac{n(2n^{2} + 1)}{3}Okay, first, let me recall what mathematical induction is. It's a method of proving that a statement holds true for all natural numbers. It involves two main steps: the base case and the inductive step. The base case is usually proving the statement for n = 1, and the inductive step assumes the statement is true for some arbitrary natural number k and then proves it for k + 1.In this problem, I don't need to prove the entire formula, just figure out what needs to be added to the left side when moving from n = k to n = k + 1. The options are given as A, B, C, D, so I need to figure out which one is correct.Let me think about how the left side of the equation changes when n increases by 1. For n = k, the left side is:{1^{2}} + {2}^{2} + dots + {(k-1)}^{2} + {k}^{2} + {(k-1)}^{2} + dots + {2}^{2} + {1}^{2}And for n = k + 1, it should be:{1^{2}} + {2}^{2} + dots + {k}^{2} + {(k+1)}^{2} + {k}^{2} + dots + {2}^{2} + {1}^{2}So, comparing the two, when moving from n = k to n = k + 1, we add two new terms: one at the middle, which is {(k + 1)}^{2}, and another one just before it, which is {k}^{2}. Wait, is that right?Wait, no. Let me visualize it. For n = k, the sequence goes up to k^2 and then back down to 1^2. For n = k + 1, it goes up to (k + 1)^2 and then back down to 1^2. So, compared to n = k, n = k + 1 adds one term at the top, which is (k + 1)^2, and also adds another term just below it, which is k^2, because the sequence now has to go from (k + 1)^2 back down to k^2, and so on.Wait, but in the original sequence for n = k, after k^2, it goes back to (k - 1)^2. So, for n = k + 1, after (k + 1)^2, it goes back to k^2, which was already present in the n = k case. So, actually, the only new term added is (k + 1)^2, and then the rest follows as before.But hold on, in the n = k case, the sequence is symmetric around k^2. For n = k + 1, it's symmetric around (k + 1)^2. So, to go from n = k to n = k + 1, you have to add (k + 1)^2 on the way up and then also add another k^2 on the way down? Hmm, that seems a bit confusing.Let me think differently. The total number of terms for n = k is 2k - 1. Because it goes from 1 to k and back to 1, excluding the middle term which is only counted once. So, for n = k, it's 2k - 1 terms. For n = k + 1, it's 2(k + 1) - 1 = 2k + 1 terms. So, the difference is 2 terms. Therefore, when moving from n = k to n = k + 1, we need to add two terms to the left side.Looking back at the sequences, for n = k, the last term before going back down is k^2, and for n = k + 1, before going back down, it's (k + 1)^2 and then k^2. So, compared to n = k, for n = k + 1, we have added (k + 1)^2 and k^2. Therefore, the expression to be added is (k + 1)^2 + k^2.Looking at the options, that's option B: (k + 1)^2 + k^2.Wait, but let me double-check. If I have the left side for n = k:Sum = 1^2 + 2^2 + ... + (k-1)^2 + k^2 + (k-1)^2 + ... + 2^2 + 1^2And for n = k + 1:Sum = 1^2 + 2^2 + ... + k^2 + (k + 1)^2 + k^2 + ... + 2^2 + 1^2So, compared to n = k, n = k + 1 has two additional terms: (k + 1)^2 and k^2. So yes, the expression to be added is (k + 1)^2 + k^2, which is option B.Alternatively, I can think about the formula for the sum. The formula given is:Sum = n(2n^2 + 1)/3So, for n = k, it's k(2k^2 + 1)/3, and for n = k + 1, it's (k + 1)(2(k + 1)^2 + 1)/3.If I subtract the two, I should get the expression that needs to be added to the left side.Let me compute:Sum for n = k + 1 - Sum for n = k = [(k + 1)(2(k + 1)^2 + 1)/3] - [k(2k^2 + 1)/3]Simplify this:= [ (k + 1)(2(k^2 + 2k + 1) + 1) - k(2k^2 + 1) ] / 3= [ (k + 1)(2k^2 + 4k + 2 + 1) - k(2k^2 + 1) ] / 3= [ (k + 1)(2k^2 + 4k + 3) - k(2k^2 + 1) ] / 3Now, expand (k + 1)(2k^2 + 4k + 3):= k(2k^2 + 4k + 3) + 1(2k^2 + 4k + 3)= 2k^3 + 4k^2 + 3k + 2k^2 + 4k + 3Combine like terms:= 2k^3 + (4k^2 + 2k^2) + (3k + 4k) + 3= 2k^3 + 6k^2 + 7k + 3Now, subtract k(2k^2 + 1):= 2k^3 + 6k^2 + 7k + 3 - (2k^3 + k)= 2k^3 + 6k^2 + 7k + 3 - 2k^3 - k= (2k^3 - 2k^3) + 6k^2 + (7k - k) + 3= 0 + 6k^2 + 6k + 3So, the difference is (6k^2 + 6k + 3)/3 = 2k^2 + 2k + 1Wait, but according to the earlier reasoning, the expression to be added is (k + 1)^2 + k^2 = (k^2 + 2k + 1) + k^2 = 2k^2 + 2k + 1Which matches the result from subtracting the sums. So, that confirms that the expression to be added is indeed (k + 1)^2 + k^2, which is option B.So, I think option B is correct.