answer：First, I need to determine the speed of the boat in still water. I'll denote this speed as ( V_b ) km/hr.When the boat is moving downstream, the speed of the stream adds to the boat's speed. Therefore, the effective downstream speed is ( V_b + 5 ) km/hr.The boat travels 54 km downstream in 2 hours. Using the formula for distance, which is ( text{Distance} = text{Speed} times text{Time} ), I can set up the equation:[ 54 = (V_b + 5) times 2 ]To solve for ( V_b ), I'll first expand the equation:[ 54 = 2V_b + 10 ]Next, I'll subtract 10 from both sides to isolate the term with ( V_b ):[ 44 = 2V_b ]Finally, I'll divide both sides by 2 to find the value of ( V_b ):[ V_b = 22 ]Therefore, the speed of the boat in still water is 22 km/hr.

answer：Alright, so I've got this expression to evaluate:[left(frac{(x+1)^{2}(x^2-x+2)^{2}}{(x^{3}+1)^{2}}right)^{2}cdotleft(frac{(x-1)^{2}(x^2+x+2)^{2}}{(x^{3}-2)^{2}}right)^{2}.]And the options are:**A)** (x+1)^4 **B)** (x^3+1)^4 **C)** (x-1)^4 **D)** None of the aboveOkay, first, I need to simplify this expression step by step. Let's break it down.Starting with the first fraction:[frac{(x+1)^{2}(x^2 - x + 2)^{2}}{(x^3 + 1)^{2}}]I remember that x^3 + 1 can be factored using the sum of cubes formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2). So, applying that here:[x^3 + 1 = (x + 1)(x^2 - x + 1)]Wait, but in the numerator, we have (x^2 - x + 2)^2, which is different from (x^2 - x + 1)^2. Hmm, that might be a problem. Let me check if I did that correctly.Yes, the sum of cubes formula is correct. So, x^3 + 1 factors into (x + 1)(x^2 - x + 1). But the numerator has (x^2 - x + 2)^2. So, they're not the same. That means I can't cancel out the (x^2 - x + 1) term in the denominator with the numerator. Hmm, that complicates things.Let me write the first fraction again:[frac{(x+1)^{2}(x^2 - x + 2)^{2}}{(x + 1)^2(x^2 - x + 1)^2}]Oh, wait! I can cancel out the (x + 1)^2 terms in the numerator and denominator. So, that simplifies to:[frac{(x^2 - x + 2)^{2}}{(x^2 - x + 1)^{2}}]Okay, so the first fraction simplifies to that. Now, the entire first part is squared, so:[left(frac{(x^2 - x + 2)^{2}}{(x^2 - x + 1)^{2}}right)^2 = left(frac{(x^2 - x + 2)}{(x^2 - x + 1)}right)^4]Alright, moving on to the second fraction:[frac{(x - 1)^{2}(x^2 + x + 2)^{2}}{(x^3 - 2)^{2}}]I wonder if x^3 - 2 can be factored similarly. The difference of cubes formula is a^3 - b^3 = (a - b)(a^2 + ab + b^2). So, applying that:[x^3 - 2 = (x - sqrt[3]{2})(x^2 + sqrt[3]{2}x + (sqrt[3]{2})^2)]Hmm, that's more complicated because it involves cube roots, which aren't as straightforward as the sum of cubes with integer coefficients. The numerator has (x^2 + x + 2)^2, which doesn't seem to match the denominator's factorization. So, similar to the first fraction, I can't cancel out terms easily here.Let me write the second fraction again:[frac{(x - 1)^{2}(x^2 + x + 2)^{2}}{(x^3 - 2)^{2}}]I don't see an obvious cancellation here because the denominator doesn't factor neatly with the numerator. So, I'll leave it as is for now.The entire second part is squared, so:[left(frac{(x - 1)^{2}(x^2 + x + 2)^{2}}{(x^3 - 2)^{2}}right)^2 = left(frac{(x - 1)(x^2 + x + 2)}{x^3 - 2}right)^4]Now, putting it all together, the original expression becomes:[left(frac{(x^2 - x + 2)}{(x^2 - x + 1)}right)^4 cdot left(frac{(x - 1)(x^2 + x + 2)}{x^3 - 2}right)^4]Hmm, this still looks pretty complicated. Maybe I can combine these two fractions into one:[left(frac{(x^2 - x + 2)(x - 1)(x^2 + x + 2)}{(x^2 - x + 1)(x^3 - 2)}right)^4]I wonder if the numerator and denominator can be simplified further. Let's look at the numerator:[(x^2 - x + 2)(x - 1)(x^2 + x + 2)]And the denominator:[(x^2 - x + 1)(x^3 - 2)]Is there any way to factor or simplify these terms? Let's see.First, let's multiply out the numerator:First, multiply (x^2 - x + 2) and (x^2 + x + 2):[(x^2 - x + 2)(x^2 + x + 2) = x^4 + x^3 + 2x^2 - x^3 - x^2 - 2x + 2x^2 + 2x + 4]Simplify term by term:- x^4- x^3 - x^3 = 0- 2x^2 - x^2 + 2x^2 = 3x^2- -2x + 2x = 0- +4So, it simplifies to:[x^4 + 3x^2 + 4]Now, multiply this by (x - 1):[(x^4 + 3x^2 + 4)(x - 1) = x^5 - x^4 + 3x^3 - 3x^2 + 4x - 4]So, the numerator is x^5 - x^4 + 3x^3 - 3x^2 + 4x - 4.Now, the denominator is (x^2 - x + 1)(x^3 - 2). Let's multiply these out:First, multiply x^2 by (x^3 - 2):[x^2 cdot x^3 = x^5][x^2 cdot (-2) = -2x^2]Next, multiply -x by (x^3 - 2):[-x cdot x^3 = -x^4][-x cdot (-2) = 2x]Finally, multiply 1 by (x^3 - 2):[1 cdot x^3 = x^3][1 cdot (-2) = -2]Now, combine all these terms:[x^5 - 2x^2 - x^4 + 2x + x^3 - 2]Arrange them in descending order:[x^5 - x^4 + x^3 - 2x^2 + 2x - 2]So, the denominator is x^5 - x^4 + x^3 - 2x^2 + 2x - 2.Now, let's compare the numerator and the denominator:Numerator: x^5 - x^4 + 3x^3 - 3x^2 + 4x - 4Denominator: x^5 - x^4 + x^3 - 2x^2 + 2x - 2Hmm, they look similar but not identical. Let's subtract the denominator from the numerator to see if there's a common factor or something:[(x^5 - x^4 + 3x^3 - 3x^2 + 4x - 4) - (x^5 - x^4 + x^3 - 2x^2 + 2x - 2) = ][0x^5 + 0x^4 + 2x^3 - x^2 + 2x - 2]So, the difference is 2x^3 - x^2 + 2x - 2. That doesn't seem to factor nicely or cancel out anything. Therefore, it looks like the numerator and denominator don't share any common factors beyond what's already been simplified.So, putting it all together, the expression simplifies to:[left(frac{x^5 - x^4 + 3x^3 - 3x^2 + 4x - 4}{x^5 - x^4 + x^3 - 2x^2 + 2x - 2}right)^4]This doesn't seem to simplify any further into a nice polynomial like (x+1)^4, (x^3+1)^4, or (x-1)^4. Therefore, the answer must be **D) None of the above**.

answer：First, I need to determine the longest possible rod that can fit inside a cubical room with a side length of 4 meters. The longest rod would align with the space diagonal of the cube.To find the space diagonal, I'll use the formula for the diagonal of a cube, which is derived from the Pythagorean theorem in three dimensions. The formula is:d = √(s² + s² + s²)where d is the diagonal and s is the side length of the cube.Substituting the given side length of 4 meters into the formula:d = √(4² + 4² + 4²)d = √(16 + 16 + 16)d = √48d = √(16 * 3)d = 4√3Therefore, the longest rod that can fit in the cubical room is 4√3 meters.

answer：Okay, so I have this problem here where I'm given a function f that satisfies f(xy) = f(x) + f(y) for any positive numbers x and y. They also tell me that f(8) = -3, and I need to find the value of a such that f(a) = 1/2. Hmm, interesting. Let me try to figure this out step by step.First, I remember that functions satisfying f(xy) = f(x) + f(y) are logarithmic functions. Because logarithms have that property where log(xy) = log(x) + log(y). So, maybe f is some kind of logarithm function. But I don't know the base yet. Maybe I can figure that out using the information given.They told me that f(8) = -3. If f is a logarithm, then f(8) would be log base b of 8, right? So, log_b(8) = -3. I can write that as an equation: log_b(8) = -3. To solve for b, I can rewrite this in exponential form. Remember, log_b(a) = c means that b^c = a. So, in this case, b^(-3) = 8.Hmm, so b^(-3) = 8. That means 1/(b^3) = 8. So, b^3 = 1/8. Taking the cube root of both sides, b = (1/8)^(1/3). The cube root of 1 is 1, and the cube root of 8 is 2, so b = 1/2. So, the base of the logarithm is 1/2.Therefore, f(x) is log base 1/2 of x. So, f(x) = log_{1/2}(x). Let me verify that. If f(8) = log_{1/2}(8), then what is that? Well, log base 1/2 of 8 is the exponent you need to raise 1/2 to get 8. So, (1/2)^k = 8. Let's solve for k.(1/2)^k = 8. That's the same as 2^(-k) = 2^3. So, -k = 3, which means k = -3. So, yes, log_{1/2}(8) = -3, which matches the given information. So, that's correct.Now, I need to find a such that f(a) = 1/2. So, log_{1/2}(a) = 1/2. Let me solve for a. Again, using the definition of logarithms, log_{1/2}(a) = 1/2 means that (1/2)^(1/2) = a.So, (1/2)^(1/2) is the square root of 1/2, which is sqrt(1)/sqrt(2) = 1/sqrt(2). But usually, we rationalize the denominator, so that's sqrt(2)/2. So, a = sqrt(2)/2.Wait, let me double-check that. If I plug a = sqrt(2)/2 into f(a), I should get 1/2. So, f(sqrt(2)/2) = log_{1/2}(sqrt(2)/2). Let me compute that.First, sqrt(2)/2 is equal to 2^(-1/2). Because sqrt(2) is 2^(1/2), so sqrt(2)/2 is 2^(1/2) * 2^(-1) = 2^(-1/2). So, log_{1/2}(2^(-1/2)).Since log base 1/2 of 2^(-1/2) is equal to (log base 2 of 2^(-1/2)) / (log base 2 of 1/2). Because of the change of base formula: log_b(a) = log_c(a)/log_c(b). So, if I choose base 2, it becomes log_2(2^(-1/2)) / log_2(1/2).Compute numerator: log_2(2^(-1/2)) = -1/2.Denominator: log_2(1/2) = log_2(2^(-1)) = -1.So, overall, it's (-1/2)/(-1) = 1/2. Perfect, that's the value we wanted. So, f(sqrt(2)/2) = 1/2.So, the value of a is sqrt(2)/2. That makes sense.Alternatively, I can think about the function f(x) = log_{1/2}(x) and its properties. Since the base is less than 1, the function is decreasing. So, as x increases, f(x) decreases. So, when x is 8, f(x) is -3, which is a negative number. When x is 1, f(x) is 0, because log_{1/2}(1) = 0. When x is less than 1, f(x) becomes positive. So, since we're looking for f(a) = 1/2, which is positive, a must be less than 1. And indeed, sqrt(2)/2 is approximately 0.707, which is less than 1. So, that fits.Another way to approach this problem is by using the properties of the function f without explicitly identifying it as a logarithm. Let me try that method as well to confirm.Given f(xy) = f(x) + f(y) for any positive x and y. This is Cauchy's functional equation, and since we're dealing with positive real numbers, the solutions are logarithmic functions. So, f(x) = k log(x) for some constant k. But in our case, we can determine k using the given information.Given f(8) = -3. Let's express 8 as 2^3. So, f(2^3) = f(2*2*2) = f(2) + f(2) + f(2) = 3f(2) = -3. So, 3f(2) = -3 implies f(2) = -1.Now, let's find f(sqrt(2)). Since sqrt(2) is 2^(1/2), we can write f(2^(1/2)) = (1/2)f(2) because f(2^(1/2)) = f(2^(1/2)) = (1/2)f(2). Wait, is that correct? Let me think.Actually, f(2^(1/2)) can be found by considering that 2^(1/2) * 2^(1/2) = 2. So, f(2^(1/2) * 2^(1/2)) = f(2) = -1. But f(2^(1/2) * 2^(1/2)) = f(2^(1/2)) + f(2^(1/2)) = 2f(2^(1/2)). So, 2f(2^(1/2)) = -1, which implies f(2^(1/2)) = -1/2.Similarly, let's find f(1/sqrt(2)). Since 1/sqrt(2) is 2^(-1/2). Let me express f(1/sqrt(2)) in terms of f(2). Let's note that 1/sqrt(2) * sqrt(2) = 1. So, f(1/sqrt(2) * sqrt(2)) = f(1). But f(1) can be found by considering f(1*1) = f(1) + f(1), which implies 2f(1) = f(1), so f(1) = 0.So, f(1/sqrt(2) * sqrt(2)) = f(1) = 0. Therefore, f(1/sqrt(2)) + f(sqrt(2)) = 0. We already found that f(sqrt(2)) = -1/2, so f(1/sqrt(2)) + (-1/2) = 0. Therefore, f(1/sqrt(2)) = 1/2.So, we have f(1/sqrt(2)) = 1/2, which means a = 1/sqrt(2). Rationalizing the denominator, that's sqrt(2)/2. So, same result as before.Therefore, regardless of whether I approach it by recognizing the logarithmic nature of the function or by using the functional equation directly, I arrive at the same conclusion that a = sqrt(2)/2.Just to recap, the steps were:1. Recognize that f is a logarithmic function because it satisfies f(xy) = f(x) + f(y).2. Use the given value f(8) = -3 to determine the base of the logarithm. This led to finding that the base is 1/2.3. Express f(a) = 1/2 in terms of logarithms and solve for a, which turned out to be sqrt(2)/2.4. Verified the result using the properties of the function without explicitly using logarithms, which also led to the same answer.So, I'm confident that the correct value of a is sqrt(2)/2.**Final Answer**The value of ( a ) is boxed{dfrac{sqrt{2}}{2}}.

answer：Okay, so I have this clock problem to solve: "When, between three and four o'clock, is the minute hand at the same distance from VIII as the hour hand from XII?" Hmm, let's break this down step by step.First, I need to visualize the clock. At three o'clock, the hour hand is exactly on the 3, and the minute hand is on the 12. As time progresses towards four o'clock, the hour hand moves towards the 4, and the minute hand moves around the clock from the 12.The problem is asking for the time between three and four o'clock when the minute hand is the same distance from VIII (which is 8 on the clock) as the hour hand is from XII (which is 12). So, I need to find a time where the distance of the minute hand from 8 is equal to the distance of the hour hand from 12.Let me think about how to measure these distances. On a clock, each number represents a certain degree. Since a full circle is 360 degrees, each hour mark is 30 degrees apart (360 degrees / 12 hours = 30 degrees per hour). So, the distance between two numbers can be calculated by the number of hours between them multiplied by 30 degrees.First, let's figure out the positions of the hour and minute hands in terms of degrees.The hour hand at three o'clock is at 90 degrees (3 * 30 degrees). As time passes, the hour hand moves towards the 4. The minute hand starts at 0 degrees (12 o'clock) and moves as time passes.Let's denote the number of minutes past three o'clock as 'm'. The minute hand moves 360 degrees in 60 minutes, so it moves 6 degrees per minute. Therefore, the position of the minute hand in degrees is 6m.The hour hand moves 30 degrees per hour, which is 0.5 degrees per minute (30 degrees / 60 minutes). So, the position of the hour hand in degrees is 90 + 0.5m.Now, the distance of the minute hand from VIII (which is 8). The 8 is at 240 degrees (8 * 30 degrees). So, the distance between the minute hand and 8 is the absolute difference between 6m and 240 degrees.Similarly, the distance of the hour hand from XII (which is 12) is the absolute difference between the hour hand's position and 0 degrees, which is |90 + 0.5m - 0| = |90 + 0.5m|.According to the problem, these two distances should be equal. So, we can set up the equation:|6m - 240| = |90 + 0.5m|This absolute value equation can have two cases:1. 6m - 240 = 90 + 0.5m2. 6m - 240 = -(90 + 0.5m)Let's solve both equations.Case 1:6m - 240 = 90 + 0.5mSubtract 0.5m from both sides:5.5m - 240 = 90Add 240 to both sides:5.5m = 330Divide both sides by 5.5:m = 330 / 5.5m = 60But wait, m = 60 minutes would mean it's exactly four o'clock. However, the problem specifies between three and four o'clock, so m should be less than 60. Therefore, this solution is at the boundary and might not be within the interval we're considering.Case 2:6m - 240 = -90 - 0.5mAdd 0.5m to both sides:6.5m - 240 = -90Add 240 to both sides:6.5m = 150Divide both sides by 6.5:m = 150 / 6.5m = 23.076923...So, m ≈ 23.0769 minutes.To express this as a fraction, 0.076923 is approximately 1/13. So, m ≈ 23 1/13 minutes.Therefore, the time is approximately 3 hours and 23 1/13 minutes.Let me double-check this result to ensure it's correct.At m = 23 1/13 minutes past three:- The minute hand is at 6 * 23 1/13 ≈ 140 degrees.- The hour hand is at 90 + 0.5 * 23 1/13 ≈ 90 + 11.538 ≈ 101.538 degrees.Distance of minute hand from VIII (240 degrees):|140 - 240| = 100 degrees.Distance of hour hand from XII (0 degrees):|101.538 - 0| = 101.538 degrees.Hmm, these are not exactly equal. There seems to be a slight discrepancy. Maybe my approximation is off.Let me calculate more precisely.First, m = 150 / 6.5 = 23.076923 minutes.So, the minute hand is at 6 * 23.076923 = 138.4615 degrees.The hour hand is at 90 + 0.5 * 23.076923 = 90 + 11.5384615 = 101.5384615 degrees.Distance of minute hand from VIII (240 degrees):|138.4615 - 240| = 101.5385 degrees.Distance of hour hand from XII (0 degrees):|101.5384615 - 0| = 101.5384615 degrees.Ah, so they are indeed equal when calculated precisely. My earlier approximation was just a rounding error. So, m = 23 1/13 minutes is the correct solution.Therefore, the time is 3 hours and 23 1/13 minutes, or 3:23 1/13.

answer：Let ( x ) be the number of days Mary performs adequately and ( y ) be the number of days she performs outstandingly.From the information given, we have the following equations:1. The total number of days: [ x + y = 15 ]2. The total amount earned: [ 4x + 7y = 85 ]To solve for ( y ), we can use the method of elimination. First, multiply the first equation by 4: [ 4x + 4y = 60 ]Now, subtract this from the second equation: [ (4x + 7y) - (4x + 4y) = 85 - 60 ] [ 3y = 25 ]Solving for ( y ): [ y = frac{25}{3} approx 8.33 ]Since the number of days must be an integer, we check the closest integer values. Testing ( y = 8 ) and ( x = 7 ): [ 4(7) + 7(8) = 28 + 56 = 84 ] This is close but not exact. Testing ( y = 9 ) and ( x = 6 ): [ 4(6) + 7(9) = 24 + 63 = 87 ] This is also not exact. However, considering possible rounding or approximation, the closest integer solution is ( y = 8 ).Thus, the number of days Mary performed outstandingly is:[boxed{C) 8}]