💡Okay, so I have this problem where a cross-section of a cube is a regular hexagon, and the area of that hexagon is Q. I need to find the total surface area of the cube. Hmm, let me try to visualize this.First, I know that a cube has all sides equal, and all angles are right angles. A cross-section of a cube can be various shapes depending on how you slice it. In this case, it's a regular hexagon, which means all sides and angles of the hexagon are equal. That must mean the cross-section is taken in such a way that it intersects six edges of the cube, right?Now, the area of this regular hexagon is Q. I remember that the area of a regular hexagon can be calculated using the formula:[ text{Area} = frac{3sqrt{3}}{2} s^2 ]where ( s ) is the side length of the hexagon. So, if I set this equal to Q, I can solve for ( s ):[ Q = frac{3sqrt{3}}{2} s^2 ][ s^2 = frac{2Q}{3sqrt{3}} ][ s = sqrt{frac{2Q}{3sqrt{3}}} ]But wait, I don't think I need the side length of the hexagon directly. Instead, I need to relate this to the cube's edge length. Let me think about how the hexagon is formed inside the cube.When you take a cross-section of a cube that results in a regular hexagon, the hexagon is formed by cutting through the midpoints of the cube's edges. So, each side of the hexagon is actually equal to the edge length of the cube divided by (sqrt{2}). Hmm, is that right?Let me draw a cube and imagine slicing it diagonally. If I slice through the midpoints of six edges, the resulting cross-section should be a regular hexagon. Each edge of the hexagon would be the distance between two midpoints of the cube's edges. Since the cube's edges are all equal, let's denote the cube's edge length as ( a ).The distance between two midpoints on adjacent edges of the cube can be found using the Pythagorean theorem. If I consider two adjacent edges of the cube meeting at a vertex, their midpoints are each ( frac{a}{2} ) away from the vertex. The distance between these midpoints would be the hypotenuse of a right triangle with legs of ( frac{a}{2} ) each.So, the distance ( d ) between the midpoints is:[ d = sqrt{left(frac{a}{2}right)^2 + left(frac{a}{2}right)^2} = sqrt{frac{a^2}{4} + frac{a^2}{4}} = sqrt{frac{a^2}{2}} = frac{a}{sqrt{2}} ]Therefore, each side of the regular hexagon is ( frac{a}{sqrt{2}} ). So, the side length ( s ) of the hexagon is ( frac{a}{sqrt{2}} ).Now, going back to the area of the hexagon. We have:[ Q = frac{3sqrt{3}}{2} s^2 ][ Q = frac{3sqrt{3}}{2} left(frac{a}{sqrt{2}}right)^2 ][ Q = frac{3sqrt{3}}{2} cdot frac{a^2}{2} ][ Q = frac{3sqrt{3}}{4} a^2 ]So, solving for ( a^2 ):[ a^2 = frac{4Q}{3sqrt{3}} ]Now, the total surface area of the cube is ( 6a^2 ), since a cube has six faces, each with area ( a^2 ).Substituting ( a^2 ):[ text{Total Surface Area} = 6 cdot frac{4Q}{3sqrt{3}} ][ = frac{24Q}{3sqrt{3}} ][ = frac{8Q}{sqrt{3}} ]But, it's usually rationalized to have no square roots in the denominator. So, multiplying numerator and denominator by ( sqrt{3} ):[ = frac{8Qsqrt{3}}{3} ]So, the total surface area of the cube is ( frac{8Qsqrt{3}}{3} ).Wait, let me double-check my steps to make sure I didn't make a mistake.1. I found the side length of the hexagon in terms of the cube's edge length: ( s = frac{a}{sqrt{2}} ). That seems correct because the distance between midpoints is ( frac{a}{sqrt{2}} ).2. Plugged into the area formula for the hexagon: ( Q = frac{3sqrt{3}}{2} s^2 ). Yes, that's the standard formula.3. Substituted ( s = frac{a}{sqrt{2}} ) into the area formula:[ Q = frac{3sqrt{3}}{2} cdot frac{a^2}{2} = frac{3sqrt{3}}{4} a^2 ]That looks right.4. Solved for ( a^2 ):[ a^2 = frac{4Q}{3sqrt{3}} ]Yes, that follows.5. Calculated total surface area:[ 6a^2 = 6 cdot frac{4Q}{3sqrt{3}} = frac{24Q}{3sqrt{3}} = frac{8Q}{sqrt{3}} ]Then rationalized:[ frac{8Qsqrt{3}}{3} ]All steps seem correct. So, I think the final answer is ( frac{8Qsqrt{3}}{3} ).

💡Alright, so I've got this problem here about translating an expression from a hypothetical programming language into ordinary algebraic notation. The expression is a bit complicated: it's a × b ÷ c - d + e × f. The catch is that in this programming language, operations are performed from right to left unless there are parentheses telling me otherwise. Hmm, that's interesting because in regular math, we usually follow the order of operations, which is parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). But here, it's different—everything goes right to left. Okay, let's try to break this down step by step.First, let me write down the expression again to make sure I have it right: a × b ÷ c - d + e × f. So, without any parentheses, I need to evaluate this from right to left. That means I start with the rightmost operation and work my way back to the left. Let's list out the operations in the expression:1. × between b and c2. ÷ between a and the result of b × c3. - between c and d4. + between d and e × f5. × between e and fWait, hold on, that doesn't seem right. Maybe I should list the operations in the order they appear from right to left. Let's see:Starting from the right, the last operation is e × f. Then, moving left, the next operation is + between d and e × f. Then, moving further left, it's - between c and (d + e × f). Then, moving left again, it's ÷ between a × b and c - (d + e × f). Finally, the leftmost operation is × between a and b. Okay, that makes more sense.So, following the right-to-left evaluation, I should group the operations starting from the right. Let's try to parenthesize this expression accordingly:Starting with e × f, then d + (e × f), then c - (d + e × f), then a × b ÷ (c - (d + e × f)), and finally a × (b ÷ (c - (d + e × f))). Wait, is that correct? Let me double-check.If I start from the right, the first operation is e × f. Then, moving left, I have d + (e × f). Next, moving left, I have c - (d + e × f). Then, moving left again, I have b ÷ (c - (d + e × f)). Finally, I have a × (b ÷ (c - (d + e × f))). Okay, that seems consistent.So, in algebraic notation, this would be written as a multiplied by (b divided by (c minus (d plus e times f))). In LaTeX, that would be: a times frac{b}{c - (d + e times f)}. Hmm, looking at the options provided, let's see which one matches this.Option A is a frac{b}{c - d + e times f}. That's similar but doesn't have the parentheses around d + e × f. Option B is a frac{b times e times f}{c - d}. That seems different because it's multiplying b, e, and f together in the numerator. Option C is a frac{b}{c - e times f - d}. That looks similar to what I have, but the order of d and e × f is different. In my expression, it's c minus (d plus e × f), which is the same as c - d - e × f. So, actually, Option C is c - e × f - d, which is the same as c - d - e × f. So, that matches. Option D is frac{a times b - c + d}{e times f}, which is entirely different because the denominator is e × f, and the numerator is a × b - c + d.So, based on this, Option C seems to be the correct one. But let me make sure I didn't make any mistakes in my grouping. Starting from the right, e × f is first, then d + (e × f), then c - (d + e × f), then b ÷ (c - (d + e × f)), and finally a × (b ÷ (c - (d + e × f))). Yes, that seems right. So, in algebraic terms, it's a multiplied by b divided by (c minus d minus e times f), which is exactly what Option C shows: a frac{b}{c - e times f - d}.I think I've got it. Option C is the correct translation of the given expression into ordinary algebraic notation under the right-to-left evaluation rule.

💡Okay, so I have this piecewise function f(x) defined as (x-1)² when x is less than 2, and 2/x when x is greater than or equal to 2. I need to figure out where this function is increasing. Hmm, okay. Let me break this down step by step.First, I remember that to determine where a function is increasing or decreasing, I should look at its derivative. If the derivative is positive, the function is increasing; if it's negative, the function is decreasing. So, I'll need to find the derivative of f(x) for both pieces of the function.Starting with the first piece: when x < 2, f(x) = (x - 1)². Let me find the derivative of this. Using the power rule, the derivative of (x - 1)² is 2(x - 1). So, f'(x) = 2(x - 1) for x < 2. Now, I need to figure out where this derivative is positive. So, 2(x - 1) > 0. Dividing both sides by 2 (which is positive, so the inequality sign doesn't change), I get x - 1 > 0, which simplifies to x > 1. So, for x < 2, the function is increasing when x > 1. Therefore, the interval where the first piece is increasing is (1, 2).Now, moving on to the second piece: when x ≥ 2, f(x) = 2/x. Let me find the derivative here. The derivative of 2/x is -2/x². So, f'(x) = -2/x² for x ≥ 2. Now, I need to check where this derivative is positive. But looking at -2/x², since x² is always positive (because squaring any real number gives a positive result), the entire derivative is negative. So, for x ≥ 2, the derivative is always negative, which means the function is decreasing on this interval.Putting it all together: the function is increasing on (1, 2) and decreasing on [2, ∞). But wait, I should also check what happens at the point where the function changes, which is at x = 2. Is the function continuous there? Let me verify.For x approaching 2 from the left, f(x) approaches (2 - 1)² = 1. For x approaching 2 from the right, f(x) approaches 2/2 = 1. So, the function is continuous at x = 2. However, since the function is decreasing for x ≥ 2, the point x = 2 itself is not part of the increasing interval. Therefore, the interval where the function is increasing is [1, 2). Wait, why [1, 2) and not (1, 2)? Let me think.At x = 1, the derivative is 2(1 - 1) = 0. So, at x = 1, the slope is zero, which means it's a critical point. But since the function is increasing for x > 1 up to x = 2, and at x = 1, it's just the start of the increasing interval. So, including x = 1 is okay because the function isn't decreasing there; it's just a stationary point. Therefore, the interval is [1, 2).Let me just recap to make sure I didn't miss anything. For x < 2, the function is a parabola opening upwards with vertex at x = 1. So, it decreases until x = 1 and then increases from x = 1 onwards. For x ≥ 2, it's a hyperbola that decreases as x increases. So, putting it all together, the function increases from x = 1 to x = 2 and then decreases beyond that. Therefore, the interval where f(x) is increasing is [1, 2).I think that's it. I don't see any mistakes in my reasoning. The key was to check each piece separately, find where each is increasing or decreasing, and then combine the results, making sure to consider the point where the function changes definition.

💡First, I need to determine the speed of the stream. The man swims downstream 48 km in 3 hours, so his downstream speed is 16 km/h. Since his speed in still water is 11 km/h, the stream's speed is 5 km/h.Next, I calculate the man's upstream speed by subtracting the stream's speed from his still water speed, which gives 6 km/h. Finally, I multiply this upstream speed by the 3 hours he spent swimming upstream to find the distance he swam upstream, which is 18 km.

💡First, I recognize that the perimeter of a semicircle consists of the curved part and the diameter. The formula for the perimeter is:Perimeter = πr + 2rGiven the perimeter is approximately 35.99114857512855, I set up the equation:πr + 2r = 35.99114857512855Next, I factor out the radius r:r(π + 2) = 35.99114857512855To solve for r, I divide both sides by (π + 2):r = 35.99114857512855 / (π + 2)Using the approximate value of π as 3.14159, I calculate:r = 35.99114857512855 / (3.14159 + 2) = 35.99114857512855 / 5.14159 ≈ 7Therefore, the radius of the semicircle is approximately 7 units.

💡To determine the probability that at least one of Person A or Person B solves the problem correctly, we can use the principle of complementary probability. First, calculate the probability that neither Person A nor Person B solves the problem correctly. This is the product of their individual probabilities of failing to solve the problem:[(1 - p_1) times (1 - p_2)]Next, subtract this probability from 1 to find the probability that at least one of them solves the problem correctly:[1 - (1 - p_1)(1 - p_2)]