💡Okay, so I have this problem about a tetrahedron P-ABC where all four vertices are on the surface of a sphere O. The triangle ABC is an equilateral triangle with each side measuring √3. The surface area of the sphere is given as 36π. I need to find the maximum distance from point P to the plane formed by triangle ABC.First, let me recall some basic formulas. The surface area of a sphere is 4πr², where r is the radius. Given that the surface area is 36π, I can set up the equation:4πr² = 36πDividing both sides by π gives:4r² = 36Then, dividing both sides by 4:r² = 9Taking the square root of both sides:r = 3So, the radius of the sphere is 3 units.Now, since points A, B, C, and P are all on the surface of this sphere, they are all 3 units away from the center of the sphere, which I'll call O.Next, I need to consider triangle ABC. It's an equilateral triangle with side length √3. I remember that in an equilateral triangle, the circumradius (the radius of the circumscribed circle) can be found using the formula:R = (a) / (√3)where a is the length of a side. Plugging in the given side length:R = √3 / √3 = 1So, the circumradius of triangle ABC is 1 unit. This means that the center of the circumscribed circle around triangle ABC is 1 unit away from each of the vertices A, B, and C.Now, I need to relate this to the sphere. The triangle ABC lies on a plane, and the center of the sphere O is somewhere in space relative to this plane. The distance from O to the plane ABC can be found using the Pythagorean theorem in three dimensions.Let me denote the distance from O to the plane ABC as d. Since the circumradius of triangle ABC is 1 and the radius of the sphere is 3, these form a right triangle with d as one leg, the circumradius as the other leg, and the sphere's radius as the hypotenuse.So, according to the Pythagorean theorem:d² + R² = r²Plugging in the known values:d² + (1)² = (3)²Which simplifies to:d² + 1 = 9Subtracting 1 from both sides:d² = 8Taking the square root:d = √8 = 2√2So, the distance from the center of the sphere O to the plane ABC is 2√2 units.Now, the problem asks for the maximum distance from point P to the plane ABC. Since P is also on the surface of the sphere, the maximum distance would occur when P is on the opposite side of the sphere relative to the plane ABC. In other words, the distance from P to the plane ABC would be the distance from O to the plane plus the distance from O to P.But wait, O is the center of the sphere, so the distance from O to P is just the radius of the sphere, which is 3. Therefore, the maximum distance from P to the plane ABC is:d_max = d + r = 2√2 + 3So, the maximum distance is 3 + 2√2 units.Let me just double-check my steps to make sure I didn't make a mistake. I found the radius of the sphere correctly using the surface area formula. Then, I calculated the circumradius of the equilateral triangle, which seems right. Then, using the Pythagorean theorem, I found the distance from the center of the sphere to the plane. That also seems correct. Finally, adding the radius of the sphere to that distance gives the maximum distance from P to the plane. Yeah, that all makes sense.I think I'm confident with this answer. It's always good to visualize it too. Imagine the sphere with radius 3, and the triangle ABC lying on a plane that's 2√2 units away from the center. Point P can be on the opposite side of the sphere, so it's 3 units away from the center in the opposite direction, making the total distance from P to the plane ABC equal to 3 + 2√2. That seems right.**Final Answer**The maximum distance from P to plane ABC is boxed{3 + 2sqrt{2}}.

💡Okay, so I have this problem about parallel lines and a coin. Let me try to visualize it first. There are parallel lines spaced 2a apart on a plane. So, imagine like a bunch of straight, parallel lines, each 2a units apart from each other. Now, we're tossing a coin with radius r, and r is less than a. So, the coin is smaller than the distance between the lines. We need to find the probability that the coin doesn't touch any of the lines when it's tossed randomly onto the plane.Hmm, okay. So, probability problems often involve figuring out the ratio of favorable outcomes to total possible outcomes. In this case, the favorable outcome is the coin landing without touching any lines, and the total possible outcomes are all possible positions the coin could land on the plane.Since the lines are parallel and equally spaced, I think the problem has some symmetry. Maybe I can model this by looking at just one strip between two lines and figure out the probability within that strip, then it should be the same for all strips.So, the distance between two consecutive lines is 2a. If I consider the area where the center of the coin can land so that the coin doesn't touch either of the two lines, that area would determine the favorable outcomes.Let me think about the position of the coin's center. If the coin has radius r, then the center must be at least r units away from any line to ensure the coin doesn't touch it. So, in the strip between two lines, which is 2a wide, the center of the coin can only be in a smaller strip that's 2a - 2r wide. Because we subtract r from both sides of the strip.Wait, is that right? If the total width between two lines is 2a, and the center needs to be at least r away from each line, then the available space for the center is 2a - 2r. So, the length of the favorable area is 2a - 2r, and the total possible area is 2a.But wait, actually, since we're dealing with probability, it's the ratio of the favorable length to the total length, right? Because the lines are equally spaced and parallel, the probability should just depend on the lengths, not the areas, since the lines are infinitely long.So, the probability that the coin doesn't touch any line is the ratio of the favorable length to the total length. The favorable length is 2a - 2r, and the total length is 2a. So, the probability is (2a - 2r)/(2a). Simplifying that, we get (a - r)/a.Wait, but looking back at the answer choices, option C is (a - r)/a, which matches what I just got. So, is that the answer? Hmm.But let me double-check. Maybe I made a mistake in considering the lengths versus areas. If the lines are parallel and equally spaced, and the coin is tossed randomly, the position of the center is uniformly distributed over the plane. So, actually, the probability should be the area where the center can land without touching any lines divided by the total area.But since the lines are infinitely long and the plane is infinite, we can consider a unit length along the direction perpendicular to the lines. So, in that case, the probability would indeed be the ratio of lengths, because the area would just scale with the length, and the direction along the lines doesn't affect the probability.So, if we consider a unit length along the direction of the lines, the width perpendicular to the lines is 2a. The center of the coin must be at least r units away from each line, so the favorable width is 2a - 2r. Therefore, the probability is (2a - 2r)/(2a) = (a - r)/a, which is option C.Wait, but another thought: maybe I should consider the distance from the center to the nearest line. Since the lines are spaced 2a apart, the distance from the center to the nearest line can range from 0 to a. For the coin not to touch any line, the distance must be greater than r. So, the favorable distance is from r to a, which is a - r. The total possible distance is from 0 to a, which is a. So, the probability is (a - r)/a.Yes, that makes sense too. So, both ways of thinking lead to the same answer, (a - r)/a, which is option C.I think I'm confident now that the answer is C.

💡First, I need to determine the side length of the original cube. Since the volume of a cube is given by ( V = s^3 ), where ( s ) is the side length, I can find ( s ) by taking the cube root of the volume.Given that the volume ( V ) is 343 cubic feet:[s^3 = 343][s = sqrt[3]{343} = 7 text{ feet}]Next, I'll calculate the new dimensions of the similar cube:- The length is tripled: ( 3 times 7 = 21 ) feet- The width is 1.5 times the original: ( 1.5 times 7 = 10.5 ) feet- The height is 2.5 times the original: ( 2.5 times 7 = 17.5 ) feetFinally, I'll find the volume of the new cuboid by multiplying its length, width, and height:[V_{text{new}} = 21 times 10.5 times 17.5 = 38587.5 text{ cubic feet}]

💡Okay, so I have this problem where Kai has an ice sheet shaped like an "angle," which I think is like an "L" shape, and he needs to cut it into four equal parts. The Snow Queen wants this done, so Kai better figure it out!First, I need to visualize what an "angle" shape looks like. It's probably something like two rectangles connected at a right angle, forming an "L." Maybe it's a bigger square with a smaller square missing from one corner? Or perhaps it's three squares connected in an "L" shape. I'm not entirely sure, but I'll assume it's a standard "L" shape made up of three smaller squares.Next, I need to think about how to divide this shape into four equal parts. Equal parts mean each part should have the same area and, ideally, the same shape if possible. Since it's an "L" shape, symmetry might play a role here.Let me try to break it down step by step.1. **Understanding the Shape:** If the ice sheet is an "L" shape, it has two arms of equal length meeting at a right angle. For simplicity, let's assume each arm is made up of two smaller squares. So, the total area would be 3 units if each small square is 1 unit.2. **Calculating the Area:** Total area = 3 units (assuming each small square is 1 unit). If we need to divide this into four equal parts, each part should be 3/4 units. Hmm, that seems a bit tricky because 3 isn't divisible by 4 evenly. Maybe I'm misunderstanding the shape.3. **Re-evaluating the Shape:** Perhaps the "angle" shape is made up of four smaller squares arranged in an "L." That would make the total area 4 units, and each part would be 1 unit. That makes more sense. So, maybe it's a 2x2 square missing one smaller square, making it an "L" shape with three squares, but that still totals 3 units. I'm a bit confused here.4. **Considering Symmetry:** Maybe the key is to use symmetry to make the cuts. If I can find lines of symmetry in the "L" shape, I can make cuts along those lines to divide it into equal parts.5. **Trying to Make Cuts:** Let's say I make a vertical cut down the middle of the longer arm of the "L." That would divide it into two parts. Then, I can make a horizontal cut on the shorter arm to divide it further. But I need four equal parts, so maybe I need to make more cuts.6. **Using Coordinates:** Maybe assigning coordinates to the shape will help. Let's place the "L" shape on a grid with the corner at (0,0). The longer arm could extend to (2,0) and the shorter arm to (0,1). Then, I can try to make cuts along certain lines to divide it into four regions.7. **Calculating Areas:** If each part needs to be equal, I need to ensure that each region has the same area. Maybe I can use integration or geometric formulas to calculate the areas after making the cuts.8. **Looking for Patterns:** I recall that sometimes, dividing shapes into equal parts involves making parallel or perpendicular cuts. Maybe I can make two vertical and two horizontal cuts to create four smaller rectangles within the "L" shape.9. **Considering Rotations:** Another idea is to rotate the "L" shape and see if it becomes easier to divide. Sometimes, changing the perspective helps in finding symmetrical cuts.10. **Testing the Cuts:** Let me try to imagine making three cuts: one vertical and two horizontal, or vice versa. I need to ensure that each resulting piece has the same area and, ideally, the same shape.11. **Verifying the Solution:** After making the cuts, I should check if each part indeed has the same area. If not, I'll need to adjust the cuts accordingly.12. **Finalizing the Answer:** Once I'm confident that the cuts divide the "L" shape into four equal parts, I can describe the process and the final cuts in detail.I think I need to draw this out or use a diagram to visualize the cuts better. Without a figure, it's a bit challenging, but I'll do my best to describe the cuts based on my understanding.

💡Alright, so I have this geometry problem here with four propositions about lines and planes. Let me try to figure out which ones are correct. I'm a bit rusty on some of these concepts, but I'll take it step by step.First, let's list out the propositions again to keep them fresh in my mind:① If α is parallel to β and α is parallel to γ, then β is parallel to γ.② If α is perpendicular to β and m is parallel to α, then m is perpendicular to β.③ If m is parallel to n and n is a subset of α, then m is parallel to α.④ If m is perpendicular to α and m is parallel to β, then α is perpendicular to β.And the options are:A: ①③ B: ①④ C: ②③ D: ②④Alright, let's tackle each proposition one by one.**Proposition ①:** If α || β and α || γ, then β || γ.Hmm, okay. So if two planes α and β are parallel, and α is also parallel to another plane γ, does that mean β and γ are parallel? Intuitively, in three-dimensional space, if two planes are both parallel to the same plane, they should be parallel to each other. I think this is a standard property of parallel planes. So I'm leaning towards ① being correct.**Proposition ②:** If α ⊥ β and m || α, then m ⊥ β.Alright, so if plane α is perpendicular to plane β, and line m is parallel to α, does that mean m is perpendicular to β? Hmm, not necessarily. If m is parallel to α, it could be that m lies within a direction that's perpendicular to β, but it could also be parallel to β or even lie within β. Wait, no, if α is perpendicular to β, then any line parallel to α would have to be perpendicular to β, right? Because the direction of α is perpendicular to β. Hmm, maybe I was overcomplicating it. I think ② is correct.Wait, no, actually, if m is parallel to α, it doesn't necessarily mean m is perpendicular to β. For example, imagine α is the xy-plane and β is the xz-plane, which are perpendicular. If m is a line parallel to the xy-plane, it could be pointing in the x or y direction. If it's pointing in the x direction, it's not perpendicular to the xz-plane; it's actually lying within it. So m could be in β, or parallel to β, or even at some angle. So ② is not necessarily true. So ② is incorrect.**Proposition ③:** If m || n and n ⊂ α, then m || α.Okay, so if line m is parallel to line n, and line n is contained within plane α, does that mean m is parallel to α? Hmm, not necessarily. If m is parallel to n, m could be parallel to α, but it could also lie within α. So m might not just be parallel; it could be in the plane. So the proposition says "m || α," but it could also be that m is in α. So the conclusion isn't necessarily just that m is parallel; it might be either. Therefore, ③ is incorrect.**Proposition ④:** If m ⊥ α and m || β, then α ⊥ β.Alright, so if line m is perpendicular to plane α and m is parallel to plane β, does that mean α is perpendicular to β? Let me visualize this. If m is perpendicular to α, then m is orthogonal to every line in α. If m is parallel to β, then m lies within a direction that's parallel to β. So, if m is perpendicular to α and parallel to β, that would mean that β contains a direction that's perpendicular to α, which by definition means α is perpendicular to β. Yeah, that makes sense. So ④ is correct.So, summarizing:① Correct ② Incorrect ③ Incorrect ④ CorrectTherefore, the correct propositions are ① and ④, which corresponds to option B.**Final Answer**boxed{B}

💡Alright, so I've got this trigonometry problem here. It says that sin(2α) equals 2/3, and I need to find the value of sin squared of (α minus π/4). The options are A: 2/3, B: 1/2, C: 1/3, and D: 1/6. Hmm, okay, let's see how to approach this.First off, I remember that sin(2α) is a double-angle identity. The formula for sin(2α) is 2 sinα cosα. So, if sin(2α) is 2/3, that means 2 sinα cosα equals 2/3. Maybe I can use this to find either sinα or cosα, but I'm not sure yet. Let me think.The problem asks for sin squared of (α minus π/4). That expression looks like it can be expanded using another trigonometric identity. I recall that sin(A - B) equals sinA cosB minus cosA sinB. So, if I apply that identity here, I can write sin(α - π/4) as sinα cos(π/4) minus cosα sin(π/4).Okay, cos(π/4) and sin(π/4) are both √2/2. So, substituting those values in, sin(α - π/4) becomes sinα*(√2/2) minus cosα*(√2/2). I can factor out the √2/2, so it's √2/2 times (sinα - cosα). That simplifies things a bit.Now, the problem is asking for sin squared of that expression. So, I need to square the entire thing. That would be (√2/2)^2 times (sinα - cosα)^2. Let's compute that. (√2/2)^2 is (2/4), which simplifies to 1/2. So, now I have 1/2 times (sinα - cosα)^2.Expanding (sinα - cosα)^2, I get sin squared α plus cos squared α minus 2 sinα cosα. I remember that sin squared α plus cos squared α equals 1, which is a fundamental identity. So, substituting that in, the expression becomes 1 minus 2 sinα cosα.Wait a minute, 2 sinα cosα is exactly what we have in the double-angle identity for sin(2α). And we know that sin(2α) is 2/3. So, 2 sinα cosα is 2/3. Therefore, the expression simplifies to 1 minus 2/3, which is 1/3.But hold on, I had that 1/2 factor earlier. So, multiplying 1/2 by 1/3 gives me 1/6. Therefore, sin squared of (α minus π/4) is 1/6.Let me double-check my steps to make sure I didn't make any mistakes. Starting from sin(2α) = 2/3, I used the double-angle identity correctly. Then, I applied the sine difference identity properly, substituted the known values of cos(π/4) and sin(π/4), and simplified correctly. Squaring the expression and expanding it also seems right. Finally, substituting back the known value of sin(2α) and simplifying gave me 1/6. Yeah, that seems solid.I think I'm confident with this answer. It's option D: 1/6.