💡Okay, so I have this equation here: the square root of (5 minus 4z) equals 10. Hmm, I need to find all solutions for z and express it as a common fraction. Alright, let's start by understanding what this equation is saying.First, the square root of something equals 10. That means the expression inside the square root must be equal to 10 squared, right? Because if the square root of a number is 10, then the number itself must be 100. So, I can square both sides of the equation to get rid of the square root. Let me write that down.So, if I square both sides, I get:√(5 - 4z) = 10Squaring both sides:(√(5 - 4z))² = 10²Which simplifies to:5 - 4z = 100Okay, now I have a linear equation. I need to solve for z. Let's see, I can start by isolating the term with z. So, I have 5 minus 4z equals 100. I can subtract 5 from both sides to get rid of the constant term on the left side.5 - 4z - 5 = 100 - 5Simplifying that:-4z = 95Now, I need to solve for z. I have -4 multiplied by z equals 95. To get z by itself, I should divide both sides by -4.(-4z)/(-4) = 95/(-4)Which simplifies to:z = -95/4Wait, let me double-check that. If I divide 95 by -4, that's the same as -95/4, right? Yeah, that makes sense because a positive divided by a negative is a negative.So, z equals -95 over 4. Hmm, is that the simplest form? Let me see, 95 and 4 don't have any common factors besides 1, so I can't reduce the fraction any further. So, that should be the solution.But just to make sure I didn't make any mistakes, let me plug this value back into the original equation and see if it works.Original equation:√(5 - 4z) = 10Plugging in z = -95/4:√(5 - 4*(-95/4)) = √(5 + 95) = √100 = 10Yes, that works out perfectly. The square root of 100 is indeed 10, so my solution checks out.I think that's it. I squared both sides, solved for z, and verified the solution by plugging it back into the original equation. Everything seems to make sense, and I didn't run into any contradictions or errors along the way. So, I'm confident that z equals -95/4 is the correct solution.**Final Answer**The solution is boxed{-dfrac{95}{4}}.

💡Okay, so I have this problem where two places, A and B, are 150 kilometers apart. A person drives a car from A to B at a speed of 60 km/h, stays at B for 1 hour, and then returns to A at a speed of 50 km/h. I need to express the distance x that the car is away from A as a function of time t (in hours). The options are given as A, B, C, D, and I need to figure out which one is correct.First, let me try to visualize the situation. The person starts at point A, drives to point B, which is 150 km away, at 60 km/h. Then, they stay at B for 1 hour before returning to A at 50 km/h. So, the journey is divided into three parts: going from A to B, staying at B, and then returning from B to A.I need to model the distance from A, x(t), as a function of time t. Since the journey has different phases, the function will likely be piecewise, meaning it has different expressions in different time intervals.Let me break it down step by step.1. **Driving from A to B:** - Distance: 150 km - Speed: 60 km/h - Time taken: Time = Distance / Speed = 150 / 60 = 2.5 hours. So, during the first 2.5 hours, the car is moving away from A towards B. The distance from A increases linearly with time. The expression for this part should be x(t) = 60t, where t is between 0 and 2.5 hours.2. **Staying at B:** - After reaching B at t = 2.5 hours, the person stays there for 1 hour. So, from t = 2.5 to t = 3.5 hours, the car is stationary at B. Therefore, the distance from A remains constant at 150 km during this interval.3. **Returning from B to A:** - After staying at B for 1 hour, the person starts returning to A at a speed of 50 km/h. - The distance to cover is again 150 km, but now moving towards A. - Time taken for the return trip: Time = Distance / Speed = 150 / 50 = 3 hours. So, the return journey starts at t = 3.5 hours and takes until t = 6.5 hours. During this time, the distance from A decreases linearly. The expression for this part needs to account for the starting point at B (150 km from A) and the time elapsed since the return started. Let me think about how to model this. At t = 3.5 hours, the distance from A is 150 km. Then, as time increases beyond 3.5 hours, the distance decreases by 50 km each hour. So, the distance at time t (where t > 3.5) can be expressed as: x(t) = 150 - 50*(t - 3.5) This is because the car has been traveling back towards A for (t - 3.5) hours, covering 50*(t - 3.5) km, so subtracting that from the total distance.Putting it all together, the function x(t) is:- x(t) = 60t, for 0 ≤ t ≤ 2.5- x(t) = 150, for 2.5 ≤ t ≤ 3.5- x(t) = 150 - 50*(t - 3.5), for 3.5 < t ≤ 6.5Looking at the options provided:A: x = 60t – This only covers the first part, not the entire journey.B: x = 60t + 50t – This seems to combine both speeds, which doesn't make sense because the person isn't moving towards and away from A at the same time.C: x= { 60t, (0 ≤ t ≤ 2.5); 150 - 50t, (t > 3.5) } – This misses the middle part where the person is stationary at B. Also, the expression for the return trip is incorrect because it doesn't account for the 1-hour stop at B.D: x= { 60t, (0 ≤ t ≤ 2.5); 150, (2.5 ≤ t ≤ 3.5); 150 - 50(t - 3.5), (3.5 < t ≤ 6.5) } – This correctly models all three parts of the journey: going to B, staying at B, and returning to A.Therefore, the correct answer is D.**Final Answer**boxed{D}

💡Alright, so I've got this problem about a cube and a plane cutting through it. The cube has vertices at (0,0,0), (0,0,6), (0,6,0), (0,6,6), (6,0,0), (6,0,6), (6,6,0), and (6,6,6). The plane intersects the cube at points P = (0,3,0), Q = (2,0,0), R = (2,6,6), and two other points. I need to find the distance between these two additional points.First, I think I should figure out the equation of the plane that passes through points P, Q, and R. To do that, I remember that the general equation of a plane is ax + by + cz = d. I can find the coefficients a, b, c by using the cross product of two vectors lying on the plane.So, let's define vectors PQ and PR. Vector PQ is Q - P, which is (2-0, 0-3, 0-0) = (2, -3, 0). Vector PR is R - P, which is (2-0, 6-3, 6-0) = (2, 3, 6).Now, I need to compute the cross product of PQ and PR to get the normal vector of the plane. The cross product formula is:If vector PQ = (a1, a2, a3) and vector PR = (b1, b2, b3), then the cross product PQ × PR = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).Plugging in the values:First component: (-3)(6) - (0)(3) = -18 - 0 = -18Second component: (0)(2) - (2)(6) = 0 - 12 = -12Third component: (2)(3) - (-3)(2) = 6 - (-6) = 12So, the cross product is (-18, -12, 12). Hmm, that seems a bit messy. Maybe I can simplify it by dividing by a common factor. Let's see, all components are divisible by -6. Dividing each component by -6 gives (3, 2, -2). Wait, actually, if I divide (-18, -12, 12) by -6, I get (3, 2, -2). But actually, the cross product is (-18, -12, 12), so dividing by -6 gives (3, 2, -2). But I think I made a mistake in the signs because cross product should be perpendicular to both vectors, but let me double-check.Wait, actually, cross product is anti-commutative, so PQ × PR = - (PR × PQ). Maybe I should have done PR × PQ instead. Let me recalculate:Vector PR = (2, 3, 6)Vector PQ = (2, -3, 0)So, PR × PQ:First component: (3)(0) - (6)(-3) = 0 + 18 = 18Second component: (6)(2) - (2)(0) = 12 - 0 = 12Third component: (2)(-3) - (3)(2) = -6 - 6 = -12So, PR × PQ = (18, 12, -12). This seems better. So, the normal vector is (18, 12, -12). I can simplify this by dividing by 6, which gives (3, 2, -2). So, the normal vector is (3, 2, -2). Therefore, the equation of the plane is 3x + 2y - 2z = d.Now, to find d, I can plug in one of the points, say P = (0,3,0):3(0) + 2(3) - 2(0) = d => 0 + 6 - 0 = d => d = 6.So, the equation of the plane is 3x + 2y - 2z = 6.Wait, but in the initial problem, the user had a different normal vector, (3, -2, 1). Did I make a mistake? Let me check.Wait, in my calculation, I got the normal vector as (3, 2, -2), but the user had (3, -2, 1). Hmm, maybe I messed up the cross product direction. Let me recast the vectors.Wait, actually, the cross product PQ × PR was (-18, -12, 12). If I factor out a -6, I get (3, 2, -2). Alternatively, if I factor out a 6, I get (-3, -2, 2). So, depending on the direction, the normal vector can be (3, 2, -2) or (-3, -2, 2). But the plane equation should be the same regardless of the direction of the normal vector, just the sign of d changes.Wait, let me plug in point P into both possibilities:If normal vector is (3, 2, -2), then 3x + 2y - 2z = d. Plugging in P (0,3,0): 0 + 6 - 0 = d => d=6.If normal vector is (-3, -2, 2), then -3x -2y + 2z = d. Plugging in P: 0 -6 + 0 = d => d=-6.But the plane equation is the same, just multiplied by -1. So, both are correct, but the user had (3, -2, 1). Wait, that's different. Let me see.Wait, maybe I made a mistake in the cross product. Let me recalculate PQ × PR.Vector PQ = (2, -3, 0)Vector PR = (2, 3, 6)Cross product PQ × PR:i component: (-3)(6) - (0)(3) = -18 - 0 = -18j component: (0)(2) - (2)(6) = 0 - 12 = -12k component: (2)(3) - (-3)(2) = 6 - (-6) = 12So, PQ × PR = (-18, -12, 12). So, the normal vector is (-18, -12, 12). If I factor out -6, I get (3, 2, -2). So, the normal vector is (3, 2, -2). Therefore, the plane equation is 3x + 2y - 2z = d. Plugging in P (0,3,0): 0 + 6 - 0 = d => d=6.So, the plane equation is 3x + 2y - 2z = 6.Wait, but the user had a normal vector of (3, -2, 1). That's different. Let me check if that's correct.Wait, perhaps I should have used a different pair of vectors. Maybe instead of PQ and PR, I should have used vectors from Q to R or something else. Let me try that.Vector QR is R - Q = (2-2, 6-0, 6-0) = (0,6,6). Vector QP is P - Q = (0-2, 3-0, 0-0) = (-2,3,0).So, QR × QP:i component: (6)(0) - (6)(3) = 0 - 18 = -18j component: (6)(-2) - (0)(6) = -12 - 0 = -12k component: (0)(3) - (6)(-2) = 0 + 12 = 12So, QR × QP = (-18, -12, 12), same as before. So, the normal vector is still (-18, -12, 12), which simplifies to (3, 2, -2). So, the plane equation is 3x + 2y - 2z = 6.Wait, but the user had a normal vector of (3, -2, 1). That's different. Let me see if I can reconcile this.Wait, maybe I made a mistake in the cross product. Let me recast the vectors.Wait, perhaps I should have used vectors from P to Q and from P to R, which I did initially, and got the normal vector as (-18, -12, 12). So, simplifying, it's (3, 2, -2). So, the plane equation is 3x + 2y - 2z = 6.But the user had a normal vector of (3, -2, 1). Let me see if that's a scalar multiple of (3, 2, -2). If I multiply (3, 2, -2) by -1, I get (-3, -2, 2). That's not the same as (3, -2, 1). So, perhaps the user made a mistake in their cross product.Alternatively, maybe I made a mistake. Let me check the cross product again.Vector PQ = (2, -3, 0)Vector PR = (2, 3, 6)Cross product:i component: (-3)(6) - (0)(3) = -18 - 0 = -18j component: (0)(2) - (2)(6) = 0 - 12 = -12k component: (2)(3) - (-3)(2) = 6 - (-6) = 12So, yes, PQ × PR = (-18, -12, 12). So, the normal vector is (-18, -12, 12). Simplifying by dividing by -6 gives (3, 2, -2). So, the plane equation is 3x + 2y - 2z = 6.Wait, but the user had 3x - 2y + z = 6. Let me see if that's equivalent.If I take 3x + 2y - 2z = 6 and multiply both sides by -1, I get -3x -2y + 2z = -6. That's not the same as 3x - 2y + z = 6. So, they are different planes. Therefore, the user must have made a mistake in their cross product.But regardless, I think my calculation is correct. So, the plane equation is 3x + 2y - 2z = 6.Now, I need to find where this plane intersects the cube edges. The cube has edges along the x, y, and z axes, each of length 6. The plane already intersects the cube at P, Q, and R, which are on the edges:- P = (0,3,0) is on the edge from (0,0,0) to (0,6,0).- Q = (2,0,0) is on the edge from (0,0,0) to (6,0,0).- R = (2,6,6) is on the edge from (0,6,6) to (6,6,6).So, the plane intersects the cube at these three points, and we need to find the other two intersection points.To find the other intersection points, I need to check the plane equation against the other edges of the cube. The cube has 12 edges, and we've already accounted for three of them. So, we need to check the remaining edges to see where the plane intersects them.Let me list all the edges of the cube:1. From (0,0,0) to (0,0,6)2. From (0,0,0) to (0,6,0)3. From (0,0,0) to (6,0,0)4. From (0,0,6) to (0,6,6)5. From (0,0,6) to (6,0,6)6. From (0,6,0) to (0,6,6)7. From (0,6,0) to (6,6,0)8. From (6,0,0) to (6,0,6)9. From (6,0,0) to (6,6,0)10. From (6,0,6) to (6,6,6)11. From (6,6,0) to (6,6,6)12. From (0,6,6) to (6,6,6)We already have intersections on edges 2, 3, and 12 (points P, Q, R). So, we need to check the other edges to see if the plane intersects them.Let's go through each edge and see if the plane intersects it.Edge 1: From (0,0,0) to (0,0,6). Parametrize this edge as (0,0,t) where t goes from 0 to 6.Plug into plane equation: 3(0) + 2(0) - 2(t) = 6 => -2t = 6 => t = -3. But t must be between 0 and 6, so no intersection here.Edge 4: From (0,0,6) to (0,6,6). Parametrize as (0, t, 6), t from 0 to 6.Plug into plane: 3(0) + 2(t) - 2(6) = 6 => 2t - 12 = 6 => 2t = 18 => t = 9. But t must be ≤6, so no intersection.Edge 5: From (0,0,6) to (6,0,6). Parametrize as (t,0,6), t from 0 to 6.Plug into plane: 3(t) + 2(0) - 2(6) = 6 => 3t - 12 = 6 => 3t = 18 => t=6. So, t=6 is the endpoint (6,0,6). So, the plane intersects this edge at (6,0,6). But wait, is (6,0,6) on the plane? Let's check: 3(6) + 2(0) - 2(6) = 18 - 12 = 6. Yes, it is. So, this is another intersection point, let's call it S = (6,0,6).Edge 6: From (0,6,0) to (0,6,6). Parametrize as (0,6,t), t from 0 to 6.Plug into plane: 3(0) + 2(6) - 2(t) = 6 => 12 - 2t = 6 => -2t = -6 => t=3. So, the intersection point is (0,6,3). Let's call this point T = (0,6,3).Edge 7: From (0,6,0) to (6,6,0). Parametrize as (t,6,0), t from 0 to 6.Plug into plane: 3(t) + 2(6) - 2(0) = 6 => 3t + 12 = 6 => 3t = -6 => t = -2. Not in [0,6], so no intersection.Edge 8: From (6,0,0) to (6,0,6). Parametrize as (6,0,t), t from 0 to 6.Plug into plane: 3(6) + 2(0) - 2(t) = 6 => 18 - 2t = 6 => -2t = -12 => t=6. So, intersection at (6,0,6), which is the same as point S.Edge 9: From (6,0,0) to (6,6,0). Parametrize as (6,t,0), t from 0 to 6.Plug into plane: 3(6) + 2(t) - 2(0) = 6 => 18 + 2t = 6 => 2t = -12 => t=-6. Not in [0,6], so no intersection.Edge 10: From (6,0,6) to (6,6,6). Parametrize as (6,t,6), t from 0 to 6.Plug into plane: 3(6) + 2(t) - 2(6) = 6 => 18 + 2t - 12 = 6 => 6 + 2t = 6 => 2t=0 => t=0. So, intersection at (6,0,6), which is point S.Edge 11: From (6,6,0) to (6,6,6). Parametrize as (6,6,t), t from 0 to 6.Plug into plane: 3(6) + 2(6) - 2(t) = 6 => 18 + 12 - 2t = 6 => 30 - 2t = 6 => -2t = -24 => t=12. Not in [0,6], so no intersection.Edge 12: From (0,6,6) to (6,6,6). Parametrize as (t,6,6), t from 0 to 6.Plug into plane: 3(t) + 2(6) - 2(6) = 6 => 3t + 12 - 12 = 6 => 3t = 6 => t=2. So, intersection at (2,6,6), which is point R.So, from this, the plane intersects the cube at points P, Q, R, S, and T. Wait, but the problem says it intersects at P, Q, R, and two other points. So, S and T are the two additional points.Wait, but in my calculation, I found S = (6,0,6) and T = (0,6,3). So, these are the two additional points.Now, I need to find the distance between S and T.Point S is (6,0,6) and point T is (0,6,3).The distance formula in 3D is sqrt[(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2].So, plugging in the coordinates:x1=6, y1=0, z1=6x2=0, y2=6, z2=3So,Distance = sqrt[(0 - 6)^2 + (6 - 0)^2 + (3 - 6)^2] = sqrt[(-6)^2 + 6^2 + (-3)^2] = sqrt[36 + 36 + 9] = sqrt[81] = 9.Wait, that's 9. But the user's initial answer was 6√3, which is approximately 10.39. Hmm, so I must have made a mistake.Wait, let me double-check the points.Point S is (6,0,6). Point T is (0,6,3). So, the differences are:x: 0 - 6 = -6y: 6 - 0 = 6z: 3 - 6 = -3So, squared differences:(-6)^2 = 366^2 = 36(-3)^2 = 9Total: 36 + 36 + 9 = 81Square root of 81 is 9. So, the distance is 9.But wait, in the initial problem, the user had points (0,0,6) and (6,6,0), which are different from my points. So, perhaps I made a mistake in identifying the intersection points.Wait, let me go back. The user's initial answer considered intersections at (0,0,6) and (6,6,0), but in my calculation, I found intersections at (6,0,6) and (0,6,3). So, which one is correct?Wait, let's check the plane equation again. I had 3x + 2y - 2z = 6.Let me plug in (0,0,6): 3(0) + 2(0) - 2(6) = -12 ≠ 6. So, (0,0,6) is not on the plane.Plug in (6,0,6): 3(6) + 2(0) - 2(6) = 18 - 12 = 6. Yes, that's correct.Plug in (0,6,3): 3(0) + 2(6) - 2(3) = 0 + 12 - 6 = 6. Yes, that's correct.So, the two additional points are (6,0,6) and (0,6,3), and the distance between them is 9.But the user's initial answer was 6√3, which is approximately 10.39, which is larger than 9. So, perhaps the user made a mistake in their cross product, leading to an incorrect plane equation, and thus incorrect intersection points.Alternatively, maybe I made a mistake in my cross product. Let me check again.Wait, in my initial calculation, I had vectors PQ = (2, -3, 0) and PR = (2, 3, 6). Their cross product was (-18, -12, 12), which simplifies to (3, 2, -2). So, the plane equation is 3x + 2y - 2z = 6.But the user had a normal vector of (3, -2, 1), leading to the plane equation 3x - 2y + z = 6. Let me see if that's correct.If I use the user's normal vector (3, -2, 1), then the plane equation is 3x - 2y + z = d. Plugging in P (0,3,0): 0 - 6 + 0 = d => d = -6. So, the plane equation would be 3x - 2y + z = -6. But that's different from what I have.Wait, but the user said d=6, so 3x - 2y + z = 6. Let me check if point P is on that plane: 3(0) - 2(3) + 0 = -6 ≠ 6. So, that's incorrect. Therefore, the user's plane equation is wrong because it doesn't pass through point P.Therefore, my plane equation is correct: 3x + 2y - 2z = 6.So, the two additional intersection points are (6,0,6) and (0,6,3), and the distance between them is 9.But wait, let me visualize the cube and the plane. The plane cuts through the cube, and the intersection points are P, Q, R, S, T. So, the polygon formed by these points is a pentagon? Wait, no, in a cube, a plane can intersect at most six edges, forming a hexagon. But in this case, it's intersecting five edges, which is unusual. Wait, no, actually, a plane can intersect a cube in a triangle, quadrilateral, pentagon, or hexagon, depending on the plane.But in this case, the plane intersects five edges, which would form a pentagonal section. But in reality, a plane intersecting a cube can't form a pentagon because the cube is convex and the intersection must be a convex polygon with at most six sides. Wait, actually, it can form a pentagon if it intersects five edges, but in this case, I think it's forming a quadrilateral because three points are on adjacent edges, and the other two are on opposite edges.Wait, no, in my calculation, the plane intersects five edges: edges 2, 3, 5, 6, and 12. So, five points: P, Q, R, S, T. So, the intersection is a pentagon. But in reality, a cube can't have a pentagonal intersection because the maximum number of edges a plane can intersect in a cube is six, forming a hexagon. So, perhaps I made a mistake in identifying the intersection points.Wait, let me check again. The plane equation is 3x + 2y - 2z = 6.Let me check edge 5: from (0,0,6) to (6,0,6). Parametrize as (t,0,6), t from 0 to 6.Plug into plane: 3t + 0 - 12 = 6 => 3t = 18 => t=6. So, intersection at (6,0,6), which is the endpoint. So, that's correct.Edge 6: from (0,6,0) to (0,6,6). Parametrize as (0,6,t), t from 0 to 6.Plug into plane: 0 + 12 - 2t = 6 => -2t = -6 => t=3. So, intersection at (0,6,3).Edge 12: from (0,6,6) to (6,6,6). Parametrize as (t,6,6), t from 0 to 6.Plug into plane: 3t + 12 - 12 = 6 => 3t = 6 => t=2. So, intersection at (2,6,6), which is point R.So, that's correct.Edge 5: (6,0,6)Edge 6: (0,6,3)Edge 12: (2,6,6)So, the plane intersects five edges, but in reality, the intersection should be a convex polygon. So, perhaps the plane intersects five edges, forming a pentagon. But in a cube, the maximum number of edges a plane can intersect is six, forming a hexagon. So, maybe I made a mistake in identifying the intersection points.Wait, let me check edge 7: from (0,6,0) to (6,6,0). Parametrize as (t,6,0), t from 0 to 6.Plug into plane: 3t + 12 - 0 = 6 => 3t = -6 => t=-2. Not in [0,6], so no intersection.Edge 8: from (6,0,0) to (6,0,6). Parametrize as (6,0,t), t from 0 to 6.Plug into plane: 18 + 0 - 2t = 6 => -2t = -12 => t=6. So, intersection at (6,0,6), which is point S.Edge 9: from (6,0,0) to (6,6,0). Parametrize as (6,t,0), t from 0 to 6.Plug into plane: 18 + 2t - 0 = 6 => 2t = -12 => t=-6. Not in [0,6], so no intersection.Edge 10: from (6,0,6) to (6,6,6). Parametrize as (6,t,6), t from 0 to 6.Plug into plane: 18 + 2t - 12 = 6 => 6 + 2t = 6 => 2t=0 => t=0. So, intersection at (6,0,6), which is point S.Edge 11: from (6,6,0) to (6,6,6). Parametrize as (6,6,t), t from 0 to 6.Plug into plane: 18 + 12 - 2t = 6 => 30 - 2t = 6 => -2t = -24 => t=12. Not in [0,6], so no intersection.So, the plane intersects edges 2, 3, 5, 6, 12, and 8, but at edge 8, it intersects at the endpoint (6,0,6), which is already counted as point S. So, the unique intersection points are P, Q, R, S, and T.Therefore, the intersection is a pentagon with vertices P, Q, R, S, T.But in the cube, a plane can't form a pentagonal intersection because the cube is a convex polyhedron with six faces, and the intersection of a plane with a convex polyhedron is a convex polygon. The maximum number of sides is six, but it can also be a triangle, quadrilateral, or pentagon.Wait, actually, it's possible for a plane to intersect a cube in a pentagonal section. For example, if the plane intersects five edges, it forms a pentagon. So, in this case, the plane intersects five edges, forming a pentagon with vertices P, Q, R, S, T.But the problem states that the plane intersects the cube at P, Q, R, and two other points. So, the two additional points are S and T, which are (6,0,6) and (0,6,3).Therefore, the distance between these two points is sqrt[(6-0)^2 + (0-6)^2 + (6-3)^2] = sqrt[36 + 36 + 9] = sqrt[81] = 9.But wait, the user's initial answer was 6√3, which is approximately 10.39, which is larger than 9. So, perhaps the user made a mistake in their cross product, leading to an incorrect plane equation, and thus incorrect intersection points.Alternatively, maybe I made a mistake in my calculation. Let me double-check.Wait, in my calculation, I found the normal vector as (3, 2, -2), leading to the plane equation 3x + 2y - 2z = 6. This plane intersects the cube at points P, Q, R, S, and T, with S = (6,0,6) and T = (0,6,3). The distance between S and T is 9.But the user's initial answer considered points (0,0,6) and (6,6,0), which are not on the plane. So, their calculation was incorrect.Therefore, the correct distance between the two additional points is 9.But wait, let me check the plane equation again. If I use the user's normal vector (3, -2, 1), the plane equation would be 3x - 2y + z = 6. Let's see where this plane intersects the cube.For example, plug in point (0,0,6): 0 - 0 + 6 = 6, so it's on the plane. Similarly, plug in (6,6,0): 18 - 12 + 0 = 6, so it's on the plane. So, the user's plane intersects the cube at (0,0,6) and (6,6,0), which are the two additional points, and the distance between them is sqrt[(6-0)^2 + (6-0)^2 + (0-6)^2] = sqrt[36 + 36 + 36] = sqrt[108] = 6√3.But wait, the user's plane equation is incorrect because it doesn't pass through point P = (0,3,0). Plugging into 3x - 2y + z = 6: 0 - 6 + 0 = -6 ≠ 6. So, the user's plane equation is wrong.Therefore, the correct plane equation is 3x + 2y - 2z = 6, leading to the two additional points being (6,0,6) and (0,6,3), with a distance of 9.But wait, let me check if the plane equation 3x + 2y - 2z = 6 actually intersects the cube at five points. That seems unusual, but it's possible.Alternatively, maybe I made a mistake in identifying the intersection points. Let me check edge 5 again: from (0,0,6) to (6,0,6). Parametrize as (t,0,6). Plug into plane: 3t + 0 - 12 = 6 => 3t = 18 => t=6. So, intersection at (6,0,6).Edge 6: from (0,6,0) to (0,6,6). Parametrize as (0,6,t). Plug into plane: 0 + 12 - 2t = 6 => -2t = -6 => t=3. So, intersection at (0,6,3).Edge 12: from (0,6,6) to (6,6,6). Parametrize as (t,6,6). Plug into plane: 3t + 12 - 12 = 6 => 3t = 6 => t=2. So, intersection at (2,6,6).So, yes, the plane intersects edges 5, 6, and 12 at (6,0,6), (0,6,3), and (2,6,6), respectively. So, these are three additional points, but the problem states that the plane intersects the cube at P, Q, R, and two other points. So, perhaps I made a mistake in counting.Wait, the plane intersects the cube at P, Q, R, S, and T, which are five points. But the problem says it intersects at P, Q, R, and two other points, implying a total of five points. So, perhaps the initial problem statement is correct, and the two additional points are S and T.Therefore, the distance between S and T is 9.But wait, in the initial problem, the user had a different normal vector, leading to different intersection points. So, perhaps the correct answer is 9, not 6√3.Alternatively, maybe I made a mistake in the cross product. Let me recast the vectors.Wait, perhaps I should have used vectors from Q to P and from Q to R instead of from P to Q and P to R. Let me try that.Vector QP = P - Q = (0-2, 3-0, 0-0) = (-2,3,0)Vector QR = R - Q = (2-2,6-0,6-0) = (0,6,6)Cross product QP × QR:i component: (3)(6) - (0)(6) = 18 - 0 = 18j component: (0)(0) - (-2)(6) = 0 + 12 = 12k component: (-2)(6) - (3)(0) = -12 - 0 = -12So, QP × QR = (18,12,-12). Simplifying by dividing by 6 gives (3,2,-2). So, same normal vector as before.Therefore, the plane equation is 3x + 2y - 2z = 6.So, my calculation is correct.Therefore, the two additional points are (6,0,6) and (0,6,3), and the distance between them is 9.But wait, the user's initial answer was 6√3, which is approximately 10.39, which is larger than 9. So, perhaps the user made a mistake in their cross product, leading to an incorrect plane equation, and thus incorrect intersection points.Therefore, the correct answer is 9.But wait, let me check the distance again.Point S = (6,0,6)Point T = (0,6,3)Distance = sqrt[(6-0)^2 + (0-6)^2 + (6-3)^2] = sqrt[36 + 36 + 9] = sqrt[81] = 9.Yes, that's correct.Therefore, the distance between the two additional points is 9.

💡Alright, so I have this problem here with three expressions: a, b, and c. I need to figure out their order from smallest to largest. Let me write them down again to make sure I have them right.a is equal to (1/2)cos8° minus (√3/2)sin8°. Hmm, that looks familiar. Maybe it's a sine or cosine of some angle? I remember there's a formula for combining sine and cosine terms into a single trigonometric function. Let me recall... Oh, right! The formula is something like sin(A - B) = sinA cosB - cosA sinB. Wait, but here I have coefficients 1/2 and √3/2. Those look like cosine and sine of 60°, right? Because cos60° is 1/2 and sin60° is √3/2. So maybe I can rewrite a as sin(60° - 8°)? Let me check:sin(60° - 8°) = sin60°cos8° - cos60°sin8°, which is exactly what a is. So, a = sin52°. Wait, hold on, 60° - 8° is 52°, but in the problem, it's (1/2)cos8° - (√3/2)sin8°, which is sin60°cos8° - cos60°sin8°, so that's sin(60° - 8°) = sin52°. So, a = sin52°. Got it.Moving on to b. It's given as 2tan14° divided by (1 - tan²14°). Hmm, that seems like the double-angle formula for tangent. Let me recall, tan(2θ) = 2tanθ / (1 - tan²θ). So, if I let θ = 14°, then tan(28°) = 2tan14° / (1 - tan²14°). Therefore, b = tan28°. Okay, so b is tan28°.Now, c is the square root of (1 - cos48°)/2. That looks like the half-angle formula for sine. The formula is sin(θ/2) = √[(1 - cosθ)/2]. So, if I let θ = 48°, then sin24° = √[(1 - cos48°)/2]. Therefore, c = sin24°.So, summarizing:- a = sin52°- b = tan28°- c = sin24°Now, I need to compare these three values: sin52°, tan28°, and sin24°.Let me recall the approximate values or think about their sizes.First, sin24°, sin52°, and tan28°.I know that sin increases from 0° to 90°, so sin24° is less than sin52°. So, c < a.Now, tan28°. Tangent increases from 0° to 90°, and tan28° is greater than sin28°, because tanθ = sinθ / cosθ, and cosθ is less than 1 for θ > 0°. So, tan28° > sin28°. But how does tan28° compare to sin52°?Hmm, let me think. I know that tan45° = 1, and sin45° is about √2/2 ≈ 0.707. So, tan28° is less than 1 because 28° is less than 45°, right? Wait, no, tan increases as the angle increases, so tan28° is less than tan45°, which is 1. So, tan28° is less than 1.What about sin52°? Sin52° is greater than sin45°, which is about 0.707, so sin52° is approximately, let me recall, sin60° is about 0.866, so sin52° is a bit less, maybe around 0.788.And tan28°, let me approximate. I know that tan30° is about 0.577, and tan28° is a bit less than that, maybe around 0.5317.So, putting it all together:c = sin24° ≈ 0.4067a = sin52° ≈ 0.7880b = tan28° ≈ 0.5317So, ordering them: c < b < aWait, but hold on, that would mean c < b < a, which is option C. But let me double-check my approximations.Wait, wait, wait. I think I made a mistake earlier. Let me go back.When I was calculating a, I thought it was sin52°, but let me check again.The expression for a was (1/2)cos8° - (√3/2)sin8°. I used the formula sin(A - B) = sinA cosB - cosA sinB, so that would be sin(60° - 8°) = sin52°, which is correct. So, a = sin52°, which is approximately 0.788.For b, 2tan14° / (1 - tan²14°). As I thought, that's tan28°, which is approximately 0.5317.For c, sqrt[(1 - cos48°)/2]. Using the half-angle formula, that's sin24°, which is approximately 0.4067.So, c ≈ 0.4067, b ≈ 0.5317, a ≈ 0.788. So, c < b < a. That would mean the order is c < b < a, which is option C.But wait, the initial solution I saw earlier said a < c < b, which is option A. That contradicts my conclusion. Did I make a mistake?Wait, let me double-check the initial problem statement.a = (1/2)cos8° - (√3/2)sin8°b = 2tan14° / (1 - tan²14°)c = sqrt[(1 - cos48°)/2]Yes, that's correct.Wait, perhaps I messed up the angle in a. Let me re-examine.I said a = sin(60° - 8°) = sin52°, but is that correct?Wait, the formula is sin(A - B) = sinA cosB - cosA sinB.But in the expression for a, it's (1/2)cos8° - (√3/2)sin8°, which is sin60°cos8° - cos60°sin8°, so that is sin(60° - 8°) = sin52°. So, that's correct.Alternatively, maybe I can think of it as a cosine of some angle? Let me see.Alternatively, using the formula cos(A + B) = cosA cosB - sinA sinB. But here, the coefficients are 1/2 and √3/2, which are cos60° and sin60°, so it's more like sin(60° - 8°).Wait, another way: perhaps it's a cosine of (60° + 8°)? Let me check.cos(A + B) = cosA cosB - sinA sinB. So, if I have cos60°cos8° - sin60°sin8°, that would be cos(60° + 8°) = cos68°, but in our case, it's (1/2)cos8° - (√3/2)sin8°, which is cos60°cos8° - sin60°sin8°, so that is cos(60° + 8°) = cos68°. Wait, so is a equal to cos68° or sin52°?Wait, cos68° is equal to sin22°, because cosθ = sin(90° - θ). So, cos68° = sin22°, which is approximately 0.3746.Wait, hold on, now I'm confused. Is a equal to sin52° or cos68°?Wait, let's compute both:sin52° ≈ 0.7880cos68° ≈ 0.3746But the expression for a is (1/2)cos8° - (√3/2)sin8°. Let me compute that numerically.Compute (1/2)cos8°: cos8° ≈ 0.9903, so (1/2)(0.9903) ≈ 0.49515Compute (√3/2)sin8°: sin8° ≈ 0.1392, so (√3/2)(0.1392) ≈ (0.8660)(0.1392) ≈ 0.1205So, a ≈ 0.49515 - 0.1205 ≈ 0.3746Wait, that's approximately 0.3746, which is equal to cos68° or sin22°, not sin52°. So, my initial thought was wrong.Wait, so where did I go wrong? I thought it was sin(60° - 8°), but actually, it's cos(60° + 8°). Because the formula is cos(A + B) = cosA cosB - sinA sinB.But in the expression for a, it's (1/2)cos8° - (√3/2)sin8°, which is cos60°cos8° - sin60°sin8°, which is cos(60° + 8°) = cos68°, which is approximately 0.3746.So, a = cos68° = sin22°, which is approximately 0.3746.Wait, so my initial mistake was thinking it was sin(60° - 8°), but actually, it's cos(60° + 8°). So, a = cos68°, which is approximately 0.3746.Okay, so that changes things. So, a ≈ 0.3746, which is less than c ≈ 0.4067.So, now, let's recast the values:a ≈ 0.3746c ≈ 0.4067b ≈ 0.5317So, ordering them: a < c < b, which is option A.Wait, so my initial mistake was in the angle addition formula. I thought it was sin(A - B), but it's actually cos(A + B). So, that's why a is smaller than I initially thought.So, to recap:a = cos68° ≈ 0.3746c = sin24° ≈ 0.4067b = tan28° ≈ 0.5317Therefore, a < c < b, which is option A.I think that's correct now. So, the answer should be A.But just to make sure, let me verify each step again.For a:(1/2)cos8° - (√3/2)sin8°= cos60°cos8° - sin60°sin8°= cos(60° + 8°)= cos68°= sin22° ≈ 0.3746For b:2tan14° / (1 - tan²14°) = tan28° ≈ 0.5317For c:sqrt[(1 - cos48°)/2] = sin24° ≈ 0.4067So, a ≈ 0.3746 < c ≈ 0.4067 < b ≈ 0.5317Yes, that seems correct. So, the order is a < c < b, which is option A.I think I got confused earlier because I misapplied the angle addition formula, but after recalculating, it makes sense now.

💡Okay, so I have this problem about four small square corrals, each with a side length of ( s ). They're going to rearrange these four small corrals to form one larger square corral, and they're using the exact same amount of fencing. I need to find the ratio of the total area of the four small corrals to the area of the new large corral.Alright, let me break this down step by step. First, I need to figure out the total amount of fencing used for the four small corrals. Since each corral is a square, each one has four sides, each of length ( s ). So, the perimeter of one small corral is ( 4s ). But there are four of these small corrals, so the total fencing used would be ( 4 times 4s ), which is ( 16s ). That makes sense because each corral contributes ( 4s ) to the total fencing, and we have four of them.Now, this total fencing is being used to form one larger square corral. So, the perimeter of the larger square must also be ( 16s ). Since a square has four equal sides, each side of the larger square must be ( frac{16s}{4} = 4s ). So, the side length of the large corral is ( 4s ).Next, I need to find the areas. The area of one small square corral is ( s^2 ), so the total area for four small corrals would be ( 4 times s^2 = 4s^2 ).Now, the area of the larger square corral. Since its side length is ( 4s ), the area is ( (4s)^2 = 16s^2 ).Finally, I need the ratio of the total area of the four small corrals to the area of the large corral. That would be ( frac{4s^2}{16s^2} ). Simplifying that, the ( s^2 ) terms cancel out, leaving ( frac{4}{16} ), which simplifies further to ( frac{1}{4} ).Wait, let me double-check that. The total fencing is the same, so the perimeters are equal. Each small corral has a perimeter of ( 4s ), four of them make ( 16s ). The large corral's perimeter is also ( 16s ), so each side is ( 4s ). Area of small is ( s^2 ), four of them is ( 4s^2 ). Area of large is ( (4s)^2 = 16s^2 ). Ratio is ( 4s^2 / 16s^2 = 1/4 ). Yeah, that seems right.I think I got it. The ratio is ( frac{1}{4} ).

💡Okay, so I've got this arithmetic sequence problem here. Let me try to figure it out step by step. First, the problem says that the sum of the first 3 terms is 4. Hmm, in an arithmetic sequence, each term increases by a common difference, right? So, if the first term is ( a_1 ), then the second term is ( a_1 + d ), and the third term is ( a_1 + 2d ). So, adding those up: ( a_1 + (a_1 + d) + (a_1 + 2d) = 4 ). Let me write that out:( a_1 + (a_1 + d) + (a_1 + 2d) = 4 )Simplifying that, I get:( 3a_1 + 3d = 4 )I can factor out the 3:( 3(a_1 + d) = 4 )So, dividing both sides by 3:( a_1 + d = frac{4}{3} )Okay, so that's one equation. Now, the problem also mentions that the sum of the last 3 terms is 7. Hmm, the last three terms would be ( a_{n-2} ), ( a_{n-1} ), and ( a_n ). Since it's an arithmetic sequence, each term is the previous term plus the common difference ( d ). So, ( a_{n-2} = a_1 + (n-3)d ), ( a_{n-1} = a_1 + (n-2)d ), and ( a_n = a_1 + (n-1)d ).Adding those together:( (a_1 + (n-3)d) + (a_1 + (n-2)d) + (a_1 + (n-1)d) = 7 )Simplifying:( 3a_1 + (3n - 6)d = 7 )Factor out the 3:( 3(a_1 + (n - 2)d) = 7 )Divide both sides by 3:( a_1 + (n - 2)d = frac{7}{3} )Alright, so now I have two equations:1. ( a_1 + d = frac{4}{3} )2. ( a_1 + (n - 2)d = frac{7}{3} )Hmm, maybe I can subtract the first equation from the second to eliminate ( a_1 ). Let's try that:( [a_1 + (n - 2)d] - [a_1 + d] = frac{7}{3} - frac{4}{3} )Simplify:( (n - 2)d - d = frac{3}{3} )Which becomes:( (n - 3)d = 1 )So, ( (n - 3)d = 1 ). Let me note that down.Now, the problem also says that the sum of all terms is 22. The formula for the sum of an arithmetic sequence is ( S_n = frac{n}{2}(a_1 + a_n) ). So, ( S_n = 22 ), which gives:( frac{n}{2}(a_1 + a_n) = 22 )I need to find ( a_1 + a_n ). Let's see, ( a_n = a_1 + (n - 1)d ), so ( a_1 + a_n = a_1 + [a_1 + (n - 1)d] = 2a_1 + (n - 1)d ).So, plugging that into the sum formula:( frac{n}{2}[2a_1 + (n - 1)d] = 22 )Simplify:( n[2a_1 + (n - 1)d] = 44 )Hmm, okay. Now, let me see if I can express ( 2a_1 + (n - 1)d ) in terms of the equations I already have.From the first equation, ( a_1 + d = frac{4}{3} ), so ( a_1 = frac{4}{3} - d ).Let me substitute ( a_1 ) into ( 2a_1 + (n - 1)d ):( 2left(frac{4}{3} - dright) + (n - 1)d )Simplify:( frac{8}{3} - 2d + (n - 1)d )Combine like terms:( frac{8}{3} + (n - 3)d )Wait, from earlier, I have ( (n - 3)d = 1 ). So, substituting that in:( frac{8}{3} + 1 = frac{8}{3} + frac{3}{3} = frac{11}{3} )So, ( 2a_1 + (n - 1)d = frac{11}{3} )Therefore, going back to the sum equation:( n times frac{11}{3} = 44 )Multiply both sides by 3:( 11n = 132 )Divide both sides by 11:( n = 12 )So, the number of terms is 12. Let me double-check my steps to make sure I didn't make any mistakes.1. Sum of first 3 terms: ( 3a_1 + 3d = 4 ) leading to ( a_1 + d = frac{4}{3} ). That seems right.2. Sum of last 3 terms: ( 3a_1 + (3n - 6)d = 7 ) leading to ( a_1 + (n - 2)d = frac{7}{3} ). That also looks correct.3. Subtracting the two equations: ( (n - 3)d = 1 ). Makes sense.4. Sum of all terms: ( frac{n}{2}(a_1 + a_n) = 22 ). Correct formula.5. Expressing ( a_1 + a_n ) as ( 2a_1 + (n - 1)d ). Yes, because ( a_n = a_1 + (n - 1)d ).6. Substituting ( a_1 = frac{4}{3} - d ) into ( 2a_1 + (n - 1)d ) gives ( frac{8}{3} - 2d + (n - 1)d ), which simplifies to ( frac{8}{3} + (n - 3)d ). Since ( (n - 3)d = 1 ), this becomes ( frac{11}{3} ). Perfect.7. Plugging back into the sum formula: ( n times frac{11}{3} = 44 ) leads to ( n = 12 ). All steps check out.I think I did everything correctly. So, the answer should be 12.**Final Answer**boxed{A}