answer：Okay, so I have this problem where a corner of a unit cube is chopped off, and the cut goes through the midpoints of the three edges that are adjacent to the chosen vertex. I need to find the height of the remaining solid when this freshly-cut face is placed on a table. Hmm, let me try to visualize this.First, a unit cube has edges of length 1. If I pick one corner, say the origin (0,0,0), and then the three edges adjacent to this corner go along the x, y, and z axes. The midpoints of these edges would then be at (0.5, 0, 0), (0, 0.5, 0), and (0, 0, 0.5). So, the cut is a plane that passes through these three midpoints.I think this plane will form a triangular face on the remaining solid. Since the midpoints are each 0.5 units away from the origin, the sides of this triangle should all be equal. Let me check that. The distance between (0.5, 0, 0) and (0, 0.5, 0) is sqrt[(0.5)^2 + (-0.5)^2] = sqrt[0.25 + 0.25] = sqrt[0.5] ≈ 0.707. Similarly, the distance between (0.5, 0, 0) and (0, 0, 0.5) is sqrt[(0.5)^2 + (0.5)^2] = sqrt[0.25 + 0.25] = sqrt[0.5] ≈ 0.707. And the same for the other side. So, yes, it's an equilateral triangle with side length sqrt(0.5).Wait, but in 3D space, is it an equilateral triangle? Because in 3D, even if the sides are equal in length, the angles might not be 60 degrees. Hmm, maybe it's not a planar equilateral triangle? Or is it? Let me think.Actually, in 3D, if all the edges are equal, the triangle is equilateral. So, yes, the cut face is an equilateral triangle with side length sqrt(0.5). That makes sense.Now, I need to find the height of the remaining solid when this triangular face is placed on a table. So, the height would be the maximum distance from the table (which is now the plane of the triangle) to the highest point of the cube.But wait, the cube is a unit cube, so the original height from the corner to the opposite corner is the space diagonal, which is sqrt(3) ≈ 1.732. But since we've cut off a corner, the height should be less than that.Alternatively, maybe the height is measured from the table (the cut face) to the farthest vertex of the cube. Let me think about that.The cube has vertices at (0,0,0), (1,0,0), (0,1,0), (0,0,1), etc. The cut removes the corner at (0,0,0), and the cut face is the triangle connecting (0.5,0,0), (0,0.5,0), and (0,0,0.5). So, the remaining solid includes the rest of the cube.When we place the cut face on the table, the table is now the plane containing (0.5,0,0), (0,0.5,0), and (0,0,0.5). The height of the solid would be the maximum distance from this plane to any point on the solid.So, the farthest point from this plane in the remaining solid would be the opposite corner of the cube, which is (1,1,1). So, I need to find the distance from the point (1,1,1) to the plane defined by (0.5,0,0), (0,0.5,0), and (0,0,0.5).To find the distance from a point to a plane, I can use the formula:Distance = |ax + by + cz + d| / sqrt(a^2 + b^2 + c^2)First, I need the equation of the plane. Let me find that.Given three points on the plane: A(0.5,0,0), B(0,0.5,0), and C(0,0,0.5).Let me find two vectors on the plane: AB = B - A = (-0.5, 0.5, 0) and AC = C - A = (-0.5, 0, 0.5).Then, the normal vector to the plane is the cross product of AB and AC.Calculating AB × AC:|i j k||-0.5 0.5 0||-0.5 0 0.5|= i*(0.5*0.5 - 0*0) - j*(-0.5*0.5 - 0*(-0.5)) + k*(-0.5*0 - 0.5*(-0.5))= i*(0.25) - j*(-0.25) + k*(0.25)= (0.25, 0.25, 0.25)So, the normal vector is (0.25, 0.25, 0.25). We can write the plane equation as:0.25(x - 0.5) + 0.25(y - 0) + 0.25(z - 0) = 0Simplifying:0.25x - 0.125 + 0.25y + 0.25z = 00.25x + 0.25y + 0.25z = 0.125Multiplying both sides by 8 to eliminate decimals:2x + 2y + 2z = 1So, the plane equation is 2x + 2y + 2z = 1.Now, the distance from the point (1,1,1) to this plane is:|2*1 + 2*1 + 2*1 - 1| / sqrt(2^2 + 2^2 + 2^2)= |2 + 2 + 2 - 1| / sqrt(12)= |5| / (2*sqrt(3))= 5 / (2*sqrt(3))Rationalizing the denominator:= (5*sqrt(3)) / 6So, the distance is (5*sqrt(3))/6 ≈ 1.443.But wait, the cube is a unit cube, so the maximum distance from the cut face to the opposite corner is about 1.443, which is less than the space diagonal of sqrt(3) ≈ 1.732. That makes sense because we've cut off a corner.But is this the height of the remaining solid? Or is there another way to interpret the height?Alternatively, maybe the height is the distance from the cut face to the highest point on the cube, which is (1,1,1). So, yes, that distance is (5*sqrt(3))/6.But let me double-check my calculations.First, the normal vector was (0.25, 0.25, 0.25). So, the plane equation is 0.25x + 0.25y + 0.25z = d. Plugging in point A(0.5,0,0):0.25*0.5 + 0 + 0 = d => d = 0.125So, the plane equation is 0.25x + 0.25y + 0.25z = 0.125, which is equivalent to x + y + z = 0.5 when multiplied by 4.Wait, earlier I multiplied by 8 to get 2x + 2y + 2z = 1. That's correct because 0.25*4 = 1, so 0.25x + 0.25y + 0.25z = 0.125 becomes x + y + z = 0.5 when multiplied by 4. But I multiplied by 8 to get rid of decimals, resulting in 2x + 2y + 2z = 1. Both are correct, just scaled differently.So, using the plane equation x + y + z = 0.5, the distance from (1,1,1) is:|1 + 1 + 1 - 0.5| / sqrt(1^2 + 1^2 + 1^2) = |2.5| / sqrt(3) = 2.5 / sqrt(3) = (5/2)/sqrt(3) = (5*sqrt(3))/6, which matches my earlier result.So, the height is (5*sqrt(3))/6.Wait, but the problem says "the height of the remaining solid when the freshly-cut face is placed on a table." So, is this the height? Or is there another way to interpret it?Alternatively, maybe the height is the distance from the cut face to the farthest point on the cube, which is indeed (1,1,1). So, yes, that distance is (5*sqrt(3))/6.But let me think if there's another approach. Maybe using volumes or something else.The volume of the cube is 1. The volume of the tetrahedron that was cut off can be calculated, and then the remaining volume is 1 minus that. But how does that help with the height?Wait, if I consider the remaining solid as a polyhedron with the cut face as its base, then the height would be the perpendicular distance from the base to the opposite vertex. So, that's exactly what I calculated earlier, which is (5*sqrt(3))/6.But let me confirm if that's correct.Alternatively, maybe I can think of the centroid of the cut face and find the distance from there to the opposite vertex, but that might not be the height.Wait, the height is the maximum distance from the base to any point on the solid, which in this case is the distance from the cut face to (1,1,1). So, yes, that should be the height.But let me calculate it again to be sure.Plane equation: x + y + z = 0.5Point: (1,1,1)Distance = |1 + 1 + 1 - 0.5| / sqrt(1 + 1 + 1) = |2.5| / sqrt(3) = 2.5 / sqrt(3) = (5/2)/sqrt(3) = (5*sqrt(3))/6 ≈ 1.443Yes, that seems correct.But wait, the unit cube has edges of length 1, so the space diagonal is sqrt(3) ≈ 1.732. So, the height of the remaining solid is less than that, which makes sense because we've cut off a corner.Alternatively, maybe the height is measured differently, like from the table to the highest point on the original cube, but I think the way I calculated it is correct.Wait, another way to think about it: when the cut face is placed on the table, the solid is resting on that triangular face. The height would be the distance from the table to the highest point of the solid, which is (1,1,1). So, yes, that distance is (5*sqrt(3))/6.But let me check if there's a simpler way to express this.(5*sqrt(3))/6 is approximately 1.443, which is less than sqrt(3) ≈ 1.732, so that seems reasonable.Alternatively, maybe the height is the distance from the centroid of the cut face to the opposite vertex. Let me calculate that.The centroid of the triangle is the average of the three points: ((0.5 + 0 + 0)/3, (0 + 0.5 + 0)/3, (0 + 0 + 0.5)/3) = (0.5/3, 0.5/3, 0.5/3) = (1/6, 1/6, 1/6)Then, the distance from (1/6, 1/6, 1/6) to (1,1,1) is sqrt[(1 - 1/6)^2 + (1 - 1/6)^2 + (1 - 1/6)^2] = sqrt[(5/6)^2 + (5/6)^2 + (5/6)^2] = sqrt[3*(25/36)] = sqrt[75/36] = sqrt[25/12] = 5/(2*sqrt(3)) = (5*sqrt(3))/6, which is the same as before.So, whether I calculate the distance from the plane or from the centroid, I get the same result. That makes sense because the centroid lies on the plane, and the distance from the centroid to the opposite vertex is the same as the distance from the plane to the vertex.Therefore, the height of the remaining solid when the freshly-cut face is placed on a table is (5*sqrt(3))/6.But let me check if this is the correct interpretation. The problem says "the height of the remaining solid when the freshly-cut face is placed on a table." So, yes, the height is the maximum distance from the table (the cut face) to the highest point of the solid, which is (1,1,1). So, the distance is (5*sqrt(3))/6.I think that's the correct answer. Let me just recap:1. Identified the cut face as an equilateral triangle with vertices at midpoints of edges.2. Found the equation of the plane using three points.3. Calculated the distance from the opposite corner (1,1,1) to this plane.4. Confirmed that this distance is indeed the height of the remaining solid when placed on the cut face.So, I'm confident that the height is (5*sqrt(3))/6.

answer：Okay, so I have this problem about constructing an insulation layer for a building. The goal is to figure out the optimal thickness of the insulation to minimize the total cost, which includes both the construction cost and the energy consumption cost over 20 years. First, let me try to understand the problem step by step. The building needs an insulation layer that can last 20 years. The construction cost is 60,000 yuan per centimeter thickness. So, if I build an insulation layer of x centimeters, the construction cost would be 60,000 times x yuan. Then, there's the annual energy consumption cost, which is given by the function C(x) = k / (3x + 5), where k is a constant we need to find. It also says that if no insulation layer is built (so x = 0), the annual energy consumption cost is 80,000 yuan. Wait, the problem mentions that C(x) is in ten thousand yuan, right? So, actually, when x = 0, C(0) should be 8 (since 80,000 yuan is 8 ten thousand yuan). That makes sense because 80,000 divided by 10,000 is 8. So, plugging x = 0 into C(x), we get C(0) = k / (3*0 + 5) = k / 5. We know that this equals 8, so k / 5 = 8. Solving for k, we multiply both sides by 5, so k = 40. Okay, that gives us the value of k. So now, the annual energy consumption cost is C(x) = 40 / (3x + 5). Next, we need to find the expression for f(x), which is the sum of the construction cost and the energy consumption cost over 20 years. The construction cost is straightforward: it's 60,000 yuan per cm, so for x cm, it's 60,000x yuan. But since the energy consumption cost is given in ten thousand yuan, we should convert the construction cost to the same units to keep things consistent. 60,000 yuan is 6 ten thousand yuan, so the construction cost in ten thousand yuan is 6x. Now, the energy consumption cost over 20 years would be 20 times the annual cost. Since C(x) is already in ten thousand yuan, multiplying by 20 gives us 20 * (40 / (3x + 5)) = 800 / (3x + 5). So, putting it all together, f(x) is the sum of the construction cost and the energy consumption cost over 20 years. Therefore, f(x) = 6x + 800 / (3x + 5). Alright, that takes care of part (I). Now, moving on to part (II), which asks for the thickness x that minimizes f(x) and the minimum value of f(x). To find the minimum, I think I need to use calculus. Specifically, I can take the derivative of f(x) with respect to x, set it equal to zero, and solve for x. That should give me the critical points, and then I can check if it's a minimum. So, let's compute f'(x). f(x) = 6x + 800 / (3x + 5)The derivative of 6x is 6. For the second term, 800 / (3x + 5), I can use the chain rule. Let me rewrite it as 800*(3x + 5)^(-1). The derivative of that is 800 * (-1) * (3x + 5)^(-2) * 3. Simplifying, that's -2400 / (3x + 5)^2. So, putting it together, f'(x) = 6 - 2400 / (3x + 5)^2. To find the critical points, set f'(x) = 0:6 - 2400 / (3x + 5)^2 = 0Let's solve for x:6 = 2400 / (3x + 5)^2Multiply both sides by (3x + 5)^2:6*(3x + 5)^2 = 2400Divide both sides by 6:(3x + 5)^2 = 400Take the square root of both sides:3x + 5 = sqrt(400) = 20So, 3x + 5 = 20Subtract 5:3x = 15Divide by 3:x = 5So, x = 5 cm is a critical point. Now, to make sure it's a minimum, I can check the second derivative or analyze the behavior of f'(x) around x = 5. Let's compute the second derivative f''(x). We have f'(x) = 6 - 2400 / (3x + 5)^2The derivative of 6 is 0. For the second term, -2400 / (3x + 5)^2, we can use the chain rule again. Let me write it as -2400*(3x + 5)^(-2). The derivative is -2400*(-2)*(3x + 5)^(-3)*3. Simplifying, that's 14400 / (3x + 5)^3. So, f''(x) = 14400 / (3x + 5)^3Since (3x + 5)^3 is always positive for x >= 0, f''(x) is positive. Therefore, the function is concave upward at x = 5, which means it's a local minimum. Since the function f(x) is defined on a closed interval [0,10], and we've found that the only critical point is at x = 5, which is a local minimum, this must be the global minimum on the interval. Now, let's compute f(5) to find the minimum total cost. f(5) = 6*5 + 800 / (3*5 + 5) = 30 + 800 / (15 + 5) = 30 + 800 / 20 = 30 + 40 = 70So, the minimum total cost is 70 ten thousand yuan, which is 700,000 yuan. Wait, let me double-check my calculations to make sure I didn't make a mistake. f(5) = 6*5 = 303*5 + 5 = 15 + 5 = 20800 / 20 = 4030 + 40 = 70. Yep, that seems correct. Just to be thorough, let me check the endpoints as well, even though we found a minimum inside the interval. At x = 0:f(0) = 6*0 + 800 / (0 + 5) = 0 + 800 / 5 = 160At x = 10:f(10) = 6*10 + 800 / (30 + 5) = 60 + 800 / 35 ≈ 60 + 22.857 ≈ 82.857So, f(0) = 160, f(10) ≈ 82.857, and f(5) = 70. So, indeed, 70 is the minimum. Therefore, the optimal thickness is 5 cm, and the minimum total cost is 70 ten thousand yuan. I think that covers both parts (I) and (II) of the problem. I found k by plugging in x = 0 and solving for k, then constructed f(x) as the sum of the construction cost and the energy cost over 20 years. Then, I used calculus to find the critical point, verified it's a minimum, and calculated the minimum value. Just to recap, the steps were:1. Find k using the given condition when x = 0.2. Express f(x) as the sum of construction cost and 20 years of energy cost.3. Take the derivative of f(x) to find critical points.4. Verify the critical point is a minimum using the second derivative.5. Calculate the minimum value by plugging the critical point back into f(x).6. Check the endpoints to ensure it's the global minimum.Everything seems to add up, so I'm confident in the solution.**Final Answer**The minimum total cost is achieved with an insulation layer of boxed{5} cm thickness, resulting in a total cost of boxed{70} ten thousand yuan.

answer：Alright, so I have this problem here about sets A and B, and I need to figure out the number of subsets of the set of real numbers a that satisfy a certain condition. Let me try to break this down step by step.First, let's understand what sets A and B are. Set A is defined as all real numbers x that satisfy the equation x² - 8x + 15 = 0. Okay, so I need to solve this quadratic equation to find the elements of set A. I remember that quadratic equations can often be factored, so let me try that.Looking at x² - 8x + 15, I need two numbers that multiply to 15 and add up to -8. Hmm, -3 and -5 come to mind because (-3) * (-5) = 15 and (-3) + (-5) = -8. So, the equation factors to (x - 3)(x - 5) = 0. That means the solutions are x = 3 and x = 5. So, set A is {3, 5}.Now, set B is defined as all real numbers x that satisfy ax - 1 = 0. To find the elements of set B, I need to solve for x. If I rearrange the equation, I get x = 1/a. So, set B contains the number 1/a, provided that a is not zero because division by zero is undefined. If a is zero, then the equation ax - 1 = 0 becomes -1 = 0, which is impossible, so set B would be empty in that case.The condition given is that A ∩ B = B. This means that every element in set B must also be in set A. In other words, B is a subset of A. So, if B is not empty, then the element 1/a must be one of the elements in A, which are 3 or 5. If B is empty, that's also acceptable because the empty set is a subset of any set.Let's consider the cases:1. If a ≠ 0, then B = {1/a}. For B to be a subset of A, 1/a must be either 3 or 5. So, 1/a = 3 implies a = 1/3, and 1/a = 5 implies a = 1/5.2. If a = 0, then B is empty, which is still a subset of A.So, the possible values of a are 0, 1/3, and 1/5. That gives us three real numbers: 0, 1/3, and 1/5.Now, the question asks for the number of subsets of the set of real numbers a. Since the set of a has three elements, the number of subsets is 2³ = 8. Each element can either be included or excluded from a subset, so for three elements, there are 2 × 2 × 2 = 8 possible subsets.Let me double-check to make sure I haven't missed anything. Set A is definitely {3, 5} because that's the solution to the quadratic. Set B is either empty or contains 1/a. The condition A ∩ B = B means B must be a subset of A, so 1/a has to be in A if B is not empty. That gives us a = 1/3 or 1/5. Including a = 0 gives us three values for a. The number of subsets of a set with three elements is indeed 8.I think that covers all the possibilities and checks out correctly. So, the answer should be 8 subsets.

answer：To determine which card Susan turned over to disprove John's statement, we need to analyze the given cards and John's claim.John's statement: "If a consonant is on one side of any card, then an odd number is on the other side."To disprove this, we need to find a card that has a consonant on one side and an even number on the other side.1. **Card A**: The letter 'A' is a vowel. Since John's statement is about consonants, this card does not help in disproving the statement.2. **Card B**: The letter 'B' is a consonant. If the other side of this card has an even number, it would disprove John's statement.3. **Card 7**: The number 7 is odd. The other side could have a vowel or a consonant, but since the number is odd, it does not help in disproving the statement.4. **Card 8**: The number 8 is even. If the other side of this card has a consonant, it would disprove John's statement.5. **Card 9**: The number 9 is odd. Similar to card 7, it does not help in disproving the statement.To disprove John's statement, Susan needs to turn over the card that could reveal a consonant with an even number. The best candidate is card 8, as it has the potential to have a consonant on the other side.Thus, Susan turned over card 8.The final answer is (boxed{B}).

answer：Okay, so I have this problem here: I need to find the point P₀ on the curve y = x³ + x - 2 where the tangent line is parallel to the line y = 4x. Hmm, okay, let me try to figure this out step by step.First, I remember that if two lines are parallel, they have the same slope. So, the tangent line to the curve at point P₀ must have a slope equal to the slope of y = 4x. Let me find the slope of y = 4x. Well, that's straightforward because it's already in the slope-intercept form y = mx + b, where m is the slope. So, the slope here is 4.Now, I need to find the point on the curve where the tangent has a slope of 4. To do this, I think I need to find the derivative of the curve's equation because the derivative at a point gives the slope of the tangent at that point.The curve is y = x³ + x - 2. Let me compute its derivative. The derivative of x³ is 3x², the derivative of x is 1, and the derivative of a constant like -2 is 0. So, putting it all together, the derivative dy/dx is 3x² + 1.Okay, so dy/dx = 3x² + 1. This derivative represents the slope of the tangent line at any point x on the curve. Since we want the tangent line to be parallel to y = 4x, we set this derivative equal to 4.So, I have the equation 3x² + 1 = 4. Let me solve for x. Subtracting 1 from both sides gives 3x² = 3. Then, dividing both sides by 3, I get x² = 1. Taking the square root of both sides, x = ±1. So, there are two possible x-values: x = 1 and x = -1.Now, I need to find the corresponding y-values for these x-values on the curve y = x³ + x - 2. Let's start with x = 1.Plugging x = 1 into the equation: y = (1)³ + 1 - 2 = 1 + 1 - 2 = 0. So, when x = 1, y = 0. That gives me the point (1, 0).Next, let's plug in x = -1. So, y = (-1)³ + (-1) - 2 = -1 - 1 - 2 = -4. Therefore, when x = -1, y = -4. That gives me the point (-1, -4).So, I have two points: (1, 0) and (-1, -4). Both of these points on the curve have tangent lines with a slope of 4, which means these tangent lines are parallel to the line y = 4x.Let me just double-check my calculations to make sure I didn't make any mistakes. Starting with the derivative: dy/dx = 3x² + 1. Setting that equal to 4 gives 3x² + 1 = 4, which simplifies to x² = 1, so x = ±1. Plugging x = 1 into the original equation: 1 + 1 - 2 = 0, correct. Plugging x = -1: -1 -1 -2 = -4, correct. So, both points seem right.I think that's it. So, the coordinates of point P₀ are (1, 0) and (-1, -4).**Final Answer**The coordinates of point ( P_{0} ) are boxed{(-1, -4)} and boxed{(1, 0)}.

answer：Okay, so I have this geometry problem here about a quadrilateral ABCD. Let me try to visualize it first. They say angle B is 90 degrees, and AB equals BC equals 5 units. Also, CD is 2 times the square root of 15, and DA is the square root of 10. I need to find the measure of angle BAD. The options are 90°, 120°, 135°, and 150°. Hmm, okay.First, since angle B is 90°, and AB and BC are both 5, that means triangle ABC is a right-angled isosceles triangle, right? Because both legs are equal. So, in triangle ABC, AB = BC = 5, and angle B is 90°, so AC should be the hypotenuse. I remember that in a right-angled triangle, the hypotenuse squared is equal to the sum of the squares of the other two sides. So, AC squared equals AB squared plus BC squared. Let me calculate that.AC² = AB² + BC² = 5² + 5² = 25 + 25 = 50. So, AC is the square root of 50, which simplifies to 5 times the square root of 2. Got that.Now, since triangle ABC is a right-angled isosceles triangle, the angles at A and C should each be 45°, right? Because in an isosceles right triangle, the two non-right angles are equal. So, angle BAC is 45°, and angle BCA is also 45°. That makes sense.Next, I need to look at the quadrilateral ABCD. I know the lengths of CD and DA, which are 2√15 and √10 respectively. Maybe I can consider triangle ACD. Let me see if triangle ACD is a right triangle or something. If I can figure that out, maybe I can find some angles there.So, in triangle ACD, the sides are AC = 5√2, CD = 2√15, and DA = √10. Let me check if this satisfies the Pythagorean theorem. Let's compute AC² + DA² and see if it equals CD².AC² is (5√2)² = 25 * 2 = 50. DA² is (√10)² = 10. So, AC² + DA² = 50 + 10 = 60. Now, CD² is (2√15)² = 4 * 15 = 60. Oh, that's interesting. So, AC² + DA² equals CD², which means triangle ACD is a right triangle with the right angle at A. So, angle CAD is 90°.Wait, so angle CAD is 90°, and angle BAC is 45°, right? So, angle BAD is the sum of angle BAC and angle CAD. That would be 45° + 90°, which is 135°. So, angle BAD is 135°, which is option C.But let me double-check to make sure I didn't make any mistakes. So, starting from the beginning, quadrilateral ABCD with AB = BC = 5, angle B = 90°, so triangle ABC is right-angled and isosceles, giving AC = 5√2. Then, triangle ACD has sides AC = 5√2, CD = 2√15, and DA = √10. Calculating AC² + DA² gives 50 + 10 = 60, which equals CD², so triangle ACD is right-angled at A. Therefore, angle CAD is 90°, and since angle BAC is 45°, angle BAD is 135°. Yeah, that seems correct.I think I got it. So, the measure of angle BAD is 135 degrees, which is option C.