💡Okay, so I have this function here: f(x) = (1/3)x³ + (1/2)x² - mx + 8. The question is asking about the range of m for which this function has extreme values. Hmm, extreme values... that means I need to find when the function has maximum or minimum points, right?Alright, to find extreme values, I remember that I need to take the derivative of the function and set it equal to zero. The solutions to that equation will give me the critical points, which could be maxima or minima. So, let's start by finding the derivative of f(x).The derivative of (1/3)x³ is x², because 3*(1/3) is 1, and then we reduce the exponent by 1, so x². Then, the derivative of (1/2)x² is x, since 2*(1/2) is 1, and then x¹ is just x. The derivative of -mx is -m, because the derivative of mx with respect to x is m, so with the negative sign, it's -m. And the derivative of 8, a constant, is zero. So putting it all together, f'(x) = x² + x - m.Okay, so f'(x) = x² + x - m. Now, to find the critical points, I need to solve f'(x) = 0. That means solving x² + x - m = 0. This is a quadratic equation in terms of x. For this quadratic equation to have real solutions, the discriminant must be positive. If the discriminant is zero, there's exactly one real solution, and if it's negative, there are no real solutions. Since we want extreme values, we need two distinct real solutions, which means the discriminant must be greater than zero.The discriminant of a quadratic equation ax² + bx + c = 0 is given by D = b² - 4ac. In this case, a is 1, b is 1, and c is -m. So, plugging these into the discriminant formula, we get D = (1)² - 4*(1)*(-m) = 1 + 4m.For the quadratic to have two distinct real roots, D must be greater than zero. So, 1 + 4m > 0. Let's solve this inequality for m. Subtracting 1 from both sides gives 4m > -1. Then, dividing both sides by 4, we get m > -1/4.So, m has to be greater than -1/4 for the function f(x) to have extreme values. That means the range of m is all real numbers greater than -1/4.Wait, let me double-check that. If m is greater than -1/4, then 1 + 4m is positive, so the discriminant is positive, which means two real roots. That makes sense. If m were equal to -1/4, the discriminant would be zero, so there would be exactly one critical point, which is a point of inflection, not an extreme value. And if m were less than -1/4, the discriminant would be negative, meaning no real critical points, so no extreme values. So yes, m must be greater than -1/4.I think that's it. So the range of m is m > -1/4.

💡Alright, so I have this geometry problem here. Let me try to parse it step by step. We have three distinct points A, B, and C on a circle O, where O is the origin. The angle between vectors OA and OB is 2π/3. That's 120 degrees, right? So, points A and B are separated by 120 degrees on the circle. Then, it says that vector OC is equal to λ times vector OA plus μ times vector OB, where λ and μ are real numbers. So, OC is a linear combination of OA and OB. That makes sense because in a plane, any vector can be expressed as a combination of two non-parallel vectors, which OA and OB are since the angle between them is 120 degrees, not 180.The problem asks for the ratio λ/μ when ω, defined as sqrt(3)λ + λ + μ, reaches its maximum value. So, ω is a linear function of λ and μ, and we need to maximize it under the given condition that OC is a combination of OA and OB.Hmm, okay. Let me think about how to approach this. Maybe I can parameterize the points A, B, and C on the unit circle. Since O is the origin, let's assume the circle has radius 1 for simplicity. Let me assign coordinates to points A and B. Since angle AOB is 2π/3, I can place point A at (1, 0) for simplicity. Then, point B will be at an angle of 2π/3 from A. So, in coordinates, that would be (cos(2π/3), sin(2π/3)). Calculating that, cos(2π/3) is -1/2, and sin(2π/3) is sqrt(3)/2. So, point B is at (-1/2, sqrt(3)/2).Now, point C is another point on the circle, so its coordinates can be represented as (cos α, sin α) for some angle α. The vector OC is given by λ OA + μ OB. So, in coordinates, that would be:OC = λ*(1, 0) + μ*(-1/2, sqrt(3)/2) = (λ - μ/2, 0 + μ*sqrt(3)/2) = (λ - μ/2, (μ sqrt(3))/2)But we also know that OC is (cos α, sin α). So, equating the components:cos α = λ - μ/2sin α = (μ sqrt(3))/2So, now we have two equations:1. cos α = λ - (μ)/22. sin α = (μ sqrt(3))/2Our goal is to express ω in terms of α and then find its maximum. But ω is given as sqrt(3)λ + λ + μ. Wait, that seems a bit odd. Let me check:ω = sqrt(3)λ + λ + μWait, is that sqrt(3)λ + λ + μ? That would be equivalent to (sqrt(3) + 1)λ + μ. Hmm, okay.So, let's express λ and μ in terms of α from the two equations above.From equation 2: sin α = (μ sqrt(3))/2. So, solving for μ:μ = (2 sin α)/sqrt(3)Similarly, from equation 1: cos α = λ - μ/2. Substituting μ from above:cos α = λ - ( (2 sin α)/sqrt(3) ) / 2 = λ - (sin α)/sqrt(3)So, solving for λ:λ = cos α + (sin α)/sqrt(3)So now, we have expressions for both λ and μ in terms of α:λ = cos α + (sin α)/sqrt(3)μ = (2 sin α)/sqrt(3)Now, let's substitute these into ω:ω = (sqrt(3) + 1)λ + μ= (sqrt(3) + 1)(cos α + (sin α)/sqrt(3)) + (2 sin α)/sqrt(3)Let me expand this:= (sqrt(3) + 1)cos α + (sqrt(3) + 1)(sin α)/sqrt(3) + (2 sin α)/sqrt(3)Simplify each term:First term: (sqrt(3) + 1)cos αSecond term: (sqrt(3) + 1)/sqrt(3) * sin αThird term: 2/sqrt(3) * sin αLet me combine the second and third terms:= [ (sqrt(3) + 1)/sqrt(3) + 2/sqrt(3) ] sin αFactor out 1/sqrt(3):= [ (sqrt(3) + 1 + 2) / sqrt(3) ] sin α= [ (sqrt(3) + 3) / sqrt(3) ] sin αSimplify numerator:sqrt(3) + 3 = sqrt(3)(1 + sqrt(3))So,= [ sqrt(3)(1 + sqrt(3)) / sqrt(3) ] sin α= (1 + sqrt(3)) sin αSo, putting it all together, ω is:ω = (sqrt(3) + 1)cos α + (1 + sqrt(3)) sin αFactor out (sqrt(3) + 1):ω = (sqrt(3) + 1)(cos α + sin α)Interesting. So, ω is proportional to (cos α + sin α). To maximize ω, we need to maximize (cos α + sin α).I remember that the maximum value of cos α + sin α is sqrt(2), achieved when α = π/4, because cos α + sin α = sqrt(2) sin(α + π/4), which has a maximum of sqrt(2).So, the maximum value of ω is (sqrt(3) + 1)*sqrt(2). But we need to find the ratio λ/μ when ω is maximized.So, we need to find λ and μ when α = π/4.Let me compute λ and μ at α = π/4.First, sin(π/4) = sqrt(2)/2, cos(π/4) = sqrt(2)/2.Compute μ:μ = (2 sin α)/sqrt(3) = (2*(sqrt(2)/2))/sqrt(3) = sqrt(2)/sqrt(3) = sqrt(6)/3Compute λ:λ = cos α + (sin α)/sqrt(3) = sqrt(2)/2 + (sqrt(2)/2)/sqrt(3) = sqrt(2)/2 + sqrt(2)/(2 sqrt(3))Factor sqrt(2)/2:= sqrt(2)/2 [1 + 1/sqrt(3)] = sqrt(2)/2 [ (sqrt(3) + 1)/sqrt(3) ]Simplify:= sqrt(2)(sqrt(3) + 1)/(2 sqrt(3)) = sqrt(2)(sqrt(3) + 1)/(2 sqrt(3))Multiply numerator and denominator by sqrt(3) to rationalize:= sqrt(2)(sqrt(3) + 1)sqrt(3)/(2*3) = sqrt(2)(3 + sqrt(3))/6So, λ = sqrt(2)(3 + sqrt(3))/6And μ = sqrt(6)/3So, the ratio λ/μ is:[ sqrt(2)(3 + sqrt(3))/6 ] / [ sqrt(6)/3 ] = [ sqrt(2)(3 + sqrt(3))/6 ] * [3 / sqrt(6) ]Simplify:= [ sqrt(2)(3 + sqrt(3)) * 3 ] / [6 sqrt(6) ]Simplify numerator and denominator:3/6 = 1/2, so:= [ sqrt(2)(3 + sqrt(3)) ] / [2 sqrt(6) ]Note that sqrt(6) = sqrt(2)*sqrt(3), so:= [ sqrt(2)(3 + sqrt(3)) ] / [2 sqrt(2) sqrt(3) ] = [ (3 + sqrt(3)) ] / [2 sqrt(3) ]Multiply numerator and denominator by sqrt(3):= [ (3 + sqrt(3)) sqrt(3) ] / [2 * 3 ] = [ 3 sqrt(3) + 3 ] / 6 = [ 3(sqrt(3) + 1) ] / 6 = (sqrt(3) + 1)/2So, the ratio λ/μ is (sqrt(3) + 1)/2 when ω is maximized.Let me just double-check my steps to make sure I didn't make any mistakes.1. Assigned coordinates correctly for A and B.2. Expressed OC as a combination of OA and OB, leading to equations for cos α and sin α in terms of λ and μ.3. Solved for λ and μ in terms of α.4. Substituted into ω and simplified, factoring out (sqrt(3) + 1).5. Recognized that to maximize ω, we need to maximize (cos α + sin α), which is sqrt(2) at α = π/4.6. Computed λ and μ at α = π/4.7. Calculated the ratio λ/μ, simplifying step by step.Everything seems to check out. So, the final answer should be (sqrt(3) + 1)/2.**Final Answer**The ratio (frac{lambda}{mu}) when (omega) reaches its maximum value is boxed{dfrac{sqrt{3} + 1}{2}}.

💡First, I need to determine the side length of the square field. Given the area of the square is 3136 square meters, I can find the side length by taking the square root of the area.Next, I'll calculate the perimeter of the square field, which is four times the side length. This will give me the total length of barbed wire needed if there were no gates.However, there are two gates, each 1 meter wide, that need to be accounted for. I'll subtract the total width of the gates from the perimeter to find the actual length of barbed wire required.Finally, I'll multiply the length of the barbed wire needed by the cost per meter to find the total cost of drawing the barbed wire around the field.

💡Okay, so I have this problem here about the Belt and Road Initiative, and it's asking me to express the number 4,500,000,000 in scientific notation. The options are A: 4.5×10⁷, B: 45×10⁸, C: 4.5×10⁹, and D: 0.45×10¹⁰. Hmm, I need to figure out which one is correct.First, I remember that scientific notation is a way to write really big or really small numbers more simply. It's usually written as a number between 1 and 10 multiplied by 10 raised to some power. So, in this case, I need to take 4,500,000,000 and write it in that form.Okay, let's start by looking at 4,500,000,000. I need to move the decimal point so that there's only one non-zero digit in front of it. Right now, the number is 4,500,000,000, which has a decimal point at the end, like 4,500,000,000.0. If I move the decimal point to the left, I can count how many places I move it until there's only one digit in front.So, starting at the end, I move the decimal point one place to the left: 450,000,000.0. That's one place. Move it again: 45,000,000.0. That's two places. Again: 4,500,000.0. Three places. Again: 450,000.0. Four places. Again: 45,000.0. Five places. Again: 4,500.0. Six places. Again: 450.0. Seven places. Again: 45.0. Eight places. And finally: 4.5. Nine places.So, I moved the decimal point nine places to the left to get from 4,500,000,000 to 4.5. In scientific notation, that would be 4.5 multiplied by 10 raised to the power of how many places I moved the decimal, which is 9. So, that would be 4.5×10⁹.Let me check the options again. Option C is 4.5×10⁹, which matches what I just figured out. Option A is 4.5×10⁷, which would be 450,000,000, way too small. Option B is 45×10⁸, which is actually the same as 4.5×10⁹ because 45×10⁸ is 4,500,000,000. Wait, that's also correct. Hmm, so both B and C seem correct?But wait, in scientific notation, the number before the multiplication should be between 1 and 10. So, 45×10⁸ is not in proper scientific notation because 45 is greater than 10. It should be adjusted to 4.5×10⁹. So, even though 45×10⁸ equals 4,500,000,000, it's not the correct way to write it in scientific notation. Therefore, option C is the right one.Option D is 0.45×10¹⁰. That would also equal 4,500,000,000, but again, 0.45 is less than 1, which doesn't fit the standard form of scientific notation where the coefficient should be between 1 and 10. So, D is also not correct.So, after considering all the options, I think the correct answer is C: 4.5×10⁹.

💡Okay, so I have this problem where I need to find the probability of selecting two plates of the same color when I randomly pick two plates from a collection. Let me try to break this down step by step.First, I need to understand the total number of plates I have. The problem says there are 7 red plates, 5 blue plates, and 3 green plates. So, adding those up: 7 + 5 + 3 equals 15 plates in total. That seems straightforward.Now, I need to find the probability that both plates I pick are the same color. Probability is generally calculated as the number of favorable outcomes divided by the total number of possible outcomes. So, in this case, the favorable outcomes are picking two plates of the same color, and the total possible outcomes are picking any two plates from the 15.Let me think about how to calculate the total number of ways to pick two plates from 15. I remember that combinations are used here because the order in which I pick the plates doesn't matter. The formula for combinations is C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items to choose.So, the total number of ways to pick two plates from 15 is C(15, 2). Plugging in the numbers: 15! / (2! * (15 - 2)!) = (15 * 14) / (2 * 1) = 105. Okay, so there are 105 possible pairs of plates.Next, I need to find the number of favorable outcomes, which are the pairs where both plates are the same color. I'll need to calculate this for each color and then add them up.Starting with red plates: there are 7 red plates. The number of ways to pick two red plates is C(7, 2). Using the combination formula again: (7 * 6) / (2 * 1) = 21. So, there are 21 ways to pick two red plates.Moving on to blue plates: there are 5 blue plates. The number of ways to pick two blue plates is C(5, 2). Calculating that: (5 * 4) / (2 * 1) = 10. So, there are 10 ways to pick two blue plates.Now, for green plates: there are 3 green plates. The number of ways to pick two green plates is C(3, 2). That would be (3 * 2) / (2 * 1) = 3. So, there are 3 ways to pick two green plates.Adding up all the favorable outcomes: 21 (red) + 10 (blue) + 3 (green) = 34. So, there are 34 favorable pairs where both plates are the same color.Finally, to find the probability, I divide the number of favorable outcomes by the total number of possible outcomes: 34 / 105. That simplifies to approximately 0.3238, or 32.38%.Wait, let me double-check my calculations to make sure I didn't make any mistakes. Total plates: 7 + 5 + 3 = 15. Total combinations: C(15, 2) = 105. Red combinations: C(7, 2) = 21. Blue combinations: C(5, 2) = 10. Green combinations: C(3, 2) = 3. Total same-color combinations: 21 + 10 + 3 = 34. Probability: 34 / 105. Yep, that seems correct.I think that's it. The probability of picking two plates of the same color is 34 out of 105, which can also be written as a fraction in its simplest form. Let me see if 34 and 105 have any common factors. 34 divided by 2 is 17, and 105 divided by 5 is 21. There are no common factors between 34 and 105 besides 1, so the fraction is already in its simplest form.So, the final answer is 34/105.

💡Alright, so I have this problem about a cube with side length 8 cm. The vertices are alternately colored black and red, and I need to find the volume of the tetrahedron formed by the red vertices. Hmm, okay. Let me try to visualize this.First, a cube has 8 vertices. If they're alternately colored black and red, that means half of them are red and half are black. So, there are 4 red vertices and 4 black vertices. These red vertices form a tetrahedron inside the cube. I remember that a tetrahedron is a three-dimensional shape with four triangular faces. So, connecting these four red vertices should give me that shape.Now, to find the volume of this tetrahedron. I know that the volume of a tetrahedron can be calculated if I know the coordinates of its vertices, but I don't have coordinates here. Maybe I can figure out the coordinates based on the cube's structure.Let me assign coordinates to the cube. Let's say the cube is positioned in a 3D coordinate system with one corner at the origin (0,0,0) and extending to (8,8,8). Since the vertices are alternately colored, the red vertices would be at positions where the sum of the coordinates is even or something like that? Wait, actually, in a cube, if you color vertices alternately, it's similar to a checkerboard pattern but in three dimensions.So, in a cube, each vertex can be represented by a binary coordinate (x,y,z) where each of x, y, z is either 0 or 8. If we color a vertex red if the number of 8s in its coordinates is even, and black otherwise, or something like that. Let me think.Alternatively, maybe it's simpler to consider that in a cube, the two sets of four vertices (red and black) form two tetrahedrons that are duals of each other. So, each tetrahedron is inscribed within the cube. That makes sense.I remember that the volume of such a tetrahedron can be calculated using the formula for the volume of a regular tetrahedron, but wait, is this tetrahedron regular? In a cube, the edges of the tetrahedron would be the face diagonals or space diagonals of the cube.Let me see. If I take two opposite vertices of the cube, the distance between them is the space diagonal, which is 8√3 cm. But the edges of the tetrahedron would actually be the face diagonals, right? Because each edge of the tetrahedron connects two vertices that are on the same face of the cube but not adjacent. So, the edge length of the tetrahedron is the face diagonal of the cube.The face diagonal of a cube with side length a is a√2. So, in this case, it's 8√2 cm. So, each edge of the tetrahedron is 8√2 cm.Wait, but is that correct? Let me think again. If I have a cube, and I connect every other vertex, the edges of the tetrahedron would actually be the space diagonals of the cube's faces, which are indeed face diagonals. So, yes, each edge is 8√2 cm.But hold on, a regular tetrahedron has all edges equal, which is the case here, so this is a regular tetrahedron. Therefore, I can use the formula for the volume of a regular tetrahedron, which is (edge length³)/(6√2). Let me write that down.Volume = (a³)/(6√2), where a is the edge length.So, plugging in a = 8√2 cm:Volume = ( (8√2)³ ) / (6√2 )Let me compute (8√2)³ first.(8√2)³ = 8³ * (√2)³ = 512 * (2√2) = 1024√2So, Volume = 1024√2 / (6√2) = 1024 / 6 = 170.666...Hmm, that's approximately 170.67 cm³. But wait, the cube's volume is 8³ = 512 cm³. If the tetrahedron's volume is 170.67, then the remaining volume is 512 - 170.67 = 341.33 cm³, which would be occupied by the other tetrahedron and some other spaces? Wait, no, actually, in the cube, the two tetrahedrons (red and black) each have the same volume, right? Because the cube is symmetric.Wait, but 170.67 * 2 = 341.34, which is less than 512. So, that doesn't make sense. There must be something wrong with my approach.Maybe I'm miscalculating the edge length. Let me double-check. If the edge length of the tetrahedron is the face diagonal of the cube, which is 8√2 cm, then the formula should be correct. But perhaps the formula for the volume of a regular tetrahedron is different?Wait, no, the formula is correct. Volume = (edge length³)/(6√2). So, let me recalculate:(8√2)³ = 8³ * (√2)³ = 512 * (2√2) = 1024√2Then, 1024√2 divided by 6√2 is indeed 1024/6 = 170.666...But wait, I think I'm confusing something here. Maybe the edge length isn't 8√2? Let me think about the cube again.In a cube, the distance between two vertices that are connected by an edge is 8 cm. The distance between two vertices on the same face but not connected by an edge is the face diagonal, which is 8√2 cm. The distance between two opposite vertices of the cube (space diagonal) is 8√3 cm.But in the tetrahedron formed by the red vertices, each edge is a face diagonal, right? Because each red vertex is connected to three other red vertices, each on a different face. So, yes, the edge length is 8√2 cm.Wait, but then the volume calculation gives me 170.67 cm³, which is less than half of the cube's volume. But if the cube is divided into two tetrahedrons, each should occupy half the cube's volume, right? Because the cube can be split into two congruent tetrahedrons.Wait, no, actually, a cube can be divided into five tetrahedrons, but in this case, with the alternating colors, it's divided into two tetrahedrons and some other spaces? Hmm, maybe I'm wrong.Alternatively, perhaps the tetrahedron formed by the red vertices doesn't occupy half the cube. Let me think differently.Maybe instead of calculating the volume directly, I can use the cube's volume and subtract the volumes of the other parts.Wait, in the cube, the red tetrahedron is one of the two possible regular tetrahedrons you can inscribe in a cube. The other one is formed by the black vertices. So, together, they don't fill the entire cube, but rather, each occupies a certain portion.Wait, actually, I think each of these tetrahedrons has a volume of 1/3 of the cube's volume. Let me check.If the cube's volume is 512 cm³, then 1/3 of that is approximately 170.67 cm³, which matches my earlier calculation. So, maybe each tetrahedron has a volume of 170.67 cm³, and the remaining volume is occupied by other structures.But wait, that doesn't seem right because if you have two tetrahedrons, each with 1/3 of the cube's volume, that would only account for 2/3 of the cube's volume. The remaining 1/3 must be something else.Alternatively, perhaps the volume of the tetrahedron is 1/6 of the cube's volume? Let me see.If I take the cube and divide it into six square pyramids, each with a face as the base and the center of the cube as the apex. Each pyramid would have a volume of (8*8*8)/6 = 512/6 ≈ 85.33 cm³. But that's not directly related to the tetrahedron.Wait, maybe I should consider the cube being divided into five tetrahedrons, but that's more complicated.Alternatively, perhaps I can use coordinates to calculate the volume.Let me assign coordinates to the cube. Let's say the cube has vertices at (0,0,0), (8,0,0), (0,8,0), (0,0,8), and so on up to (8,8,8). Now, if I color the vertices alternately, the red vertices would be, for example, (0,0,0), (8,8,0), (8,0,8), and (0,8,8). Let me check if these are indeed alternately colored.Yes, because each of these vertices has an even number of 8s in their coordinates, or something like that. Wait, actually, in a cube, the parity of the number of 1s in the binary representation of the coordinates determines the color. So, if we consider the cube as a 3D grid with coordinates (x,y,z) where x, y, z are either 0 or 8, then the parity (even or odd number of 8s) determines the color. So, vertices with even number of 8s are red, and odd are black, or vice versa.So, (0,0,0) has zero 8s, which is even, so red. (8,8,0) has two 8s, even, red. (8,0,8) has two 8s, red. (0,8,8) has two 8s, red. So, these four are red. The other four vertices have one or three 8s, which are odd, so black.Okay, so the red vertices are (0,0,0), (8,8,0), (8,0,8), and (0,8,8). Now, to find the volume of the tetrahedron formed by these four points.I can use the formula for the volume of a tetrahedron given by four points in space. The formula is:Volume = | ( (B - A) · [ (C - A) × (D - A) ] ) | / 6Where A, B, C, D are the four points.Let me assign A = (0,0,0), B = (8,8,0), C = (8,0,8), D = (0,8,8).So, vectors:B - A = (8,8,0)C - A = (8,0,8)D - A = (0,8,8)Now, compute the cross product of (C - A) and (D - A):(8,0,8) × (0,8,8) = determinant of the matrix:|i   j   k||8    0    8||0    8    8|= i*(0*8 - 8*8) - j*(8*8 - 0*8) + k*(8*8 - 0*0)= i*(0 - 64) - j*(64 - 0) + k*(64 - 0)= (-64, -64, 64)Now, take the dot product of (B - A) with this cross product:(8,8,0) · (-64, -64, 64) = 8*(-64) + 8*(-64) + 0*64 = -512 -512 + 0 = -1024Take the absolute value and divide by 6:| -1024 | / 6 = 1024 / 6 ≈ 170.666...So, the volume is 1024/6 cm³, which simplifies to 512/3 cm³, approximately 170.67 cm³.But wait, earlier I thought that the cube's volume is 512, and this tetrahedron is 512/3, which is about 170.67, and the other tetrahedron would also be 512/3, totaling 1024/3 ≈ 341.33, which is less than 512. So, what's the remaining volume?Ah, I see. The cube is actually divided into six smaller tetrahedrons, each with volume 512/6 ≈ 85.33 cm³. But in this case, the red tetrahedron is one of the larger ones, occupying 512/3 cm³, and the black tetrahedron is another 512/3 cm³, and the remaining space is filled by other structures.Wait, no, actually, the cube can be divided into five tetrahedrons, but that's a different approach. In this case, the two tetrahedrons (red and black) each have a volume of 512/3 cm³, and the remaining volume is occupied by other parts of the cube.But regardless, according to the calculation, the volume of the red tetrahedron is 512/3 cm³, which is approximately 170.67 cm³. However, the problem asks for the volume, so I should present it as an exact value rather than a decimal.So, 512 divided by 3 is 170 and 2/3, which is 170.666... cm³. But since the problem might expect an exact value, I can write it as 512/3 cm³.Wait, but let me double-check my calculation because earlier I thought the edge length was 8√2, and using the regular tetrahedron formula gave me the same result, 512/3 cm³. So, that seems consistent.Alternatively, another way to think about it is that the cube can be divided into six square pyramids, each with a face as the base and the cube's center as the apex. Each pyramid has a volume of (8*8*8)/6 = 512/6 ≈ 85.33 cm³. But that's not directly related to the tetrahedron.Wait, actually, the tetrahedron formed by the red vertices can be seen as combining four of these pyramids. Let me see.Each pyramid has a volume of 512/6, so four pyramids would be 4*(512/6) = 2048/6 ≈ 341.33 cm³, which is more than the cube's volume. That doesn't make sense.Alternatively, maybe the tetrahedron is made up of smaller tetrahedrons. Wait, perhaps not. Maybe it's better to stick with the coordinate method.So, using the coordinates, I calculated the volume as 512/3 cm³, which is approximately 170.67 cm³. That seems correct.But wait, another way to think about it is that the tetrahedron is a regular tetrahedron inscribed in the cube, and its volume is 1/3 of the cube's volume. Since the cube's volume is 512, 1/3 of that is 512/3, which is the same result.So, I think that's the correct answer. Therefore, the volume of the tetrahedron formed by the red vertices is 512/3 cm³, which is approximately 170.67 cm³.But let me just confirm once more. If I consider the cube with side length 8, the space diagonal is 8√3. The regular tetrahedron inscribed in the cube has edge length equal to the face diagonal, which is 8√2. The volume formula for a regular tetrahedron is (edge length³)/(6√2). Plugging in 8√2:Volume = ( (8√2)³ ) / (6√2 ) = (512 * 2√2 ) / (6√2 ) = (1024√2 ) / (6√2 ) = 1024/6 = 512/3.Yes, that's consistent. So, the volume is indeed 512/3 cm³.Therefore, the final answer is 512/3 cm³, which can be written as boxed{dfrac{512}{3}} cm³.