💡Alright, so I've got this problem here: there's a cube with all its vertices on the surface of a sphere, and each edge of the cube is 2 units long. I need to find the volume of that sphere. Hmm, okay, let's break this down step by step.First off, I know that when a cube is inscribed inside a sphere, all eight of its vertices touch the sphere's surface. That means the sphere perfectly encloses the cube. So, the sphere's diameter should be related to the cube's dimensions somehow. But how exactly?I remember that for a cube, the space diagonal (which is the longest diagonal inside the cube, stretching from one vertex to the opposite vertex) is related to the edge length. Maybe that's the key here. If I can find the space diagonal of the cube, that should give me the diameter of the sphere, right? Because the space diagonal would stretch from one end of the sphere to the other, passing through the center.Okay, so let's recall the formula for the space diagonal of a cube. If the edge length is 'a', then the space diagonal 'd' is given by d = a√3. I think that's correct. Let me verify that. Yeah, in a cube, the space diagonal forms a right-angled triangle with one edge and the face diagonal. So, using the Pythagorean theorem in three dimensions, it makes sense that d = √(a² + a² + a²) = a√3.Alright, so in this case, the edge length 'a' is 2. Plugging that into the formula, the space diagonal would be 2√3. So, the space diagonal is 2√3 units long.Since the space diagonal of the cube is equal to the diameter of the sphere, that means the diameter of the sphere is also 2√3. Therefore, the radius 'r' of the sphere would be half of that, which is √3. Got it, so r = √3.Now, to find the volume of the sphere, I need to use the formula for the volume of a sphere, which is V = (4/3)πr³. I remember this formula, but let me make sure I'm applying it correctly. Yes, the volume depends on the cube of the radius, multiplied by 4/3 and π.So, plugging in the radius we found, r = √3, into the formula, we get:V = (4/3)π(√3)³.Now, let's compute (√3)³. That's the same as (√3)*(√3)*(√3). I know that √3 times √3 is 3, so then we have 3*(√3) = 3√3. So, (√3)³ = 3√3.Substituting that back into the volume formula:V = (4/3)π*(3√3).Simplifying this, the 3 in the numerator and the 3 in the denominator cancel out, leaving us with:V = 4π√3.So, the volume of the sphere is 4π√3 cubic units.Wait a second, let me double-check my calculations to make sure I didn't make any mistakes. Starting from the space diagonal: edge length 2, space diagonal 2√3, radius √3. Volume formula: (4/3)πr³. Plugging in √3 for r: (√3)³ is indeed 3√3, so (4/3)*π*3√3 simplifies to 4π√3. Yep, that seems correct.I was a bit confused at first about whether the space diagonal was the diameter or not, but thinking about it, since all the vertices are on the sphere, the farthest two vertices would be at opposite ends of the sphere, so their distance apart is the diameter. That makes sense.I also considered if there was another way to approach this problem, maybe by finding the center of the cube and then calculating the distance from the center to a vertex, which should also give the radius. Let's try that method to confirm.The center of the cube is equidistant from all its vertices. If we consider the cube with edge length 2, we can place it in a coordinate system where the center is at (1,1,1), assuming one vertex is at (0,0,0). Then, the distance from the center to any vertex, say (0,0,0), would be √[(1-0)² + (1-0)² + (1-0)²] = √(1 + 1 + 1) = √3. So, that's the radius, which matches what we found earlier. Good, that confirms our radius is indeed √3.Therefore, using either method—calculating the space diagonal or finding the distance from the center to a vertex—we arrive at the same radius, which gives us confidence in our answer.In summary, the key steps are:1. Recognize that the space diagonal of the cube is the diameter of the sphere.2. Calculate the space diagonal using the formula d = a√3, where 'a' is the edge length.3. Determine the radius by dividing the space diagonal by 2.4. Use the volume formula for a sphere, plugging in the radius.5. Simplify the expression to find the volume.I think I've covered all the necessary steps and double-checked my work to ensure there are no errors. It's always good to verify using different methods to confirm the answer, especially when dealing with geometric problems involving multiple dimensions.Another thing I thought about is visualizing the cube inside the sphere. If I imagine a cube perfectly fitting inside a sphere, all its corners just touching the sphere's surface, it helps me understand why the space diagonal is the diameter. It's like stretching a string from one corner of the cube to the opposite corner, and that string would pass through the center of the sphere, effectively being the diameter.I also considered if there's a general formula for the volume of a sphere circumscribed around a cube. It seems like it would be a function of the cube's edge length. If I denote the edge length as 'a', then the space diagonal is a√3, radius is (a√3)/2, and volume would be (4/3)π*(a√3/2)³. Simplifying that, it would be (4/3)π*(a³*(3√3)/8) = (4/3)*(3√3/8)*π*a³ = (√3/2)*π*a³. So, for any cube with edge length 'a', the volume of its circumscribed sphere is (√3/2)*π*a³.Applying this general formula to our specific case where a = 2, we get (√3/2)*π*(2)³ = (√3/2)*π*8 = 4√3π, which matches our earlier result. This gives me even more confidence that our solution is correct.I also wondered if there's a way to derive the space diagonal formula from scratch, just to understand it better. Let's see, in a cube, each edge is perpendicular to the others, so the space diagonal forms a right-angled triangle with one edge and the face diagonal. The face diagonal can be found using the Pythagorean theorem in two dimensions: if each edge is 'a', then the face diagonal is a√2. Then, using the Pythagorean theorem again in three dimensions, the space diagonal 'd' satisfies d² = a² + (a√2)² = a² + 2a² = 3a², so d = a√3. Yep, that's how the space diagonal formula is derived.Understanding the derivation helps solidify why the space diagonal is a√3 and not some other value. It's all about applying the Pythagorean theorem step by step, first in two dimensions to find the face diagonal, and then in three dimensions to find the space diagonal.I also thought about the relationship between the cube and the sphere in terms of symmetry. A cube has octahedral symmetry, and the sphere has perfect spherical symmetry. The fact that all vertices of the cube lie on the sphere means that the sphere is the circumscribed sphere or the circum sphere of the cube. This term "circumscribed" means that the sphere passes through all the vertices of the cube, which is exactly the case here.It's interesting to note that not all polyhedrons can be circumscribed around a sphere, but regular polyhedrons like the cube can. This is because their vertices are symmetrically placed, allowing a single sphere to pass through all of them.I also considered if there's a way to find the surface area of the sphere instead of the volume, just to see how that would compare. The surface area of a sphere is given by 4πr². Using our radius √3, the surface area would be 4π*(√3)² = 4π*3 = 12π. But since the problem specifically asks for the volume, we'll stick with that.Thinking about the units, since the edge length is given as 2 units, the volume of the sphere will be in cubic units. So, our final answer, 4√3π, is in cubic units, which makes sense.I also thought about approximating the volume numerically to get a sense of its magnitude. Since √3 is approximately 1.732, then 4√3 is about 6.928. Multiplying that by π (approximately 3.1416), we get roughly 6.928 * 3.1416 ≈ 21.765. So, the volume is approximately 21.765 cubic units. This helps me visualize how big the sphere is relative to the cube.But of course, since the problem doesn't specify to approximate, we'll keep the answer in its exact form, 4√3π.I also considered if there's any alternative formula or method that could be used to find the volume without directly calculating the radius. For example, maybe using the cube's volume and some proportional reasoning. But I don't think that's straightforward because the volume of the sphere isn't directly proportional to the volume of the cube in an obvious way. The relationship is more about the spatial dimensions rather than the volumes themselves.So, sticking with the method of finding the radius through the space diagonal seems to be the most efficient and accurate approach.In conclusion, by recognizing that the space diagonal of the cube is the diameter of the sphere, calculating that diagonal using the formula d = a√3, finding the radius, and then applying the volume formula for a sphere, we arrive at the volume being 4√3π cubic units. Double-checking through alternative methods and verifying each step ensures that our solution is correct.

💡First, I recognize that the mean of the distribution is 17.5, and the value that is exactly 2 standard deviations less than the mean is 12.5.To find the standard deviation (σ), I set up the equation based on the given information:Mean - 2σ = 12.5Substituting the known mean into the equation:17.5 - 2σ = 12.5Next, I solve for σ by isolating it on one side of the equation:2σ = 17.5 - 12.52σ = 5Finally, I divide both sides by 2 to find the value of σ:σ = 5 / 2σ = 2.5Therefore, the standard deviation of the distribution is 2.5.

💡Alright, so I have this problem here about a geometric sequence. Let me try to figure it out step by step. First, the problem says that all the terms of the geometric sequence {a_n} are positive numbers. It gives me that the first term, a₁, is 81, and the fifth term, a₅, is 16. I need to find the sum of the first five terms, and the options are given as A: 179, B: 211, C: 243, D: 275.Okay, so since it's a geometric sequence, I remember that each term is found by multiplying the previous term by a common ratio, which is usually denoted as 'r'. So, in general, the nth term of a geometric sequence can be written as aₙ = a₁ * r^(n-1). Given that a₁ is 81, and a₅ is 16, I can plug these into the formula to find the common ratio 'r'. Let me write that out:a₅ = a₁ * r^(5-1)16 = 81 * r⁴Hmm, so I need to solve for 'r'. Let me rearrange this equation:r⁴ = 16 / 81Alright, so r⁴ is 16 over 81. To find 'r', I need to take the fourth root of both sides. But since all the terms are positive, I don't have to worry about negative roots or anything like that. So, the fourth root of 16 is 2, because 2⁴ is 16, and the fourth root of 81 is 3, because 3⁴ is 81. So, r is 2/3.Let me double-check that. If I take (2/3)⁴, that's (2⁴)/(3⁴) which is 16/81. Yep, that's correct. So, the common ratio 'r' is 2/3.Now, I need to find the sum of the first five terms. The formula for the sum of the first n terms of a geometric sequence is Sₙ = a₁ * (1 - rⁿ) / (1 - r), provided that r ≠ 1. So, plugging in the values I have:S₅ = 81 * (1 - (2/3)⁵) / (1 - 2/3)Let me compute each part step by step. First, let's calculate (2/3)⁵. That's 2⁵ divided by 3⁵. 2⁵ is 32 and 3⁵ is 243, so (2/3)⁵ is 32/243.So, 1 - (32/243) is equal to (243/243 - 32/243) = 211/243.Now, the denominator is 1 - 2/3, which is 1/3.So, putting it all together:S₅ = 81 * (211/243) / (1/3)Dividing by (1/3) is the same as multiplying by 3, so:S₅ = 81 * (211/243) * 3Let me simplify this. 81 divided by 243 is 1/3 because 81 * 3 is 243. So, 81 * 3 is 243, which cancels out with the denominator 243. So, we're left with 1 * 211 * 1, which is 211.Wait, let me make sure I did that correctly. So, 81 multiplied by (211/243) is equal to (81/243) * 211. Since 81 is a third of 243, 81/243 is 1/3. So, 1/3 * 211 is 211/3. Then, multiplying by 3, as per the division by (1/3), gives 211/3 * 3 = 211. Yep, that seems right.So, the sum of the first five terms is 211. Looking back at the options, that's option B.Just to recap, the steps were:1. Recognize it's a geometric sequence with a₁ = 81 and a₅ = 16.2. Use the formula for the nth term to find the common ratio r.3. Calculate r⁴ = 16/81, so r = 2/3.4. Use the sum formula S₅ = a₁*(1 - r⁵)/(1 - r).5. Plug in the numbers, simplify step by step, and arrive at 211.I think that makes sense. I don't see any mistakes in my calculations, so I feel confident that the answer is B: 211.**Final Answer**boxed{B}

💡Okay, so I have this problem where there are two positive integers. Their difference is 8, and their product is 180. I need to find their sum. Hmm, let me think about how to approach this.First, I'll denote the two integers as x and y, where x is the larger number and y is the smaller one. That way, I can write the difference as x - y = 8. Also, their product is given as xy = 180.So, I have two equations:1. x - y = 82. xy = 180I need to find x + y. Hmm, how can I get the sum from these equations? Maybe I can express one variable in terms of the other and substitute.From the first equation, x = y + 8. That seems straightforward. Now, I can substitute this into the second equation.So, substituting x in the second equation gives:(y + 8) * y = 180Let me expand this:y^2 + 8y = 180Now, I can rearrange this to form a quadratic equation:y^2 + 8y - 180 = 0Okay, now I have a quadratic equation. I can try to solve this using the quadratic formula. The quadratic formula is y = [-b ± sqrt(b^2 - 4ac)] / (2a). In this equation, a = 1, b = 8, and c = -180.Calculating the discriminant first:b^2 - 4ac = 8^2 - 4*1*(-180) = 64 + 720 = 784That's a perfect square, which is good. The square root of 784 is 28.So, plugging back into the quadratic formula:y = [-8 ± 28] / 2This gives two possible solutions:1. y = (-8 + 28)/2 = 20/2 = 102. y = (-8 - 28)/2 = -36/2 = -18But since we're dealing with positive integers, y can't be negative. So, y = 10.Now, using x = y + 8, x = 10 + 8 = 18.So, the two numbers are 18 and 10. Their sum is 18 + 10 = 28.Wait, let me double-check to make sure I didn't make a mistake. If x is 18 and y is 10, then x - y is 8, which matches the first condition. Their product is 18*10 = 180, which matches the second condition. So, everything checks out.Alternatively, I could have used another method, like factoring the quadratic equation y^2 + 8y - 180 = 0. Let me see if that's possible.Looking for two numbers that multiply to -180 and add up to 8. Hmm, 18 and -10: 18 * (-10) = -180 and 18 + (-10) = 8. Perfect.So, the equation factors as (y + 18)(y - 10) = 0, giving y = -18 or y = 10. Again, since y must be positive, y = 10, and x = 18. So, the sum is 28.Another way to think about this is using the relationship between the sum and product of two numbers. If I let S = x + y and D = x - y = 8, then I know that:(x + y)^2 = x^2 + 2xy + y^2(x - y)^2 = x^2 - 2xy + y^2Subtracting these two equations:(x + y)^2 - (x - y)^2 = 4xyPlugging in the known values:S^2 - 8^2 = 4*180S^2 - 64 = 720S^2 = 720 + 64 = 784S = sqrt(784) = 28So, again, the sum is 28. This method seems a bit more abstract, but it also leads to the same answer.I think I've verified it enough ways. The sum of the two integers is definitely 28.

💡Okay, so I have this problem where I need to find the smallest and largest possible values of ( x ) given two equations involving ( x, y, z ). The equations are:[x + y + z = 5,][x^2 + y^2 + z^2 = 9.]And then, I need to find the sum of the smallest (( m )) and largest (( M )) values of ( x ). Hmm, let me think about how to approach this.First, I remember that when dealing with sums and sums of squares, sometimes it's helpful to use inequalities or maybe even think geometrically. But since this is algebra, maybe I can express ( y ) and ( z ) in terms of ( x ) and then find constraints on ( x ).Let me try that. From the first equation:[y + z = 5 - x.]And from the second equation:[y^2 + z^2 = 9 - x^2.]So now I have expressions for ( y + z ) and ( y^2 + z^2 ) in terms of ( x ). I think I can use the Cauchy-Schwarz inequality here because it relates sums and sums of squares. The Cauchy-Schwarz inequality states that for any real numbers ( a ) and ( b ):[(a^2 + b^2)(c^2 + d^2) geq (ac + bd)^2.]But in this case, maybe I can use a simpler version. I recall that for two variables, the inequality can be written as:[(y + z)^2 leq 2(y^2 + z^2).]Wait, is that right? Let me check. If I set ( a = y ), ( b = z ), ( c = 1 ), and ( d = 1 ), then Cauchy-Schwarz gives:[(y^2 + z^2)(1^2 + 1^2) geq (y cdot 1 + z cdot 1)^2,]which simplifies to:[(y^2 + z^2)(2) geq (y + z)^2.]Yes, that's correct. So, substituting the expressions I have in terms of ( x ):[2(9 - x^2) geq (5 - x)^2.]Let me write that down:[2(9 - x^2) geq (5 - x)^2.]Now, I need to solve this inequality for ( x ). Let me expand both sides.First, the left side:[2(9 - x^2) = 18 - 2x^2.]And the right side:[(5 - x)^2 = 25 - 10x + x^2.]So, putting it all together:[18 - 2x^2 geq 25 - 10x + x^2.]Hmm, let's bring all terms to one side to make it easier to solve. I'll subtract ( 25 - 10x + x^2 ) from both sides:[18 - 2x^2 - 25 + 10x - x^2 geq 0.]Simplify the terms:18 - 25 is -7, and -2x^2 - x^2 is -3x^2, and then we have +10x.So:[-3x^2 + 10x - 7 geq 0.]Hmm, this is a quadratic inequality. Let me rewrite it to make it more standard:[3x^2 - 10x + 7 leq 0.]I multiplied both sides by -1, which reverses the inequality sign.Now, I need to solve ( 3x^2 - 10x + 7 leq 0 ). To find the values of ( x ) that satisfy this inequality, I should first find the roots of the quadratic equation ( 3x^2 - 10x + 7 = 0 ).Using the quadratic formula:[x = frac{10 pm sqrt{(-10)^2 - 4 cdot 3 cdot 7}}{2 cdot 3} = frac{10 pm sqrt{100 - 84}}{6} = frac{10 pm sqrt{16}}{6} = frac{10 pm 4}{6}.]So, the roots are:[x = frac{10 + 4}{6} = frac{14}{6} = frac{7}{3},]and[x = frac{10 - 4}{6} = frac{6}{6} = 1.]So, the quadratic ( 3x^2 - 10x + 7 ) factors as ( (3x - 7)(x - 1) ). Let me verify that:[(3x - 7)(x - 1) = 3x^2 - 3x - 7x + 7 = 3x^2 - 10x + 7.]Yes, that's correct.Now, since the quadratic opens upwards (the coefficient of ( x^2 ) is positive), the inequality ( 3x^2 - 10x + 7 leq 0 ) is satisfied between the roots. So, the solution is:[1 leq x leq frac{7}{3}.]Therefore, the smallest possible value of ( x ) is ( m = 1 ), and the largest possible value is ( M = frac{7}{3} ).Hence, ( m + M = 1 + frac{7}{3} = frac{10}{3} ).Wait, let me double-check my steps to make sure I didn't make a mistake.1. I started with the given equations and expressed ( y + z ) and ( y^2 + z^2 ) in terms of ( x ). That seems correct.2. Then, I applied the Cauchy-Schwarz inequality, which in this case is a standard inequality for two variables, so that should be fine.3. Substituted the expressions into the inequality, expanded both sides, and simplified. The algebra steps seem correct.4. Solved the quadratic inequality by finding the roots, correctly noting that the quadratic opens upwards, so the inequality is satisfied between the roots. That makes sense.5. Calculated the sum ( m + M ) as ( 1 + frac{7}{3} ), which is indeed ( frac{10}{3} ).I think all the steps are correct. So, I'm confident that the answer is ( frac{10}{3} ).**Final Answer**The sum of the smallest and largest possible values of ( x ) is boxed{dfrac{10}{3}}.

💡To compute (45 times 55 + 45 times 45), I notice that both terms share a common factor of 45. This observation allows me to apply the distributive property.First, I factor out the 45 from both terms:[45 times 55 + 45 times 45 = 45 times (55 + 45)]Next, I simplify the expression inside the parentheses:[55 + 45 = 100]Now, the expression becomes:[45 times 100]Finally, I perform the multiplication:[45 times 100 = 4500]Therefore, the final answer is (boxed{4500}).