answer：Okay, so I have this equation: (2x - 3y)^2 = 4x^2 + 9y^2. I need to figure out what its graph looks like. The options are one point, the entire plane, a circle, two lines, or the empty set. Hmm, let me think step by step.First, I remember that when you have an equation like this, it might help to expand the left side and see if it simplifies to something more familiar. So, let me try expanding (2x - 3y)^2. I know that (a - b)^2 is a^2 - 2ab + b^2, so applying that here:(2x - 3y)^2 = (2x)^2 - 2*(2x)*(3y) + (3y)^2 = 4x^2 - 12xy + 9y^2.Okay, so the left side becomes 4x^2 - 12xy + 9y^2. The right side of the equation is 4x^2 + 9y^2. So, now I can set them equal:4x^2 - 12xy + 9y^2 = 4x^2 + 9y^2.Hmm, let me subtract 4x^2 + 9y^2 from both sides to see what I get:4x^2 - 12xy + 9y^2 - 4x^2 - 9y^2 = 0.Simplifying this, the 4x^2 and -4x^2 cancel out, and the 9y^2 and -9y^2 also cancel out. That leaves me with:-12xy = 0.So, -12xy = 0. Well, if I divide both sides by -12, I get:xy = 0.Now, when does the product of x and y equal zero? I remember that if a product of two things is zero, then at least one of them has to be zero. So, either x = 0 or y = 0.If x = 0, then y can be any real number. That's the equation of the y-axis, which is a vertical line. If y = 0, then x can be any real number, which is the equation of the x-axis, a horizontal line.So, the graph of this equation is the combination of the x-axis and y-axis, which are two lines intersecting at the origin.Let me double-check to make sure I didn't make any mistakes. I expanded the left side correctly, subtracted the right side, and simplified to get xy = 0. That seems right. So, the solutions are x = 0 or y = 0, which are indeed two lines.I don't think it's a circle because the equation doesn't resemble the standard form of a circle, which is (x - h)^2 + (y - k)^2 = r^2. It's also not just one point or the entire plane because we have specific lines as solutions. And it's definitely not the empty set because there are solutions.So, yeah, I'm pretty confident the answer is two lines.

answer：Okay, so I have this problem here about a curve C given by parametric equations, and I need to find the polar equation of the tangent line at a specific point. Let me try to break this down step by step.First, the parametric equations are:[begin{cases}x = sqrt{2} cos t y = sqrt{2} sin tend{cases}]where ( t ) is the parameter. Hmm, these look familiar. I think these are parametric equations for a circle because they resemble the standard parametric equations for a circle, which are ( x = r cos t ) and ( y = r sin t ). So, in this case, the radius ( r ) must be ( sqrt{2} ). Let me confirm that by squaring both equations and adding them together.Calculating ( x^2 + y^2 ):[x^2 + y^2 = (sqrt{2} cos t)^2 + (sqrt{2} sin t)^2 = 2 cos^2 t + 2 sin^2 t = 2(cos^2 t + sin^2 t) = 2(1) = 2]So, yes, the curve C is a circle centered at the origin with radius ( sqrt{2} ). Got that.Now, the problem mentions the tangent line to C at the point (1,1). I need to find the equation of this tangent line in Cartesian coordinates first, and then convert it to polar coordinates.First, let me visualize this. The circle has radius ( sqrt{2} ), so it's a bit larger than the unit circle. The point (1,1) lies on this circle because plugging into the equation:[1^2 + 1^2 = 1 + 1 = 2]Which matches the circle's equation. So, (1,1) is indeed a point on the circle.To find the tangent line at (1,1), I remember that the tangent to a circle at a given point is perpendicular to the radius at that point. So, the radius from the origin (0,0) to the point (1,1) has a certain slope, and the tangent line will have a slope that's the negative reciprocal of that.Let me find the slope of the radius first. The radius goes from (0,0) to (1,1). So, the slope ( m_r ) is:[m_r = frac{1 - 0}{1 - 0} = frac{1}{1} = 1]So, the slope of the radius is 1. Therefore, the slope of the tangent line ( m_t ) should be the negative reciprocal, which is:[m_t = -frac{1}{m_r} = -1]Alright, so the tangent line has a slope of -1 and passes through the point (1,1). Now, I can write the equation of the tangent line using the point-slope form:[y - y_1 = m(x - x_1)]Plugging in the values:[y - 1 = -1(x - 1)]Simplifying:[y - 1 = -x + 1 y = -x + 2]So, the equation of the tangent line in Cartesian coordinates is ( y = -x + 2 ). Alternatively, I can write it as:[x + y = 2]Which is the standard form.Now, I need to convert this equation into polar coordinates. The polar coordinate system has the origin as the pole and the positive x-axis as the polar axis. The conversion formulas from Cartesian to polar coordinates are:[x = rho cos theta y = rho sin theta]Where ( rho ) is the radius (distance from the origin) and ( theta ) is the angle.So, substituting ( x ) and ( y ) in the equation ( x + y = 2 ) with their polar counterparts:[rho cos theta + rho sin theta = 2]I can factor out ( rho ):[rho (cos theta + sin theta) = 2]But looking at the answer choices, they are all in the form ( rho cos theta pm rho sin theta = text{constant} ). So, my equation is already in that form:[rho cos theta + rho sin theta = 2]Which matches option A.Wait, let me double-check my steps to make sure I didn't make any mistakes.1. I converted parametric equations to Cartesian and found it's a circle with radius ( sqrt{2} ). That seems correct.2. I found the point (1,1) lies on the circle. Plugging into the equation, yes, 1 + 1 = 2, so correct.3. Calculated the slope of the radius: from (0,0) to (1,1), slope is 1. Correct.4. Therefore, the tangent slope is -1. That's the negative reciprocal, which is correct.5. Used point-slope form to find the equation of the tangent line: ( y = -x + 2 ). Correct.6. Converted to polar coordinates: substituted ( x = rho cos theta ) and ( y = rho sin theta ) into ( x + y = 2 ), resulting in ( rho (cos theta + sin theta) = 2 ). Which is the same as ( rho cos theta + rho sin theta = 2 ). So, that's option A.I don't see any mistakes in my reasoning. So, I think the correct answer is A.**Final Answer**boxed{A}

answer：Okay, so I have this problem about vectors and minimizing the dot products. Let me try to understand what's being asked here. First, the problem defines the minimum of two numbers a and b. It says that min{a, b} is a if a is less than or equal to b, otherwise it's b. So, it's just the smaller of the two numbers. Got that.Now, we have three vectors: a, b, and c. The vectors a and b have magnitudes |a| = 1 and |b| = 2. The angle between a and b is 120 degrees. Vector c is given as a linear combination of a and b: c = λa + μb, and it's also given that λ + μ = 2. The task is to find the maximum value of the minimum of the two dot products: min{c · a, c · b}. Additionally, we need to find the magnitude of vector c when this minimum is maximized.Alright, let's break this down step by step.First, I need to recall how to compute dot products. The dot product of two vectors is given by |a||b|cosθ, where θ is the angle between them. So, for vectors a and b, their dot product is |a||b|cos(120°). Since |a| is 1 and |b| is 2, this becomes 1*2*cos(120°). Cos(120°) is equal to -0.5, so the dot product a · b is 1*2*(-0.5) = -1.Next, let's think about vector c. It's given as c = λa + μb, with λ + μ = 2. So, we can express either λ or μ in terms of the other. Let's say μ = 2 - λ. Then, c = λa + (2 - λ)b.Now, we need to compute c · a and c · b. Let's compute each:1. c · a = (λa + μb) · a = λ(a · a) + μ(b · a). Since a · a is |a|² = 1, and b · a is the same as a · b, which we found to be -1. So, c · a = λ*1 + μ*(-1) = λ - μ.But since μ = 2 - λ, substitute that in: c · a = λ - (2 - λ) = 2λ - 2.2. Similarly, c · b = (λa + μb) · b = λ(a · b) + μ(b · b). a · b is -1, and b · b is |b|² = 4. So, c · b = λ*(-1) + μ*4 = -λ + 4μ.Again, substituting μ = 2 - λ: c · b = -λ + 4*(2 - λ) = -λ + 8 - 4λ = 8 - 5λ.So now, we have expressions for both c · a and c · b in terms of λ:c · a = 2λ - 2c · b = 8 - 5λOur goal is to find the maximum value of min{c · a, c · b}. So, we need to find the maximum of the smaller of these two expressions.To do this, let's first find when c · a equals c · b. That will give us the point where the two expressions cross, and beyond that point, one will be larger than the other.Set 2λ - 2 = 8 - 5λ:2λ - 2 = 8 - 5λ2λ + 5λ = 8 + 27λ = 10λ = 10/7 ≈ 1.4286So, when λ is 10/7, both c · a and c · b are equal. Let's compute that value:c · a = 2*(10/7) - 2 = 20/7 - 14/7 = 6/7 ≈ 0.8571Similarly, c · b = 8 - 5*(10/7) = 8 - 50/7 = 56/7 - 50/7 = 6/7.So, at λ = 10/7, both dot products are equal to 6/7. Now, let's analyze the behavior of min{c · a, c · b} as λ varies.1. When λ < 10/7: Let's pick a value less than 10/7, say λ = 1.Then, c · a = 2*1 - 2 = 0c · b = 8 - 5*1 = 3So, min{0, 3} = 0.2. When λ = 10/7: min{6/7, 6/7} = 6/7.3. When λ > 10/7: Let's pick λ = 2.Then, c · a = 2*2 - 2 = 2c · b = 8 - 5*2 = -2So, min{2, -2} = -2.Wait, that's interesting. So, as λ increases beyond 10/7, c · b becomes negative, which makes the minimum drop below zero.But we are looking for the maximum of the minimum. So, the maximum occurs at λ = 10/7, where the minimum is 6/7. Wait, but let's check another point. Let's take λ = 0.Then, c · a = 2*0 - 2 = -2c · b = 8 - 5*0 = 8So, min{-2, 8} = -2.Similarly, if λ approaches negative infinity, c · a becomes negative infinity, so the minimum would be negative infinity. But since λ + μ = 2, and vectors can't have negative coefficients unless we allow them, but in this problem, I think λ and μ can be any real numbers as long as their sum is 2.Wait, but in the initial problem, it's not specified whether λ and μ are positive or not. So, they can be any real numbers as long as λ + μ = 2.But in our earlier analysis, when λ = 10/7, which is approximately 1.4286, the minimum is 6/7, which is positive. When λ is less than that, the minimum is either c · a or c · b, whichever is smaller.Wait, when λ < 10/7, c · a is less than c · b. Because when λ increases, c · a increases (since it's 2λ - 2), and c · b decreases (since it's 8 - 5λ). So, for λ < 10/7, c · a < c · b, so min{c · a, c · b} = c · a.Similarly, for λ > 10/7, c · b < c · a, so min{c · a, c · b} = c · b.Therefore, the function min{c · a, c · b} is equal to:- c · a = 2λ - 2, when λ ≤ 10/7- c · b = 8 - 5λ, when λ ≥ 10/7So, to find the maximum of this function, we need to see where it peaks.For λ ≤ 10/7, the function is increasing because the coefficient of λ is positive (2). So, as λ increases, the function increases.For λ ≥ 10/7, the function is decreasing because the coefficient of λ is negative (-5). So, as λ increases beyond 10/7, the function decreases.Therefore, the maximum of the function occurs at λ = 10/7, where the function reaches 6/7.So, the maximum value of min{c · a, c · b} is 6/7.Now, we need to find the magnitude of vector c when this maximum occurs.So, when λ = 10/7, μ = 2 - λ = 2 - 10/7 = 14/7 - 10/7 = 4/7.Therefore, vector c = λa + μb = (10/7)a + (4/7)b.To find |c|, we need to compute the magnitude of this vector.First, let's express a and b in coordinate form to make the calculation easier. Let's choose a coordinate system where vector a is along the x-axis. So, a = (1, 0).Since the angle between a and b is 120 degrees, and |b| = 2, we can write b in terms of its components.The x-component of b is |b|cos(120°) = 2*(-1/2) = -1.The y-component of b is |b|sin(120°) = 2*(√3/2) = √3.So, b = (-1, √3).Therefore, vector c = (10/7)a + (4/7)b = (10/7)*(1, 0) + (4/7)*(-1, √3) = (10/7 - 4/7, 0 + 4√3/7) = (6/7, 4√3/7).Now, the magnitude of c is sqrt[(6/7)^2 + (4√3/7)^2].Compute each component:(6/7)^2 = 36/49(4√3/7)^2 = (16*3)/49 = 48/49So, |c| = sqrt[(36 + 48)/49] = sqrt[84/49] = sqrt(84)/7.Simplify sqrt(84): 84 = 4*21, so sqrt(84) = 2*sqrt(21).Therefore, |c| = (2*sqrt(21))/7.So, the maximum value of min{c · a, c · b} is 6/7, and the magnitude of c at this point is 2√21/7.Wait, let me double-check my calculations to make sure I didn't make any errors.First, when λ = 10/7, μ = 4/7.c = (10/7)a + (4/7)b.Expressed in coordinates:a = (1, 0), so (10/7)a = (10/7, 0)b = (-1, √3), so (4/7)b = (-4/7, 4√3/7)Adding these together: (10/7 - 4/7, 0 + 4√3/7) = (6/7, 4√3/7). That looks correct.Magnitude squared: (6/7)^2 + (4√3/7)^2 = 36/49 + 48/49 = 84/49 = 12/7.Wait, hold on, sqrt(84/49) is sqrt(84)/7, which is sqrt(4*21)/7 = 2*sqrt(21)/7. Yes, that's correct.So, |c| = 2√21/7.Therefore, the maximum value of the minimum is 6/7, and the magnitude of c at that point is 2√21/7.I think that's the solution.

answer：Okay, so I have this problem here: If the three numbers "lg3, lg6, lgx" are in arithmetic progression, then what is x? Hmm, arithmetic progression, right? That means the difference between consecutive terms is constant. So, the difference between the second and the first term should be the same as the difference between the third and the second term.Let me write that down. If lg3, lg6, and lgx are in arithmetic progression, then:lg6 - lg3 = lgx - lg6So, the difference between lg6 and lg3 should be equal to the difference between lgx and lg6. That makes sense because in an arithmetic progression, the difference between consecutive terms is constant.Now, I need to solve for x. Let me rearrange the equation:lg6 - lg3 = lgx - lg6If I bring the lg6 from the right side to the left side, I get:lg6 - lg3 + lg6 = lgxWhich simplifies to:2*lg6 - lg3 = lgxHmm, okay. So, 2 times lg6 minus lg3 equals lgx. I think I can use logarithmic properties to simplify this further.I remember that 2*lg6 is the same as lg(6^2) because of the power rule of logarithms, which states that a*log_b(c) = log_b(c^a). So, 2*lg6 is lg(36).Similarly, lg3 is just lg3. So, substituting back into the equation:lg(36) - lg3 = lgxNow, I can use another logarithmic property: lg(a) - lg(b) = lg(a/b). Applying that here:lg(36/3) = lgxSimplify 36 divided by 3:lg(12) = lgxSince the logarithms are equal, their arguments must be equal as well (assuming the same base, which I think is base 10 here since it's just "lg"). So, 12 = x.Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the beginning:1. The three terms are in arithmetic progression, so the difference between the second and first is equal to the difference between the third and second.2. That gives me lg6 - lg3 = lgx - lg6.3. Rearranging, I get 2*lg6 - lg3 = lgx.4. Using logarithmic properties, I converted 2*lg6 to lg(36) and then subtracted lg3, which is lg(36/3) = lg(12).5. Therefore, lgx = lg12 implies x = 12.Yeah, that seems correct. I don't think I made any calculation errors. So, x should be 12.**Final Answer**The value of ( x ) is boxed{12}.

answer：Okay, so I have to find the centers of mass for two different uniform bodies. Let me start with part a) which is a hemisphere defined by ( x^{2} + y^{2} + z^{2} leqslant R^{2} ) with ( z geqslant 0 ). Hmm, I remember that for symmetric shapes, the center of mass can be found using symmetry arguments. Since the hemisphere is symmetric around the z-axis, the x and y coordinates of the center of mass should be zero. So, I just need to find the z-coordinate.To find the z-coordinate, I think I need to set up an integral over the hemisphere. The formula for the center of mass in the z-direction is given by the integral of z over the volume divided by the total volume. So, I should first calculate the volume of the hemisphere. The volume of a full sphere is ( frac{4}{3}pi R^3 ), so the hemisphere should be half of that, which is ( frac{2}{3}pi R^3 ).Now, for the integral of z over the hemisphere. Maybe it's easier to switch to spherical coordinates since the hemisphere is symmetric. In spherical coordinates, z is ( r costheta ). The limits for r would be from 0 to R, theta from 0 to ( pi/2 ) because it's a hemisphere, and phi from 0 to ( 2pi ). So, the integral becomes:[int_{0}^{2pi} int_{0}^{pi/2} int_{0}^{R} r costheta cdot r^2 sintheta , dr , dtheta , dphi]Simplifying the integrand, I get ( r^3 costheta sintheta ). Let me separate the integrals:[int_{0}^{2pi} dphi int_{0}^{pi/2} costheta sintheta , dtheta int_{0}^{R} r^3 , dr]Calculating each integral step by step. The integral over phi is straightforward:[int_{0}^{2pi} dphi = 2pi]Next, the integral over theta. Let me make a substitution: let u = sin(theta), then du = cos(theta) d(theta). So, the integral becomes:[int_{0}^{1} u , du = frac{1}{2}]Finally, the integral over r:[int_{0}^{R} r^3 , dr = frac{R^4}{4}]Multiplying all these together:[2pi cdot frac{1}{2} cdot frac{R^4}{4} = frac{pi R^4}{4}]Now, the center of mass in the z-direction is this integral divided by the volume of the hemisphere:[z_{text{cm}} = frac{frac{pi R^4}{4}}{frac{2}{3}pi R^3} = frac{R}{4} cdot frac{3}{2} = frac{3R}{8}]So, the center of mass for the hemisphere is at ( (0, 0, frac{3R}{8}) ).Moving on to part b), which is a tetrahedron bounded by the planes ( x + 2y + z = 1 ), ( x = 0 ), ( y = 0 ), and ( z = 0 ). I need to find its center of mass. I remember that for a tetrahedron, the center of mass is the average of its vertices. But first, I should find the coordinates of the vertices.The vertices are where the planes intersect. Let's find them:1. Intersection of ( x + 2y + z = 1 ), ( y = 0 ), ( z = 0 ): ( x = 1 ). So, vertex at (1, 0, 0).2. Intersection of ( x + 2y + z = 1 ), ( x = 0 ), ( z = 0 ): ( 2y = 1 ) → ( y = frac{1}{2} ). So, vertex at (0, 1/2, 0).3. Intersection of ( x + 2y + z = 1 ), ( x = 0 ), ( y = 0 ): ( z = 1 ). So, vertex at (0, 0, 1).4. The origin (0, 0, 0).So, the four vertices are (1, 0, 0), (0, 1/2, 0), (0, 0, 1), and (0, 0, 0). The centroid (center of mass) is the average of these coordinates:[x_{text{cm}} = frac{1 + 0 + 0 + 0}{4} = frac{1}{4}][y_{text{cm}} = frac{0 + frac{1}{2} + 0 + 0}{4} = frac{1}{8}][z_{text{cm}} = frac{0 + 0 + 1 + 0}{4} = frac{1}{4}]Wait, that doesn't seem right. I think I might have made a mistake here because the plane equation is ( x + 2y + z = 1 ), so the weights might not be uniform in each direction. Maybe I should calculate the centroid using integrals instead.Let me set up the integrals for the centroid coordinates. The centroid coordinates are given by:[x_{text{cm}} = frac{1}{V} iiint x , dV][y_{text{cm}} = frac{1}{V} iiint y , dV][z_{text{cm}} = frac{1}{V} iiint z , dV]First, I need to find the volume V of the tetrahedron. The volume can be found using the triple integral over the region. Let's describe the region:From the plane equation ( x + 2y + z = 1 ), we can express z as ( z = 1 - x - 2y ). So, the limits for z are from 0 to ( 1 - x - 2y ). The projection onto the xy-plane is where ( x geq 0 ), ( y geq 0 ), and ( x + 2y leq 1 ).So, the volume integral becomes:[V = int_{x=0}^{1} int_{y=0}^{frac{1 - x}{2}} int_{z=0}^{1 - x - 2y} dz , dy , dx]Calculating the innermost integral:[int_{0}^{1 - x - 2y} dz = 1 - x - 2y]Now, the volume becomes:[V = int_{0}^{1} int_{0}^{frac{1 - x}{2}} (1 - x - 2y) , dy , dx]Let me compute the integral over y first:[int_{0}^{frac{1 - x}{2}} (1 - x - 2y) , dy = (1 - x) cdot frac{1 - x}{2} - 2 cdot frac{1}{2} left( frac{1 - x}{2} right)^2][= frac{(1 - x)^2}{2} - frac{(1 - x)^2}{4} = frac{(1 - x)^2}{4}]So, the volume integral simplifies to:[V = int_{0}^{1} frac{(1 - x)^2}{4} , dx = frac{1}{4} int_{0}^{1} (1 - 2x + x^2) , dx][= frac{1}{4} left[ x - x^2 + frac{x^3}{3} right]_{0}^{1} = frac{1}{4} left( 1 - 1 + frac{1}{3} right) = frac{1}{4} cdot frac{1}{3} = frac{1}{12}]Wait, that doesn't match my earlier thought. Maybe I made a mistake in the limits. Let me double-check. The projection onto the xy-plane is bounded by x from 0 to 1, and for each x, y goes from 0 to ( frac{1 - x}{2} ). So, the integral over y is correct. Then, integrating over x:[int_{0}^{1} frac{(1 - x)^2}{4} , dx]Let me substitute u = 1 - x, then du = -dx, and when x=0, u=1; x=1, u=0. So,[int_{1}^{0} frac{u^2}{4} (-du) = frac{1}{4} int_{0}^{1} u^2 , du = frac{1}{4} cdot frac{1}{3} = frac{1}{12}]So, the volume is ( frac{1}{12} ).Now, let's compute ( x_{text{cm}} ):[x_{text{cm}} = frac{1}{V} iiint x , dV = 12 int_{0}^{1} int_{0}^{frac{1 - x}{2}} int_{0}^{1 - x - 2y} x , dz , dy , dx]The innermost integral is:[int_{0}^{1 - x - 2y} x , dz = x(1 - x - 2y)]So,[x_{text{cm}} = 12 int_{0}^{1} int_{0}^{frac{1 - x}{2}} x(1 - x - 2y) , dy , dx]Let me expand the integrand:[x(1 - x - 2y) = x - x^2 - 2xy]So, the integral becomes:[12 int_{0}^{1} left[ int_{0}^{frac{1 - x}{2}} (x - x^2 - 2xy) , dy right] dx]Compute the inner integral term by term:1. ( int_{0}^{frac{1 - x}{2}} x , dy = x cdot frac{1 - x}{2} )2. ( int_{0}^{frac{1 - x}{2}} x^2 , dy = x^2 cdot frac{1 - x}{2} )3. ( int_{0}^{frac{1 - x}{2}} 2xy , dy = 2x cdot frac{1}{2} left( frac{1 - x}{2} right)^2 = x cdot frac{(1 - x)^2}{4} )Putting it all together:[12 int_{0}^{1} left[ x cdot frac{1 - x}{2} - x^2 cdot frac{1 - x}{2} - x cdot frac{(1 - x)^2}{4} right] dx]Simplify each term:1. ( x cdot frac{1 - x}{2} = frac{x - x^2}{2} )2. ( x^2 cdot frac{1 - x}{2} = frac{x^2 - x^3}{2} )3. ( x cdot frac{(1 - x)^2}{4} = frac{x(1 - 2x + x^2)}{4} = frac{x - 2x^2 + x^3}{4} )So, the integral becomes:[12 int_{0}^{1} left( frac{x - x^2}{2} - frac{x^2 - x^3}{2} - frac{x - 2x^2 + x^3}{4} right) dx]Combine the terms:Let me get a common denominator of 4:[12 int_{0}^{1} left( frac{2x - 2x^2 - 2x^2 + 2x^3 - x + 2x^2 - x^3}{4} right) dx]Simplify numerator:- ( 2x - x = x )- ( -2x^2 - 2x^2 + 2x^2 = -2x^2 )- ( 2x^3 - x^3 = x^3 )So, numerator is ( x - 2x^2 + x^3 ). Therefore,[12 cdot frac{1}{4} int_{0}^{1} (x - 2x^2 + x^3) , dx = 3 int_{0}^{1} (x - 2x^2 + x^3) , dx]Compute the integral:[3 left[ frac{x^2}{2} - frac{2x^3}{3} + frac{x^4}{4} right]_{0}^{1} = 3 left( frac{1}{2} - frac{2}{3} + frac{1}{4} right)]Calculate the expression inside:[frac{1}{2} - frac{2}{3} + frac{1}{4} = frac{6}{12} - frac{8}{12} + frac{3}{12} = frac{1}{12}]So,[x_{text{cm}} = 3 cdot frac{1}{12} = frac{1}{4}]Okay, that matches my initial guess for x-coordinate. Now, let's compute ( y_{text{cm}} ):[y_{text{cm}} = frac{1}{V} iiint y , dV = 12 int_{0}^{1} int_{0}^{frac{1 - x}{2}} int_{0}^{1 - x - 2y} y , dz , dy , dx]The innermost integral:[int_{0}^{1 - x - 2y} y , dz = y(1 - x - 2y)]So,[y_{text{cm}} = 12 int_{0}^{1} int_{0}^{frac{1 - x}{2}} y(1 - x - 2y) , dy , dx]Expand the integrand:[y(1 - x - 2y) = y - xy - 2y^2]So, the integral becomes:[12 int_{0}^{1} left[ int_{0}^{frac{1 - x}{2}} (y - xy - 2y^2) , dy right] dx]Compute the inner integral term by term:1. ( int_{0}^{frac{1 - x}{2}} y , dy = frac{1}{2} left( frac{1 - x}{2} right)^2 = frac{(1 - x)^2}{8} )2. ( int_{0}^{frac{1 - x}{2}} xy , dy = x cdot frac{1}{2} left( frac{1 - x}{2} right)^2 = x cdot frac{(1 - x)^2}{8} )3. ( int_{0}^{frac{1 - x}{2}} 2y^2 , dy = 2 cdot frac{1}{3} left( frac{1 - x}{2} right)^3 = frac{(1 - x)^3}{12} )Putting it all together:[12 int_{0}^{1} left( frac{(1 - x)^2}{8} - x cdot frac{(1 - x)^2}{8} - frac{(1 - x)^3}{12} right) dx]Factor out common terms:[12 int_{0}^{1} left( frac{(1 - x)^2 (1 - x)}{8} - frac{(1 - x)^3}{12} right) dx]Wait, that might not be the best approach. Let me instead combine the terms:First, let me write all terms with denominator 24 to combine:1. ( frac{(1 - x)^2}{8} = frac{3(1 - x)^2}{24} )2. ( -x cdot frac{(1 - x)^2}{8} = -frac{3x(1 - x)^2}{24} )3. ( -frac{(1 - x)^3}{12} = -frac{2(1 - x)^3}{24} )So, combining:[12 int_{0}^{1} left( frac{3(1 - x)^2 - 3x(1 - x)^2 - 2(1 - x)^3}{24} right) dx = 12 cdot frac{1}{24} int_{0}^{1} [3(1 - x)^2 - 3x(1 - x)^2 - 2(1 - x)^3] dx]Simplify the expression inside the integral:Factor out ( (1 - x)^2 ):[(1 - x)^2 [3 - 3x - 2(1 - x)] = (1 - x)^2 [3 - 3x - 2 + 2x] = (1 - x)^2 [1 - x]]So, the integral becomes:[frac{12}{24} int_{0}^{1} (1 - x)^3 dx = frac{1}{2} int_{0}^{1} (1 - x)^3 dx]Let me substitute u = 1 - x, du = -dx:[frac{1}{2} int_{1}^{0} u^3 (-du) = frac{1}{2} int_{0}^{1} u^3 du = frac{1}{2} cdot frac{1}{4} = frac{1}{8}]So, ( y_{text{cm}} = frac{1}{8} ).Finally, let's compute ( z_{text{cm}} ):[z_{text{cm}} = frac{1}{V} iiint z , dV = 12 int_{0}^{1} int_{0}^{frac{1 - x}{2}} int_{0}^{1 - x - 2y} z , dz , dy , dx]The innermost integral:[int_{0}^{1 - x - 2y} z , dz = frac{(1 - x - 2y)^2}{2}]So,[z_{text{cm}} = 12 int_{0}^{1} int_{0}^{frac{1 - x}{2}} frac{(1 - x - 2y)^2}{2} , dy , dx]Simplify:[6 int_{0}^{1} int_{0}^{frac{1 - x}{2}} (1 - x - 2y)^2 , dy , dx]Let me expand ( (1 - x - 2y)^2 ):[(1 - x - 2y)^2 = (1 - x)^2 - 4(1 - x)y + 4y^2]So, the integral becomes:[6 int_{0}^{1} int_{0}^{frac{1 - x}{2}} left( (1 - x)^2 - 4(1 - x)y + 4y^2 right) dy , dx]Compute the inner integral term by term:1. ( int_{0}^{frac{1 - x}{2}} (1 - x)^2 , dy = (1 - x)^2 cdot frac{1 - x}{2} = frac{(1 - x)^3}{2} )2. ( int_{0}^{frac{1 - x}{2}} -4(1 - x)y , dy = -4(1 - x) cdot frac{1}{2} left( frac{1 - x}{2} right)^2 = -4(1 - x) cdot frac{(1 - x)^2}{8} = -frac{(1 - x)^3}{2} )3. ( int_{0}^{frac{1 - x}{2}} 4y^2 , dy = 4 cdot frac{1}{3} left( frac{1 - x}{2} right)^3 = frac{4}{3} cdot frac{(1 - x)^3}{8} = frac{(1 - x)^3}{6} )Putting it all together:[6 int_{0}^{1} left( frac{(1 - x)^3}{2} - frac{(1 - x)^3}{2} + frac{(1 - x)^3}{6} right) dx = 6 int_{0}^{1} frac{(1 - x)^3}{6} dx = int_{0}^{1} (1 - x)^3 dx]Again, substitute u = 1 - x, du = -dx:[int_{1}^{0} u^3 (-du) = int_{0}^{1} u^3 du = frac{1}{4}]So, ( z_{text{cm}} = frac{1}{4} ).Wait, but earlier I thought it might be ( frac{1}{6} ). Hmm, let me double-check. The integral for z was:[z_{text{cm}} = 12 cdot frac{1}{2} int_{0}^{1} int_{0}^{frac{1 - x}{2}} (1 - x - 2y)^2 dy dx]But I think I might have made a mistake in the expansion or integration steps. Let me go through it again.Wait, no, the steps seem correct. The final integral gave me ( frac{1}{4} ). But I recall that for a tetrahedron, the centroid is at ( frac{1}{4} ) from each face. So, maybe it is ( frac{1}{4} ). But earlier, when I tried averaging the vertices, I got ( frac{1}{4} ) for x, ( frac{1}{8} ) for y, and ( frac{1}{4} ) for z. But I thought that was incorrect because the plane equation is ( x + 2y + z = 1 ), so the weights are different. But according to the integrals, it's ( frac{1}{4} ) for z.Wait, maybe my initial thought about the centroid being the average of the vertices is only true for a regular tetrahedron, but this one is not regular because the plane equation is ( x + 2y + z = 1 ), which stretches the tetrahedron. So, the centroid coordinates are indeed ( frac{1}{4} ) for x, ( frac{1}{8} ) for y, and ( frac{1}{4} ) for z.But wait, in the integral for z, I got ( frac{1}{4} ), but I thought earlier it might be ( frac{1}{6} ). Let me check the integral again.Wait, no, the integral for z was:[z_{text{cm}} = 12 cdot frac{1}{2} int_{0}^{1} int_{0}^{frac{1 - x}{2}} (1 - x - 2y)^2 dy dx]But after expanding and integrating, I ended up with ( frac{1}{4} ). So, I think that's correct.So, putting it all together, the center of mass for the tetrahedron is at ( left( frac{1}{4}, frac{1}{8}, frac{1}{4} right) ).Wait, but earlier I thought it might be ( frac{1}{6} ) for z. Maybe I confused it with another problem. I think the integral result is correct because it's based on the actual integration over the volume.So, final answers:a) Hemisphere: ( (0, 0, frac{3R}{8}) )b) Tetrahedron: ( left( frac{1}{4}, frac{1}{8}, frac{1}{4} right) )

answer：Okay, so I have this problem where I need to find the number of different routes from point A to point B on a 3x2 grid. I can only move to the right or down along the drawn segments. Hmm, let me try to visualize the grid first.From the Asymptote code provided, it seems like the grid is a rectangle divided into smaller rectangles. There are vertical lines at x=0, x=5, x=10, and x=15, and horizontal lines at y=0, y=5, and y=10. So, it's a 3x2 grid because there are 3 columns and 2 rows of smaller rectangles.Point A is at the top-left corner (0,10), and point B is at the bottom-right corner (15,0). So, to get from A to B, I need to move right and down. Since it's a 3x2 grid, moving from the top-left to the bottom-right would require moving right 3 times and down 2 times. That makes a total of 5 moves: 3 rights (R) and 2 downs (D).Now, the problem is similar to arranging these moves in different sequences. Each route is a unique permutation of the moves R, R, R, D, D. The number of such permutations can be calculated using combinations because the order of the same moves doesn't matter. Specifically, I need to find how many ways I can arrange 3 R's and 2 D's in a sequence of 5 moves.The formula for combinations is given by:[binom{n}{k} = frac{n!}{k!(n - k)!}]Where ( n ) is the total number of moves, and ( k ) is the number of moves in one direction. In this case, ( n = 5 ) (total moves) and ( k = 2 ) (down moves). Plugging these into the formula:[binom{5}{2} = frac{5!}{2!(5 - 2)!} = frac{5 times 4 times 3!}{2 times 1 times 3!} = frac{5 times 4}{2 times 1} = 10]So, there are 10 different routes from A to B. Let me double-check this. If I think about each step, I have to make 3 right moves and 2 down moves in some order. The number of unique sequences is indeed the number of ways to choose 2 positions out of 5 for the down moves (or equivalently, 3 positions for the right moves). Alternatively, I can think of it as a grid where each intersection is a point, and from each point, I can either go right or down. Starting from A, at each step, the number of ways to reach a point is the sum of the ways to reach the point to its left and the point above it. This is similar to Pascal's triangle.Let me try to map this out:Starting at A (0,10), which is the top-left corner. The number of ways to reach A is 1.From A, I can move right to (5,10) or down to (0,5). So, both (5,10) and (0,5) have 1 way each.From (5,10), I can move right to (10,10) or down to (5,5). Similarly, from (0,5), I can move right to (5,5) or down to (0,0). So, (10,10) has 1 way, (5,5) has 1 (from (5,10)) + 1 (from (0,5)) = 2 ways, and (0,0) has 1 way.Continuing this way:From (10,10), I can move right to (15,10) or down to (10,5). (15,10) is another corner, so it has 1 way. (10,5) has 1 way from (10,10).From (5,5), I can move right to (10,5) or down to (5,0). (10,5) already has 1 way, so adding the way from (5,5), it becomes 2. (5,0) has 1 way.From (0,0), I can move right to (5,0) or down to (0,-5), but since we can't go beyond the grid, only right is possible. So, (5,0) has 1 way from (0,0), but it already had 1 way from (5,5), so total 2 ways.From (10,5), I can move right to (15,5) or down to (10,0). (15,5) is a new point, so it gets 2 ways (from (10,5)). (10,0) gets 1 way.From (5,0), I can move right to (10,0) or down to (5,-5). Again, can't go down, so only right. (10,0) already has 1 way, adding the way from (5,0), it becomes 2.From (15,5), I can move right to (15,10) or down to (15,0). (15,10) already has 1 way, and (15,0) is point B. So, (15,0) gets 2 ways from (15,5).Wait, but according to this, point B has 2 ways? That doesn't seem right because earlier calculation gave 10 ways. Hmm, maybe I made a mistake in this grid approach.Wait, perhaps I didn't map all the points correctly. Let me try again, more carefully.Starting from A (0,10):- A: 1 wayFrom A, right to (5,10) and down to (0,5):- (5,10): 1 way- (0,5): 1 wayFrom (5,10):- Right to (10,10): 1 way- Down to (5,5): 1 wayFrom (0,5):- Right to (5,5): 1 way- Down to (0,0): 1 waySo, (10,10): 1 way(5,5): 1 (from (5,10)) + 1 (from (0,5)) = 2 ways(0,0): 1 wayFrom (10,10):- Right to (15,10): 1 way- Down to (10,5): 1 wayFrom (5,5):- Right to (10,5): 1 way- Down to (5,0): 1 wayFrom (0,0):- Right to (5,0): 1 waySo, (15,10): 1 way(10,5): 1 (from (10,10)) + 1 (from (5,5)) = 2 ways(5,0): 1 (from (5,5)) + 1 (from (0,0)) = 2 waysFrom (10,5):- Right to (15,5): 2 ways- Down to (10,0): 1 wayFrom (5,0):- Right to (10,0): 1 waySo, (15,5): 2 ways(10,0): 1 (from (10,5)) + 1 (from (5,0)) = 2 waysFrom (15,5):- Right to (15,10): already counted- Down to (15,0): 2 waysFrom (10,0):- Right to (15,0): 2 waysSo, point B (15,0) gets 2 (from (15,5)) + 2 (from (10,0)) = 4 ways?Wait, that's conflicting with the initial calculation of 10 ways. Clearly, I'm making a mistake here. Maybe the grid approach isn't being applied correctly because the grid is 3x2, meaning 4 vertical lines and 3 horizontal lines, making 3 columns and 2 rows of squares. So, the number of intersections is 4x3=12 points.Wait, perhaps I need to consider all the points and their connections properly. Let me list all the points:Vertical lines at x=0,5,10,15 (4 lines)Horizontal lines at y=0,5,10 (3 lines)So, points are (0,10), (5,10), (10,10), (15,10),(0,5), (5,5), (10,5), (15,5),(0,0), (5,0), (10,0), (15,0)Starting from A (0,10):- To (5,10): 1- To (0,5): 1From (5,10):- To (10,10): 1- To (5,5): 1From (0,5):- To (5,5): 1- To (0,0): 1From (10,10):- To (15,10): 1- To (10,5): 1From (5,5):- To (10,5): 1- To (5,0): 1From (0,0):- To (5,0): 1From (15,10):- To (15,5): 1From (10,5):- To (15,5): 1- To (10,0): 1From (5,0):- To (10,0): 1From (15,5):- To (15,0): 1From (10,0):- To (15,0): 1Now, let's compute the number of ways to each point:- A (0,10): 1- (5,10): 1- (10,10): 1- (15,10): 1- (0,5): 1- (5,5): 1 (from (5,10)) + 1 (from (0,5)) = 2- (10,5): 1 (from (10,10)) + 1 (from (5,5)) = 2- (15,5): 1 (from (15,10)) + 1 (from (10,5)) = 2- (0,0): 1- (5,0): 1 (from (5,5)) + 1 (from (0,0)) = 2- (10,0): 1 (from (10,5)) + 1 (from (5,0)) = 2- B (15,0): 1 (from (15,5)) + 1 (from (10,0)) = 2Wait, so according to this, B has only 2 ways? That contradicts the earlier calculation of 10. Clearly, something is wrong here. Maybe the grid approach isn't accounting for all possible paths because it's considering only the immediate next steps, but in reality, each point can be reached in multiple ways from different directions.Wait, no, actually, in the grid approach, each point's number of ways is the sum of the ways from the point above and the point to the left. So, let's try to compute it properly:Starting from A (0,10): 1First row (y=10):- (0,10): 1- (5,10): 1 (from (0,10))- (10,10): 1 (from (5,10))- (15,10): 1 (from (10,10))Second row (y=5):- (0,5): 1 (from (0,10))- (5,5): (from (5,10)) + (from (0,5)) = 1 + 1 = 2- (10,5): (from (10,10)) + (from (5,5)) = 1 + 2 = 3- (15,5): (from (15,10)) + (from (10,5)) = 1 + 3 = 4Third row (y=0):- (0,0): 1 (from (0,5))- (5,0): (from (5,5)) + (from (0,0)) = 2 + 1 = 3- (10,0): (from (10,5)) + (from (5,0)) = 3 + 3 = 6- (15,0): (from (15,5)) + (from (10,0)) = 4 + 6 = 10Ah, there we go! So, point B (15,0) has 10 ways. That matches the initial calculation using combinations. I must have made a mistake earlier when I was trying to compute it step by step without properly summing the ways from all possible previous points.So, the correct number of routes is 10. This makes sense because it's a 3x2 grid, requiring 3 right moves and 2 down moves, and the number of permutations is 5 choose 2, which is 10.