💡Okay, so I have this problem about a triangular pyramid, which is also called a tetrahedron. The problem says that all the lateral edges have the same length, which is 'a'. At the apex of the pyramid, the three plane angles formed by these edges are two 45-degree angles and one 60-degree angle. I need to find the volume of this pyramid.First, let me try to visualize the pyramid. It has a triangular base, and the apex is connected to each vertex of the base by edges of length 'a'. The angles between these edges at the apex are given: two of them are 45 degrees, and one is 60 degrees.I remember that the volume of a pyramid is given by one-third the area of the base multiplied by the height. So, Volume = (1/3) * Base Area * Height. So, I need to find the area of the base and the height of the pyramid.But wait, the base is a triangle, but I don't know its specific shape yet. Maybe I can figure it out using the given angles at the apex.Let me denote the apex as point F, and the base triangle as ABC. So, the edges FA, FB, and FC are all of length 'a'. The angles at F are: angle AFC is 60 degrees, and angles AFB and BFC are 45 degrees each.Hmm, so at the apex F, the edges FA, FB, FC form a sort of three-dimensional angle. Maybe I can model this using vectors or coordinates.Let me try to assign coordinates to the points. Let's place point F at the origin (0, 0, 0). Then, I can define points A, B, and C in 3D space such that the distances FA, FB, FC are all 'a', and the angles between the edges are as given.Since the angles between FA, FB, and FC are given, I can use the dot product formula to find the coordinates of A, B, and C.Let me denote vectors FA, FB, FC as vectors **a**, **b**, **c** respectively.Given that the angle between FA and FB is 45 degrees, the angle between FB and FC is 45 degrees, and the angle between FA and FC is 60 degrees.Since all edges are of length 'a', the magnitude of each vector is 'a'.The dot product formula is: **a** · **b** = |**a**||**b**|cosθ.So, for vectors FA and FB:**a** · **b** = a * a * cos45° = a² * (√2 / 2)Similarly, **b** · **c** = a² * (√2 / 2)And **a** · **c** = a² * cos60° = a² * (1/2)Now, I can assign coordinates to points A, B, and C such that these dot products are satisfied.Let me choose coordinate system such that point F is at (0, 0, 0), point A is along the x-axis, point B is in the xy-plane, and point C is somewhere in 3D space.So, let me assign:Point A: (a, 0, 0)Point B: (x, y, 0)Point C: (p, q, r)Now, since FA is along the x-axis, FA vector is (a, 0, 0). The vector FB is (x, y, 0), and vector FC is (p, q, r).Given that |FB| = a, so sqrt(x² + y²) = a.Similarly, |FC| = a, so sqrt(p² + q² + r²) = a.Now, let's compute the dot products.First, **a** · **b** = (a, 0, 0) · (x, y, 0) = a*x + 0*y + 0*0 = a*x.We know this equals a² * (√2 / 2), so:a*x = a² * (√2 / 2) => x = a * (√2 / 2)Similarly, **b** · **c** = (x, y, 0) · (p, q, r) = x*p + y*q + 0*r = x*p + y*q.This should equal a² * (√2 / 2).And **a** · **c** = (a, 0, 0) · (p, q, r) = a*p + 0*q + 0*r = a*p.This should equal a² * (1/2), so:a*p = a² * (1/2) => p = a * (1/2)So, p = a/2.Now, let's summarize what we have so far.Point A: (a, 0, 0)Point B: (a√2/2, y, 0)Point C: (a/2, q, r)We also know that |FB| = a, so sqrt( (a√2/2)^2 + y^2 ) = a.Calculating (a√2/2)^2 = (a² * 2)/4 = a²/2.So, sqrt(a²/2 + y²) = a => a²/2 + y² = a² => y² = a² - a²/2 = a²/2 => y = a√2/2 or y = -a√2/2.Since we're dealing with a pyramid, let's assume y is positive, so y = a√2/2.So, point B is (a√2/2, a√2/2, 0)Similarly, for point C, we have |FC| = a, so sqrt( (a/2)^2 + q^2 + r^2 ) = a.Calculating (a/2)^2 = a²/4.So, sqrt(a²/4 + q² + r²) = a => a²/4 + q² + r² = a² => q² + r² = (3/4)a².Now, we also have **b** · **c** = x*p + y*q = (a√2/2)*(a/2) + (a√2/2)*q.Which is (a²√2)/4 + (a√2/2)*q.This should equal a² * (√2 / 2).So, let's set up the equation:(a²√2)/4 + (a√2/2)*q = (a²√2)/2Subtract (a²√2)/4 from both sides:(a√2/2)*q = (a²√2)/2 - (a²√2)/4 = (a²√2)/4Divide both sides by (a√2/2):q = (a²√2)/4 / (a√2/2) = (a²√2)/4 * 2/(a√2) = (a²√2 * 2) / (4a√2) = (2a²√2) / (4a√2) = (2a²√2) / (4a√2) = (a)/2So, q = a/2.Therefore, point C is (a/2, a/2, r)Now, from earlier, we have q² + r² = (3/4)a².Substituting q = a/2:(a/2)^2 + r² = (3/4)a² => a²/4 + r² = 3a²/4 => r² = 3a²/4 - a²/4 = 2a²/4 = a²/2 => r = a/√2 or r = -a/√2.Again, since it's a pyramid, let's take r positive, so r = a/√2.So, point C is (a/2, a/2, a/√2)Now, we have coordinates for points A, B, C, and F.Point F: (0, 0, 0)Point A: (a, 0, 0)Point B: (a√2/2, a√2/2, 0)Point C: (a/2, a/2, a/√2)Now, to find the volume of the pyramid, we can use the scalar triple product formula.The volume is (1/6) times the absolute value of the scalar triple product of vectors FA, FB, FC.But since FA, FB, FC are vectors from F, which is the origin, the scalar triple product is the determinant of the matrix formed by these vectors as columns (or rows).So, let's write the vectors:FA = (a, 0, 0)FB = (a√2/2, a√2/2, 0)FC = (a/2, a/2, a/√2)So, the matrix is:| a a√2/2 a/2 || 0 a√2/2 a/2 || 0 0 a/√2 |Now, the determinant of this upper triangular matrix is the product of the diagonal elements.So, determinant = a * (a√2/2) * (a/√2) = a * (a√2/2) * (a/√2)Simplify:First, multiply a and a√2/2: a * (a√2)/2 = (a²√2)/2Then, multiply by a/√2: (a²√2)/2 * (a/√2) = (a³ * (√2 * 1/√2))/2 = (a³ * 1)/2 = a³/2So, determinant = a³/2Therefore, the volume is (1/6) * |determinant| = (1/6)*(a³/2) = a³/12So, the volume of the pyramid is a³/12.Wait, let me double-check my steps.First, assigning coordinates: seems correct.Calculating the dot products: yes, that's how you find the angles between vectors.Solving for x, y, p, q, r: step by step, that seems correct.Calculating the determinant: yes, for an upper triangular matrix, determinant is product of diagonals.So, a * (a√2/2) * (a/√2) = a³/2, then volume is 1/6 of that, so a³/12.Yes, that seems right.Alternatively, I could have used another method, like finding the base area and height.Let me try that approach to confirm.First, find the area of the base triangle ABC.Points A: (a, 0, 0)Point B: (a√2/2, a√2/2, 0)Point C: (a/2, a/2, a/√2)Wait, but the base is triangle ABC, which is not necessarily in the same plane as the other points. Wait, actually, in a pyramid, the base is a face, so in this case, the base is triangle ABC, and the apex is F.Wait, but in my coordinate system, points A, B, and C are not coplanar with F, but F is the apex.Wait, no, in a pyramid, the base is a face, and the apex is connected to all vertices of the base. So, in this case, the base is triangle ABC, and F is connected to A, B, C.But in my coordinate system, points A, B, C are not all in the same plane. Wait, no, actually, in my coordinate system, points A, B, and C are in 3D space, so the base is triangle ABC, which is a triangle in 3D space, and F is the apex.But to compute the volume, using the scalar triple product is the way to go.Alternatively, I could compute the area of the base ABC and then find the height from F to the base.But that might be more complicated because I need to find the equation of the plane ABC and then compute the distance from F to that plane.But let's try it.First, find the area of triangle ABC.Points A: (a, 0, 0)Point B: (a√2/2, a√2/2, 0)Point C: (a/2, a/2, a/√2)So, vectors AB and AC can be found as:AB = B - A = (a√2/2 - a, a√2/2 - 0, 0 - 0) = (a(√2/2 - 1), a√2/2, 0)AC = C - A = (a/2 - a, a/2 - 0, a/√2 - 0) = (-a/2, a/2, a/√2)Then, the area of triangle ABC is (1/2) the magnitude of the cross product of AB and AC.Compute AB × AC.Let me compute AB and AC first.AB = (a(√2/2 - 1), a√2/2, 0)AC = (-a/2, a/2, a/√2)Compute cross product:|i j k ||a(√2/2 -1) a√2/2 0 ||-a/2 a/2 a/√2 |So, determinant:i * (a√2/2 * a/√2 - 0 * a/2) - j * (a(√2/2 -1) * a/√2 - 0*(-a/2)) + k * (a(√2/2 -1) * a/2 - a√2/2*(-a/2))Compute each component:i component: (a√2/2 * a/√2) = (a² * (√2 / 2) * (1/√2)) = (a² * (1/2)) = a²/2j component: - [ (a(√2/2 -1) * a/√2 ) ] = - [ a² (√2/2 -1)/√2 ] = - [ a² ( (√2/2)/√2 - 1/√2 ) ] = - [ a² ( (1/2) - 1/√2 ) ]Wait, let me compute it step by step.First, compute (√2/2 -1) * (1/√2):= (√2/2 * 1/√2) - (1 * 1/√2)= ( (√2 * 1) / (2 * √2) ) - (1/√2)= (1/2) - (1/√2)So, the j component is - [ a² (1/2 - 1/√2 ) ]k component: [ a(√2/2 -1) * a/2 - a√2/2*(-a/2) ]= [ a² (√2/2 -1)/2 + a² √2/4 ]= [ a² (√2/4 - 1/2 ) + a² √2/4 ]= a² (√2/4 - 1/2 + √2/4 )= a² ( (√2/4 + √2/4 ) - 1/2 )= a² ( (√2/2 ) - 1/2 )So, putting it all together:AB × AC = (a²/2, -a² (1/2 - 1/√2 ), a² (√2/2 - 1/2 ) )Now, the magnitude of this cross product is sqrt( (a²/2)^2 + [ -a² (1/2 - 1/√2 ) ]^2 + [ a² (√2/2 - 1/2 ) ]^2 )This seems complicated, but let's compute each term.First term: (a²/2)^2 = a⁴/4Second term: [ -a² (1/2 - 1/√2 ) ]^2 = a⁴ (1/2 - 1/√2 )²Third term: [ a² (√2/2 - 1/2 ) ]^2 = a⁴ (√2/2 - 1/2 )²Let me compute (1/2 - 1/√2 )² and (√2/2 - 1/2 )².Compute (1/2 - 1/√2 )²:= (1/2)^2 - 2*(1/2)*(1/√2) + (1/√2)^2= 1/4 - (1/√2) + 1/2= 1/4 + 1/2 - 1/√2= 3/4 - 1/√2Similarly, (√2/2 - 1/2 )²:= (√2/2)^2 - 2*(√2/2)*(1/2) + (1/2)^2= (2/4) - (√2/2) + 1/4= 1/2 - √2/2 + 1/4= 3/4 - √2/2So, the second term is a⁴*(3/4 - 1/√2 )The third term is a⁴*(3/4 - √2/2 )So, the magnitude squared is:a⁴/4 + a⁴*(3/4 - 1/√2 ) + a⁴*(3/4 - √2/2 )= a⁴ [ 1/4 + 3/4 - 1/√2 + 3/4 - √2/2 ]Simplify inside the brackets:1/4 + 3/4 = 11 + 3/4 = 7/4So, 7/4 - 1/√2 - √2/2Now, let's combine the terms with √2:-1/√2 - √2/2 = -(1/√2 + √2/2 )Note that 1/√2 = √2/2, so:= -(√2/2 + √2/2 ) = -√2So, the total inside the brackets is 7/4 - √2Therefore, the magnitude squared is a⁴*(7/4 - √2 )So, the magnitude is a² * sqrt(7/4 - √2 )Therefore, the area of triangle ABC is (1/2)*a²*sqrt(7/4 - √2 )Hmm, that seems complicated. Maybe I made a mistake somewhere.Alternatively, perhaps using the scalar triple product was easier and gave me a clean answer of a³/12.Given that, and since the scalar triple product method is straightforward, I think that the volume is indeed a³/12.So, I think my initial calculation was correct, and the volume is a³/12.

💡Okay, so I have this problem where I need to apply two transformations to the complex number (-3 - 8i). The first transformation is a (90^circ) rotation around the origin in the counter-clockwise direction, and the second is a dilation centered at the origin with a scale factor of 2. Hmm, let me think about how to approach this step by step.First, I remember that in the complex plane, a rotation can be represented by multiplying the complex number by a specific factor. For a (90^circ) counter-clockwise rotation, I think the multiplication factor is (i), because multiplying a complex number by (i) rotates it (90^circ) counter-clockwise. Let me verify that. If I take a simple complex number like (1 + 0i) and multiply it by (i), I get (i), which is indeed a (90^circ) rotation. Okay, so that seems right.So, applying the first transformation, which is a (90^circ) rotation, to (-3 - 8i) would involve multiplying it by (i). Let me write that out:[(-3 - 8i) times i]Now, I need to perform this multiplication. Remembering how complex multiplication works, I'll distribute the (i) across both terms:[-3 times i + (-8i) times i]Calculating each term separately:- The first term is (-3i).- The second term is (-8i^2).Wait, (i^2) is equal to (-1), right? So, (-8i^2) becomes (-8 times (-1)), which is (8).Putting it all together, the result of the first transformation is:[-3i + 8 = 8 - 3i]Okay, so after the rotation, the complex number is (8 - 3i). Now, moving on to the second transformation, which is a dilation with a scale factor of 2 centered at the origin. Dilation in the complex plane is straightforward; it's just multiplying the complex number by the scale factor. So, I need to multiply (8 - 3i) by 2.Let me write that out:[2 times (8 - 3i)]Distributing the 2 across both terms:[2 times 8 + 2 times (-3i)]Calculating each term:- The first term is (16).- The second term is (-6i).So, combining these, the result after the dilation is:[16 - 6i]Let me double-check my calculations to make sure I didn't make any mistakes. Starting with the rotation:[(-3 - 8i) times i = -3i - 8i^2 = -3i + 8 = 8 - 3i]Yes, that looks correct. Then, the dilation:[2 times (8 - 3i) = 16 - 6i]That also seems right. I don't see any errors in my steps. So, the final result after both transformations is (16 - 6i).Just to visualize, the original complex number (-3 - 8i) is in the third quadrant of the complex plane. After a (90^circ) counter-clockwise rotation, it moves to the second quadrant, becoming (8 - 3i). Then, scaling it by 2 moves it further out in the same direction, resulting in (16 - 6i), which is still in the second quadrant but twice as far from the origin. That makes sense.I think I've covered all the necessary steps and verified my calculations. I don't see any reason to doubt this result, so I'm confident that (16 - 6i) is the correct answer.

💡Okay, so I need to find the arc length of the curve given by the equation ( y = sqrt{1 - x^2} + arcsin x ) from ( x = 0 ) to ( x = frac{7}{9} ). Hmm, I remember that the formula for the arc length of a curve defined by ( y = f(x) ) from ( x = a ) to ( x = b ) is:[L = int_{a}^{b} sqrt{1 + left( frac{dy}{dx} right)^2} , dx]Alright, so first things first, I need to find the derivative ( frac{dy}{dx} ) of the given function. The function is ( y = sqrt{1 - x^2} + arcsin x ). Let me break this down into two parts: the derivative of ( sqrt{1 - x^2} ) and the derivative of ( arcsin x ).Starting with ( sqrt{1 - x^2} ), I recall that the derivative of ( sqrt{u} ) with respect to ( u ) is ( frac{1}{2sqrt{u}} ). So applying the chain rule, the derivative of ( sqrt{1 - x^2} ) with respect to ( x ) should be:[frac{d}{dx} sqrt{1 - x^2} = frac{1}{2sqrt{1 - x^2}} cdot (-2x) = frac{-x}{sqrt{1 - x^2}}]Okay, that seems right. Now, moving on to the derivative of ( arcsin x ). I think the derivative of ( arcsin x ) is ( frac{1}{sqrt{1 - x^2}} ). Let me double-check that. Yes, that's correct.So putting it all together, the derivative ( frac{dy}{dx} ) is:[frac{dy}{dx} = frac{-x}{sqrt{1 - x^2}} + frac{1}{sqrt{1 - x^2}} = frac{1 - x}{sqrt{1 - x^2}}]Hmm, that simplifies nicely. I can factor out ( frac{1}{sqrt{1 - x^2}} ) from both terms, so:[frac{dy}{dx} = frac{1 - x}{sqrt{1 - x^2}}]Wait, I think this can be simplified further. Let me see. If I factor the numerator and denominator, maybe I can write this as a square root of a fraction. Let's try:[frac{1 - x}{sqrt{1 - x^2}} = sqrt{frac{(1 - x)^2}{1 - x^2}} = sqrt{frac{(1 - x)^2}{(1 - x)(1 + x)}}} = sqrt{frac{1 - x}{1 + x}}]Oh, that's a nice simplification! So now, ( frac{dy}{dx} = sqrt{frac{1 - x}{1 + x}} ).Alright, now I need to plug this into the arc length formula. So:[L = int_{0}^{frac{7}{9}} sqrt{1 + left( sqrt{frac{1 - x}{1 + x}} right)^2} , dx]Simplifying the expression inside the square root:[sqrt{1 + frac{1 - x}{1 + x}} = sqrt{frac{(1 + x) + (1 - x)}{1 + x}} = sqrt{frac{2}{1 + x}}]So now, the integral becomes:[L = int_{0}^{frac{7}{9}} sqrt{frac{2}{1 + x}} , dx]I can factor out the square root of 2:[L = sqrt{2} int_{0}^{frac{7}{9}} frac{1}{sqrt{1 + x}} , dx]Now, this integral looks straightforward. The integral of ( frac{1}{sqrt{1 + x}} ) with respect to ( x ) is ( 2sqrt{1 + x} ). Let me verify that:[frac{d}{dx} 2sqrt{1 + x} = 2 cdot frac{1}{2sqrt{1 + x}} = frac{1}{sqrt{1 + x}}]Yes, that's correct. So, evaluating the definite integral:[sqrt{2} left[ 2sqrt{1 + x} right]_{0}^{frac{7}{9}} = 2sqrt{2} left( sqrt{1 + frac{7}{9}} - sqrt{1 + 0} right)]Simplifying inside the square roots:[sqrt{1 + frac{7}{9}} = sqrt{frac{16}{9}} = frac{4}{3}][sqrt{1 + 0} = sqrt{1} = 1]So, plugging these back in:[2sqrt{2} left( frac{4}{3} - 1 right) = 2sqrt{2} left( frac{1}{3} right) = frac{2sqrt{2}}{3}]Therefore, the arc length of the curve from ( x = 0 ) to ( x = frac{7}{9} ) is ( frac{2sqrt{2}}{3} ).**Final Answer**The arc length is boxed{dfrac{2sqrt{2}}{3}}.

💡First, I need to determine the proportion of the population in each province. Given that 6/9 of the population lives in Montague province, this simplifies to 2/3 of the total population. Consequently, the remaining 1/3 of the population resides in Capulet province.Next, I'll calculate the number of supporters for each candidate in both provinces. In Montague, 80% of the residents support Romeo, which means 20% support Juliet. Similarly, in Capulet, 70% of the residents support Juliet.To find the total number of Juliet supporters, I'll add the Juliet supporters from both provinces. Finally, to determine the probability that a randomly chosen Juliet supporter resides in Capulet, I'll divide the number of Juliet supporters in Capulet by the total number of Juliet supporters and convert this fraction to a percentage.

💡First, I need to determine the percentage of marks Victor obtained out of the maximum possible marks.I will use the formula for calculating percentage: (Marks Obtained / Maximum Marks) × 100.Victor scored 184 marks, and the maximum marks are 200.Plugging these values into the formula: (184 / 200) × 100.Simplifying the division: 184 divided by 200 equals 0.92.Multiplying by 100 to find the percentage: 0.92 × 100 equals 92%.Therefore, Victor achieved 92% of the total marks.

💡Alright, so I've got this problem here about three cylinders being combined to form an object, and I need to find the surface area of that object. The cylinders all have a height of 1 meter, and their base radii are 1.5 meters, 1 meter, and 0.5 meters. They also mention to take π as 3, which should make the calculations a bit simpler.First, I need to visualize how these cylinders are combined. Since the problem mentions a diagram, but I don't have access to it, I'll have to make some assumptions. Typically, when cylinders are combined, they can be stacked on top of each other or placed side by side. Given that all cylinders have the same height, it might make sense that they are stacked vertically, with each cylinder placed on top of the previous one. Alternatively, they could be placed side by side, but that might complicate the surface area calculation more.Let me think: if they are stacked vertically, the total height of the object would be 3 meters, but the problem doesn't specify the total height, so maybe that's not the case. Alternatively, they could be arranged in a way where each smaller cylinder is placed on top of a larger one, kind of like a tiered cake. That might make sense because the radii are decreasing: 1.5, 1, and 0.5 meters. So, the largest cylinder at the bottom, then a medium one on top, and the smallest on top of that.If that's the case, then the surface area would involve the lateral surfaces of all three cylinders plus the areas of the top and bottom circles. But wait, when cylinders are stacked, the top of one cylinder covers the bottom of the next one, so those areas would be internal and not contribute to the total surface area. So, I need to be careful not to double-count those areas.Alternatively, if the cylinders are placed side by side, the total surface area would be the sum of their individual surface areas minus the areas where they overlap. But again, without the diagram, it's a bit tricky. Maybe I should consider both possibilities.But let's start with the basics. The surface area of a single cylinder is given by the formula:[ text{Surface Area} = 2pi r (r + h) ]This includes both the lateral surface area and the areas of the two circular bases. However, if the cylinders are combined, some of these areas might be covered or internal, so we need to adjust accordingly.Given that all cylinders have the same height, 1 meter, and radii 1.5, 1, and 0.5 meters, let's calculate the surface area for each cylinder individually first.For the largest cylinder (radius 1.5 m):[ text{Surface Area}_1 = 2pi (1.5) (1.5 + 1) = 2pi (1.5)(2.5) ]Calculating that:First, 1.5 times 2.5 is 3.75.So,[ text{Surface Area}_1 = 2pi times 3.75 = 7.5pi ]Since π is given as 3,[ text{Surface Area}_1 = 7.5 times 3 = 22.5 , text{square meters} ]Next, for the medium cylinder (radius 1 m):[ text{Surface Area}_2 = 2pi (1) (1 + 1) = 2pi (1)(2) = 4pi ]Again, with π = 3,[ text{Surface Area}_2 = 4 times 3 = 12 , text{square meters} ]And for the smallest cylinder (radius 0.5 m):[ text{Surface Area}_3 = 2pi (0.5) (0.5 + 1) = 2pi (0.5)(1.5) ]Calculating that:0.5 times 1.5 is 0.75.So,[ text{Surface Area}_3 = 2pi times 0.75 = 1.5pi ]With π = 3,[ text{Surface Area}_3 = 1.5 times 3 = 4.5 , text{square meters} ]So, individually, the surface areas are 22.5, 12, and 4.5 square meters.But since they are combined into one object, some areas will overlap or be internal, so we need to subtract those overlapping areas.If we assume they are stacked vertically, with each smaller cylinder on top of a larger one, then the top base of the lower cylinder will be covered by the bottom base of the upper cylinder. Therefore, for each interface between two cylinders, we lose two circular areas: one from the top of the lower cylinder and one from the bottom of the upper cylinder. However, since these are internal, they don't contribute to the external surface area.But wait, actually, when you stack a smaller cylinder on top of a larger one, the bottom base of the smaller cylinder is entirely within the top base of the larger cylinder, so the overlapping area is just the area of the smaller base. Therefore, for each interface, we lose the area of the smaller base.So, for the interface between the largest and medium cylinder, the overlapping area is the area of the medium cylinder's base, which is:[ text{Area}_2 = pi (1)^2 = pi ]Similarly, between the medium and smallest cylinder, the overlapping area is the area of the smallest cylinder's base:[ text{Area}_3 = pi (0.5)^2 = 0.25pi ]Therefore, the total overlapping area is:[ pi + 0.25pi = 1.25pi ]But since π = 3,[ 1.25pi = 1.25 times 3 = 3.75 , text{square meters} ]So, the total surface area of the composite object would be the sum of the individual surface areas minus twice the overlapping areas (since each overlapping area is counted twice in the individual surface areas: once as the top of the lower cylinder and once as the bottom of the upper cylinder).Wait, actually, no. Each overlapping area is only counted once in the total surface area because the top of the lower cylinder and the bottom of the upper cylinder are both internal and not part of the external surface. Therefore, we need to subtract both of these areas from the total.But let's think carefully. The individual surface areas include both the top and bottom bases. When we stack them, the top base of the lower cylinder and the bottom base of the upper cylinder become internal and are no longer part of the external surface. Therefore, for each interface, we need to subtract twice the area of the smaller base (since both the top of the lower and the bottom of the upper are lost).Wait, no. Actually, for each interface, the top of the lower cylinder and the bottom of the upper cylinder are both covered, so we need to subtract both areas. However, in the individual surface areas, both the top and bottom are already included. So, when combining, the top of the lower cylinder and the bottom of the upper cylinder are internal, so we need to subtract both from the total.Therefore, for each interface, we subtract 2 times the area of the smaller base.So, for the interface between the largest and medium cylinder, we subtract 2 times the area of the medium base:[ 2 times pi (1)^2 = 2pi ]And for the interface between the medium and smallest cylinder, we subtract 2 times the area of the smallest base:[ 2 times pi (0.5)^2 = 0.5pi ]Therefore, the total subtraction is:[ 2pi + 0.5pi = 2.5pi ]With π = 3,[ 2.5pi = 2.5 times 3 = 7.5 , text{square meters} ]So, the total surface area of the composite object would be:Sum of individual surface areas: 22.5 + 12 + 4.5 = 39 square metersMinus the overlapping areas: 7.5 square metersTotal surface area: 39 - 7.5 = 31.5 square metersBut wait, let's double-check this because I might have made a mistake in the subtraction.Alternatively, another approach is to calculate the total lateral surface areas and add the top and bottom areas appropriately.Since the cylinders are stacked, the total lateral surface area would be the sum of the lateral surface areas of all three cylinders.The lateral surface area of a cylinder is given by:[ 2pi r h ]So, for the largest cylinder:[ 2pi (1.5)(1) = 3pi ]Medium cylinder:[ 2pi (1)(1) = 2pi ]Smallest cylinder:[ 2pi (0.5)(1) = pi ]Total lateral surface area:[ 3pi + 2pi + pi = 6pi ]With π = 3,[ 6pi = 18 , text{square meters} ]Now, for the top and bottom areas. The bottom of the largest cylinder is still exposed, so we have:[ pi (1.5)^2 = 2.25pi ]And the top of the smallest cylinder is also exposed:[ pi (0.5)^2 = 0.25pi ]So, total top and bottom areas:[ 2.25pi + 0.25pi = 2.5pi ]With π = 3,[ 2.5pi = 7.5 , text{square meters} ]Therefore, total surface area is lateral surface area plus top and bottom areas:[ 18 + 7.5 = 25.5 , text{square meters} ]Wait, this is different from the previous calculation. So, which one is correct?In the first approach, I subtracted the overlapping areas from the total individual surface areas, resulting in 31.5 square meters.In the second approach, I calculated the lateral surface areas and added the exposed top and bottom areas, resulting in 25.5 square meters.There's a discrepancy here, so I need to figure out which approach is correct.Let me think again. The first approach considered the total surface areas of all three cylinders and then subtracted the overlapping areas. However, when subtracting, I might have subtracted too much because the overlapping areas are only the areas where the cylinders are stacked, not the entire top and bottom.Alternatively, the second approach directly calculates the external surfaces: the lateral areas and the exposed top and bottom. This seems more accurate because when stacking, only the lateral surfaces and the very top and bottom are exposed.Therefore, the second approach is likely correct, resulting in 25.5 square meters.But let's verify this by considering the individual contributions.The largest cylinder contributes its lateral surface area and its bottom base.The medium cylinder contributes its lateral surface area but not its top or bottom because they are covered by the largest and smallest cylinders, respectively.The smallest cylinder contributes its lateral surface area and its top base.Therefore:Largest cylinder: lateral surface area + bottom baseMedium cylinder: lateral surface areaSmallest cylinder: lateral surface area + top baseCalculating each:Largest cylinder:Lateral: 3πBottom base: 2.25πTotal: 3π + 2.25π = 5.25πMedium cylinder:Lateral: 2πTotal: 2πSmallest cylinder:Lateral: πTop base: 0.25πTotal: π + 0.25π = 1.25πAdding them all together:5.25π + 2π + 1.25π = 8.5πWith π = 3,8.5π = 25.5 square metersYes, this matches the second approach. So, the total surface area is 25.5 square meters.But wait, the problem statement says "three cylinders... are combined to form an object." It doesn't specify how they are combined, so maybe I should consider another possibility.Alternatively, if the cylinders are placed side by side, the total surface area would be the sum of their individual surface areas minus the areas where they overlap. However, without knowing how they are arranged, it's hard to say. But given the radii are decreasing, it's more likely they are stacked vertically.Therefore, I think the correct total surface area is 25.5 square meters.But let's check the initial individual surface areas again:Largest: 22.5Medium: 12Smallest: 4.5Total: 39If we subtract the overlapping areas, which are the top of the largest and the bottom of the medium, and the top of the medium and the bottom of the smallest.Each overlapping area is the area of the smaller base.So, between largest and medium: area of medium base = π(1)^2 = πBetween medium and smallest: area of smallest base = π(0.5)^2 = 0.25πTotal overlapping areas: π + 0.25π = 1.25πBut since these areas are covered twice in the total individual surface areas (once as the top of the lower cylinder and once as the bottom of the upper cylinder), we need to subtract them twice.Therefore, total subtraction: 2*(π + 0.25π) = 2.5πSo,Total surface area = 39 - 2.5πWith π = 3,Total surface area = 39 - 7.5 = 31.5But this contradicts the previous result.Wait, now I'm confused. Which approach is correct?The confusion arises from whether to subtract the overlapping areas once or twice.In the first approach, I subtracted the overlapping areas twice because each overlapping area is counted once in the top of the lower cylinder and once in the bottom of the upper cylinder in the total individual surface areas. Therefore, to get the correct external surface area, we need to subtract both.In the second approach, I directly calculated the external surfaces: lateral areas plus the exposed top and bottom. This method avoids double-counting because it only includes the external parts.Therefore, the second approach is more accurate because it directly accounts for what's visible, whereas the first approach might have led to an over-subtraction.Hence, the correct total surface area is 25.5 square meters.But let's think again: the individual surface areas include both the top and bottom of each cylinder. When we stack them, the top of the lower cylinder and the bottom of the upper cylinder are covered, so we lose those areas. Therefore, for each interface, we lose two areas: the top of the lower and the bottom of the upper. However, in the individual surface areas, both of these are included, so we need to subtract both.Therefore, total overlapping areas to subtract are:For largest and medium: top of largest (2.25π) and bottom of medium (π)Wait, no. The top of the largest cylinder is covered by the bottom of the medium cylinder, but the medium cylinder's bottom is smaller than the largest cylinder's top. Therefore, the overlapping area is only the area of the medium cylinder's base, which is π(1)^2 = π.Similarly, the top of the medium cylinder is covered by the bottom of the smallest cylinder, which is π(0.5)^2 = 0.25π.Therefore, the total overlapping areas are π + 0.25π = 1.25π.But since these areas are covered, they are not part of the external surface. However, in the individual surface areas, both the top of the lower cylinder and the bottom of the upper cylinder are included. Therefore, we need to subtract both.Wait, no. The top of the lower cylinder is covered by the bottom of the upper cylinder, but the bottom of the upper cylinder is smaller, so only part of the top of the lower cylinder is covered. Therefore, the overlapping area is the area of the upper cylinder's base.Therefore, for each interface, we only need to subtract the area of the upper cylinder's base once because that's the area that is covered.Wait, this is getting complicated. Maybe it's better to use the second approach where I directly calculate the external surfaces.In that case, the total surface area is the sum of the lateral areas plus the exposed top and bottom.Largest cylinder contributes its lateral area and its bottom.Medium cylinder contributes its lateral area.Smallest cylinder contributes its lateral area and its top.Therefore:Largest: 3π + 2.25π = 5.25πMedium: 2πSmallest: π + 0.25π = 1.25πTotal: 5.25π + 2π + 1.25π = 8.5πWith π = 3,8.5π = 25.5So, 25.5 square meters.Alternatively, if I use the first approach:Total individual surface areas: 22.5 + 12 + 4.5 = 39Subtract the overlapping areas: for each interface, subtract the area of the upper cylinder's base.Between largest and medium: subtract π(1)^2 = πBetween medium and smallest: subtract π(0.5)^2 = 0.25πTotal subtraction: π + 0.25π = 1.25πBut since these areas are covered, and in the individual surface areas, both the top of the lower and the bottom of the upper are included, we need to subtract both.Wait, no. The top of the lower cylinder is covered by the bottom of the upper cylinder, but the bottom of the upper cylinder is smaller, so only part of the top of the lower cylinder is covered. Therefore, the overlapping area is the area of the upper cylinder's base.Therefore, for each interface, we need to subtract the area of the upper cylinder's base once because that's the area that is covered and not part of the external surface.Therefore, total subtraction: π + 0.25π = 1.25πWith π = 3,1.25π = 3.75So, total surface area = 39 - 3.75 = 35.25But this contradicts the second approach.I think the confusion arises from whether to subtract the overlapping areas once or twice.In the first approach, the total individual surface areas include both the top and bottom of each cylinder. When we stack them, the top of the lower cylinder and the bottom of the upper cylinder are covered. Therefore, for each interface, we have two areas that are no longer external: the top of the lower and the bottom of the upper. However, the bottom of the upper cylinder is smaller than the top of the lower cylinder, so only the area of the upper cylinder's base is covered.Therefore, for each interface, we need to subtract the area of the upper cylinder's base once because that's the area that is covered and not part of the external surface.Therefore, total subtraction: π + 0.25π = 1.25πWith π = 3,1.25π = 3.75So, total surface area = 39 - 3.75 = 35.25But this still doesn't match the second approach.I think the correct way is to consider that when stacking, the top of the lower cylinder is covered by the bottom of the upper cylinder, but the bottom of the upper cylinder is smaller, so the overlapping area is the area of the upper cylinder's base. Therefore, for each interface, we lose the area of the upper cylinder's base from the top of the lower cylinder, and we also lose the area of the upper cylinder's base from its own bottom. Therefore, for each interface, we need to subtract twice the area of the upper cylinder's base.Therefore, total subtraction: 2*(π + 0.25π) = 2.5πWith π = 3,2.5π = 7.5Total surface area = 39 - 7.5 = 31.5But this still doesn't match the second approach.I think the confusion is arising because the overlapping areas are not just the areas of the upper cylinder's base, but also the areas that are covered on the lower cylinder's top.Wait, no. The overlapping area is the area where the upper cylinder sits on the lower cylinder. That area is the area of the upper cylinder's base, which is smaller than the lower cylinder's top. Therefore, the area of the lower cylinder's top that is not covered is the area of the lower cylinder's top minus the area of the upper cylinder's base.Similarly, the upper cylinder's bottom is entirely covered, so we lose that area.Therefore, for each interface:- The lower cylinder loses an area equal to the upper cylinder's base from its top.- The upper cylinder loses its entire bottom area.Therefore, total overlapping areas to subtract are:For largest and medium:- Lower cylinder (largest) loses π(1)^2 = π from its top.- Upper cylinder (medium) loses π(1)^2 = π from its bottom.Total subtraction for this interface: π + π = 2πFor medium and smallest:- Lower cylinder (medium) loses π(0.5)^2 = 0.25π from its top.- Upper cylinder (smallest) loses π(0.5)^2 = 0.25π from its bottom.Total subtraction for this interface: 0.25π + 0.25π = 0.5πTherefore, total subtraction: 2π + 0.5π = 2.5πWith π = 3,2.5π = 7.5Total surface area = 39 - 7.5 = 31.5But this still doesn't match the second approach.Wait, in the second approach, we directly calculated the external surfaces:- Largest cylinder: lateral + bottom- Medium cylinder: lateral- Smallest cylinder: lateral + topSo, largest contributes 3π + 2.25π = 5.25πMedium contributes 2πSmallest contributes π + 0.25π = 1.25πTotal: 5.25π + 2π + 1.25π = 8.5π = 25.5This seems more accurate because it directly accounts for what's visible.Therefore, the correct total surface area is 25.5 square meters.But to reconcile the two approaches, let's see:Total individual surface areas: 39Subtract overlapping areas: 7.5Total: 31.5But according to the direct method, it's 25.5The discrepancy is because in the first approach, we subtracted the overlapping areas, but in reality, the overlapping areas are not just the areas of the upper cylinder's base, but also the areas that are covered on the lower cylinder's top.Wait, no. The overlapping areas are the areas where the upper cylinder sits on the lower cylinder, which is the area of the upper cylinder's base. Therefore, for each interface, we lose the area of the upper cylinder's base from the lower cylinder's top and the entire area of the upper cylinder's base from its own bottom.Therefore, total subtraction per interface is 2 times the area of the upper cylinder's base.Therefore, total subtraction: 2*(π + 0.25π) = 2.5π = 7.5Total surface area: 39 - 7.5 = 31.5But this contradicts the direct method.I think the confusion is because in the direct method, we are not considering that the lower cylinder's top is partially covered, but in reality, the area of the lower cylinder's top that is not covered is still part of the external surface.Wait, no. When you stack a smaller cylinder on top of a larger one, the area of the larger cylinder's top that is not covered by the smaller cylinder is still exposed. Therefore, in the direct method, we should include that area.Therefore, in the direct method, the largest cylinder contributes its lateral area plus its bottom and the area of its top that is not covered by the medium cylinder.Similarly, the medium cylinder contributes its lateral area plus the area of its top that is not covered by the smallest cylinder.And the smallest cylinder contributes its lateral area plus its top.Therefore, let's recalculate:Largest cylinder:Lateral: 3πBottom: 2.25πTop: area not covered by medium cylinder = π(1.5)^2 - π(1)^2 = 2.25π - π = 1.25πTotal: 3π + 2.25π + 1.25π = 6.5πMedium cylinder:Lateral: 2πTop: area not covered by smallest cylinder = π(1)^2 - π(0.5)^2 = π - 0.25π = 0.75πTotal: 2π + 0.75π = 2.75πSmallest cylinder:Lateral: πTop: 0.25πTotal: π + 0.25π = 1.25πAdding them all together:6.5π + 2.75π + 1.25π = 10.5πWith π = 3,10.5π = 31.5 square metersNow, this matches the first approach.So, the correct total surface area is 31.5 square meters.Therefore, the initial direct method was incomplete because it didn't account for the exposed part of the largest cylinder's top.So, the correct approach is to consider that when stacking, the largest cylinder's top is partially covered, but the remaining part is still exposed. Similarly, the medium cylinder's top is partially covered.Therefore, the total surface area is the sum of:- Largest cylinder's lateral area + bottom + exposed top- Medium cylinder's lateral area + exposed top- Smallest cylinder's lateral area + topCalculating each:Largest cylinder:Lateral: 3πBottom: 2.25πExposed top: 2.25π - π = 1.25πTotal: 3π + 2.25π + 1.25π = 6.5πMedium cylinder:Lateral: 2πExposed top: π - 0.25π = 0.75πTotal: 2π + 0.75π = 2.75πSmallest cylinder:Lateral: πTop: 0.25πTotal: π + 0.25π = 1.25πTotal surface area:6.5π + 2.75π + 1.25π = 10.5πWith π = 3,10.5π = 31.5 square metersTherefore, the correct total surface area is 31.5 square meters.But let's double-check this with the first approach:Total individual surface areas: 22.5 + 12 + 4.5 = 39Subtract overlapping areas: for each interface, subtract 2 times the area of the upper cylinder's baseFirst interface (largest and medium): 2πSecond interface (medium and smallest): 0.5πTotal subtraction: 2π + 0.5π = 2.5π = 7.5Total surface area: 39 - 7.5 = 31.5Yes, this matches.Therefore, the correct total surface area is 31.5 square meters.But wait, the problem statement says "three cylinders... are combined to form an object." It doesn't specify the arrangement, but given the radii, it's likely stacked as I considered.However, another possibility is that the cylinders are placed side by side, forming a sort of triangular prism with cylindrical ends. But without the diagram, it's hard to be certain.Assuming they are stacked vertically, the total surface area is 31.5 square meters.But let's consider another scenario: if the cylinders are placed side by side, the total surface area would be the sum of their individual surface areas minus the areas where they touch each other.However, since they have different radii, placing them side by side would require that their circular bases are adjacent, but the areas where they touch would be along their curved surfaces, not their bases.In that case, the overlapping areas would be along the lateral surfaces, which complicates the calculation because the overlapping area would be a rectangle or something else, depending on how they are arranged.Given the lack of a diagram, it's safer to assume they are stacked vertically, as that is a common way to combine cylinders.Therefore, the total surface area is 31.5 square meters.But wait, earlier I had two different results: 25.5 and 31.5. Now, after a more careful consideration, I arrived at 31.5.But let's see, if I use the formula for the surface area of a composite object, it's the sum of the individual surface areas minus twice the overlapping areas.In this case, the overlapping areas are the areas where the cylinders are stacked, which are the areas of the upper cylinder's base.Therefore, total overlapping areas: π + 0.25π = 1.25πBut since these areas are covered twice in the total individual surface areas (once as the top of the lower cylinder and once as the bottom of the upper cylinder), we need to subtract them twice.Therefore, total subtraction: 2*(π + 0.25π) = 2.5π = 7.5Total surface area: 39 - 7.5 = 31.5Yes, this seems consistent.Therefore, the final answer is 31.5 square meters.But the problem asks to take π as 3, so 31.5 is already in numerical form.However, in the initial calculation, I got 25.5, but after considering the exposed parts of the larger cylinders' tops, it became 31.5.Therefore, the correct total surface area is 31.5 square meters.