💡To determine the number of toothpicks needed to construct a large equilateral triangle with a base consisting of 10 small equilateral triangles, we can follow these steps:1. **Understand the Structure**: The large triangle is composed of smaller equilateral triangles arranged in rows. The base has 10 small triangles, the next row has 9, and so on, up to the top row which has 1 small triangle.2. **Calculate the Total Number of Small Triangles**: The total number of small triangles can be found by summing the number of triangles in each row. This forms an arithmetic series: [ T = 1 + 2 + 3 + ldots + 10 = frac{10 times 11}{2} = 55 ]3. **Determine the Total Number of Sides Without Sharing**: Each small triangle has 3 sides. Therefore, the total number of sides without considering shared sides is: [ text{Total sides without sharing} = 3 times 55 = 165 ]4. **Adjust for Shared Sides**: Each internal side is shared by two triangles. The number of shared sides can be calculated by subtracting the boundary sides from the total sides and then dividing by 2. The boundary sides are the sides of the large triangle, which are 10 on each side: [ text{Boundary sides} = 3 times 10 = 30 ] [ text{Internal sides} = frac{165 - 30}{2} = 67.5 ]5. **Calculate the Total Toothpicks Needed**: The total number of toothpicks is the sum of the internal sides and the boundary sides: [ text{Total toothpicks} = 67.5 + 30 = 97.5 ]6. **Round to the Nearest Whole Number**: Since the number of toothpicks must be a whole number, we round 97.5 to the nearest whole number, which is 98.Therefore, the number of toothpicks needed is (boxed{B}).

💡First, determine the number of students in the 3rd and 4th grades.There are 19 students in the 3rd grade. Since the 4th grade has twice as many students, there are:19 × 2 = 38 students in the 4th grade.Next, calculate the total number of students in the 3rd and 4th grades:19 (3rd grade) + 38 (4th grade) = 57 students.Given that the total number of students across all three grades is 86, find the number of students in the 2nd grade:86 (total students) - 57 (3rd and 4th grade students) = 29 students in the 2nd grade.Finally, since there are 19 girls in the 2nd grade, subtract the number of girls from the total number of students in the 2nd grade to find the number of boys:29 (2nd grade students) - 19 (girls) = 10 boys in the 2nd grade.

💡Okay, so I have this problem about ten spheres inside a larger sphere. Let me try to visualize it. There's one sphere at the origin, and the other nine are arranged in some way related to a cube. Hmm, the problem says each of the remaining nine spheres is tangent to one face of a cube centered at the origin. Also, each sphere is tangent to three mutually adjacent squares on the surface of the cube. First, I need to figure out how these nine spheres are positioned. Since the cube is centered at the origin, each face is equidistant from the origin. If each sphere is tangent to one face of the cube, that means the center of each sphere is at a distance equal to its radius from that face. The radius of each small sphere is 2, so the distance from the center of each sphere to the face of the cube is 2. Wait, so if the cube is centered at the origin, the distance from the origin to each face is equal to half the side length of the cube. Let me denote the side length of the cube as 's'. Then, the distance from the origin to each face is s/2. Since the center of each sphere is 2 units away from the face, that means s/2 - 2 is the distance from the origin to the center of each sphere. But hold on, the sphere is tangent to the face, so actually, the center of the sphere must be at a distance of 2 from the face. Therefore, the distance from the origin to the center of the sphere is s/2 - 2. But wait, the sphere is also tangent to three mutually adjacent squares on the surface of the cube. That probably means that the sphere is touching three edges or something? Maybe it's sitting in a corner where three faces meet? Hmm, no, because it's tangent to one face and three mutually adjacent squares. Maybe each sphere is placed such that it's tangent to one face and also tangent to three edges or something. Wait, actually, if the cube has squares on its surface, each face is a square. So, each sphere is tangent to one face and three adjacent squares. That might mean that each sphere is placed near a corner, touching one face and the three adjacent faces? But no, because each sphere is only tangent to one face. Maybe it's touching one face and three edges? Alternatively, perhaps each sphere is placed in such a way that it's tangent to one face and also tangent to three other spheres? Hmm, not sure. Maybe I should think about the cube's structure. Since the cube is centered at the origin, each face is at a distance of s/2 from the origin. Each sphere is tangent to one face, so the center of each sphere is at a distance of s/2 - 2 from the origin. Also, since each sphere is tangent to three mutually adjacent squares, which are on the surface of the cube, maybe the sphere is placed such that it's touching three edges of the cube? Wait, perhaps the centers of the nine spheres are located at the centers of each face of the cube? But no, because each sphere is tangent to one face and three adjacent squares. If it's at the center of a face, it wouldn't be tangent to three adjacent squares. Alternatively, maybe the centers of the nine spheres are located at the midpoints of each edge of the cube? But each edge is shared by two faces, so that might not make sense. Wait, maybe each sphere is placed near a corner, but not exactly at the corner. Since the cube has eight corners, but we have nine spheres, so that doesn't fit. Hmm, maybe the spheres are placed at the centers of each face and the origin. But the origin is already occupied by one sphere. Wait, the problem says ten spheres: one at the origin, and nine others. So, the nine others must be placed in some symmetric way. Maybe they are placed at the centers of each face of the cube? But a cube has six faces, so that would only account for six spheres. Hmm, but we have nine. Wait, maybe each sphere is placed at the center of each face, but there are three spheres per face? No, that doesn't make sense. Maybe it's a 3x3x3 arrangement? But that would be 27 spheres, which is too many. Wait, maybe the cube is divided into smaller cubes, each with a sphere at their centers. But the problem says each sphere is tangent to one face and three mutually adjacent squares. Alternatively, perhaps the nine spheres are placed at the centers of each face of a larger cube, but that doesn't explain the three mutually adjacent squares. Wait, maybe each sphere is placed such that it's tangent to one face of the cube and also tangent to three edges that meet at a corner. So, each sphere is near a corner, touching one face and three edges. That would make sense for a cube, as each corner has three edges meeting. So, if each sphere is tangent to one face and three edges, then the center of each sphere must be 2 units away from the face and 2 units away from each of the three edges. Let me try to model this. Suppose we take one corner of the cube at (s/2, s/2, s/2). The three edges meeting at this corner are along the x, y, and z axes. If a sphere is tangent to the face at x = s/2 and also tangent to the edges along y and z, then the center of the sphere must be 2 units away from x = s/2, so its x-coordinate is s/2 - 2. Similarly, it must be 2 units away from the edges along y and z, so its y and z coordinates are 2 units from the edges. Wait, the edges are along the axes, so the distance from the center to each edge is 2. The distance from a point (x, y, z) to the edge along the x-axis is sqrt(y^2 + z^2). Similarly for the other edges. So, if the sphere is tangent to the three edges, then sqrt(y^2 + z^2) = 2, sqrt(x^2 + z^2) = 2, and sqrt(x^2 + y^2) = 2. But if the sphere is also tangent to the face x = s/2, then x = s/2 - 2. Similarly, if it's tangent to the face y = s/2, then y = s/2 - 2, and same for z. But wait, the problem says each sphere is tangent to one face, not necessarily all three. So, maybe each sphere is only tangent to one face and three edges. So, let's say the sphere is tangent to the face x = s/2, so x = s/2 - 2. It's also tangent to the three edges along y and z axes. So, the distance from the center to each of these edges is 2. The distance from the center (x, y, z) to the edge along the y-axis (where x=0, z=0) is sqrt(x^2 + z^2). Similarly, the distance to the edge along the z-axis is sqrt(x^2 + y^2). Since the sphere is tangent to these edges, these distances must be equal to the radius, which is 2. So, sqrt(x^2 + z^2) = 2 and sqrt(x^2 + y^2) = 2. But we already have x = s/2 - 2. So, substituting x into these equations:sqrt((s/2 - 2)^2 + z^2) = 2sqrt((s/2 - 2)^2 + y^2) = 2Squaring both sides:(s/2 - 2)^2 + z^2 = 4(s/2 - 2)^2 + y^2 = 4So, z^2 = 4 - (s/2 - 2)^2Similarly, y^2 = 4 - (s/2 - 2)^2Therefore, y = z = sqrt(4 - (s/2 - 2)^2)But since the sphere is also tangent to the three mutually adjacent squares, which are on the surface of the cube, perhaps these squares are the ones adjacent to the face x = s/2. So, the squares would be on the faces y = s/2, z = s/2, and maybe another? Wait, no, each face of the cube is a square, so the three adjacent squares to the face x = s/2 would be the faces y = s/2, z = s/2, and maybe another? Wait, no, each face is a square, and adjacent faces meet along edges.Wait, maybe the three mutually adjacent squares are the three faces meeting at a corner. So, if the sphere is near the corner, it's tangent to one face and the three adjacent faces. But the problem says it's tangent to one face and three mutually adjacent squares. Maybe the squares are the three adjacent faces.But if that's the case, then the sphere would be tangent to four faces: one face and three adjacent faces. But the problem says it's tangent to one face and three mutually adjacent squares. Hmm, maybe the squares are not the entire faces but smaller squares on the surface.Wait, maybe the cube is divided into smaller squares, and each sphere is tangent to one face and three smaller squares. But the problem doesn't specify that. It just says each sphere is tangent to one face of the cube and three mutually adjacent squares on the surface of the cube.Alternatively, maybe the three mutually adjacent squares are the three faces meeting at a corner, so the sphere is tangent to one face and the three adjacent faces. That would make sense.So, if the sphere is tangent to four faces: one face and the three adjacent faces. But the problem says it's tangent to one face and three mutually adjacent squares. Maybe the squares are the three adjacent faces.So, if the sphere is tangent to four faces, then the center of the sphere must be 2 units away from each of these four faces. Therefore, the center would be at (s/2 - 2, s/2 - 2, s/2 - 2). But that would be the case if the sphere is tangent to all four faces, but the problem says it's tangent to one face and three mutually adjacent squares. Maybe the three squares are the three adjacent faces.Wait, I'm getting confused. Let me try to approach this differently.If each of the nine spheres is tangent to one face of the cube and three mutually adjacent squares, then perhaps each sphere is placed near a corner, but not exactly at the corner. So, the center of each sphere is 2 units away from one face and 2 units away from three edges.So, let's say the sphere is near the corner (s/2, s/2, s/2). It's tangent to the face x = s/2, so its x-coordinate is s/2 - 2. It's also tangent to the three edges along y, z, and maybe another? Wait, each corner has three edges. So, the sphere is tangent to the three edges meeting at that corner.So, the distance from the center to each of these edges is 2. The distance from a point (x, y, z) to the edge along the x-axis is sqrt(y^2 + z^2). Similarly, the distance to the edge along the y-axis is sqrt(x^2 + z^2), and to the edge along the z-axis is sqrt(x^2 + y^2).Since the sphere is tangent to all three edges, these distances must be equal to 2.So, we have:sqrt(y^2 + z^2) = 2sqrt(x^2 + z^2) = 2sqrt(x^2 + y^2) = 2And we also have x = s/2 - 2 because the sphere is tangent to the face x = s/2.So, substituting x = s/2 - 2 into the equations:sqrt((s/2 - 2)^2 + z^2) = 2sqrt((s/2 - 2)^2 + y^2) = 2Squaring both sides:(s/2 - 2)^2 + z^2 = 4(s/2 - 2)^2 + y^2 = 4So, z^2 = 4 - (s/2 - 2)^2Similarly, y^2 = 4 - (s/2 - 2)^2Therefore, y = z = sqrt(4 - (s/2 - 2)^2)But we also have sqrt(x^2 + y^2) = 2, so substituting x and y:sqrt((s/2 - 2)^2 + (sqrt(4 - (s/2 - 2)^2))^2) = 2Simplify inside the sqrt:(s/2 - 2)^2 + (4 - (s/2 - 2)^2) = 4So, (s/2 - 2)^2 + 4 - (s/2 - 2)^2 = 4Which simplifies to 4 = 4, which is always true. So, that doesn't give us any new information.Hmm, maybe I need to find the side length 's' of the cube. Since the sphere is tangent to one face and three edges, and the center is at (s/2 - 2, y, z), with y and z as above.But I also know that the sphere is tangent to three mutually adjacent squares on the surface of the cube. If the squares are the three faces meeting at a corner, then the sphere is tangent to those three faces as well. So, the center must be 2 units away from each of those three faces.Therefore, the center is at (s/2 - 2, s/2 - 2, s/2 - 2). Because it's 2 units away from each of the three faces x = s/2, y = s/2, z = s/2.So, the center is at (s/2 - 2, s/2 - 2, s/2 - 2). Therefore, the distance from the origin to this center is sqrt[(s/2 - 2)^2 + (s/2 - 2)^2 + (s/2 - 2)^2] = sqrt[3*(s/2 - 2)^2] = (s/2 - 2)*sqrt(3)But we also know that the sphere is inside the larger sphere, so the distance from the origin to the center of the small sphere plus the radius of the small sphere must be less than or equal to the radius of the larger sphere.Wait, but actually, the larger sphere must enclose all ten small spheres. So, the radius of the larger sphere must be equal to the maximum distance from the origin to any point on the small spheres.The small spheres are at the origin and at (s/2 - 2, s/2 - 2, s/2 - 2) and similar positions for the other eight spheres.So, the distance from the origin to the center of each small sphere is (s/2 - 2)*sqrt(3). Then, the distance from the origin to the surface of each small sphere is (s/2 - 2)*sqrt(3) + 2.Similarly, the sphere at the origin has its surface at a distance of 2 from the origin.Therefore, the radius of the larger sphere must be the maximum of these two distances. So, we need to ensure that (s/2 - 2)*sqrt(3) + 2 is greater than or equal to 2, which it will be as long as s/2 - 2 is positive.But we need to find the side length 's' of the cube. How?Wait, the cube is centered at the origin, and the small spheres are placed such that their centers are at (s/2 - 2, s/2 - 2, s/2 - 2) and similar positions for the other eight spheres. So, the cube must be large enough to contain these centers.But actually, the cube is the one that the small spheres are tangent to. So, the cube's side length is determined by the placement of the small spheres.Wait, each small sphere is tangent to one face of the cube, so the distance from the center of the small sphere to that face is 2. Therefore, the distance from the origin to that face is s/2, and the center of the small sphere is at s/2 - 2 from the origin along that axis.But the center of the small sphere is also at (s/2 - 2, s/2 - 2, s/2 - 2), so the distance from the origin is (s/2 - 2)*sqrt(3). Wait, but how does this relate to the cube? The cube's side length is s, so the distance from the origin to each face is s/2. The center of the small sphere is at s/2 - 2 from the origin along each axis.But the cube's side length must accommodate the small spheres. Since each small sphere is tangent to one face, the cube's side length is such that s/2 = (s/2 - 2) + 2, which is just s/2 = s/2, which doesn't help.Wait, maybe I need to consider the diagonal of the cube. The space diagonal of the cube is s*sqrt(3). The centers of the small spheres are at (s/2 - 2, s/2 - 2, s/2 - 2), so the distance from the origin to this center is (s/2 - 2)*sqrt(3). But the larger sphere must enclose all the small spheres, so the radius R of the larger sphere must satisfy R = (s/2 - 2)*sqrt(3) + 2.But we need to find 's' such that the small spheres fit inside the cube. Wait, the cube is just the structure to which the small spheres are tangent, but the cube itself is inside the larger sphere.Wait, maybe the cube is inscribed within the larger sphere. So, the space diagonal of the cube would be equal to the diameter of the larger sphere. But the space diagonal is s*sqrt(3), so the diameter would be s*sqrt(3), and the radius would be (s*sqrt(3))/2.But we also have that the radius of the larger sphere must be equal to (s/2 - 2)*sqrt(3) + 2, as that's the distance from the origin to the surface of the small sphere.So, setting these equal:(s*sqrt(3))/2 = (s/2 - 2)*sqrt(3) + 2Let me solve for 's':Multiply both sides by 2 to eliminate denominators:s*sqrt(3) = (s - 4)*sqrt(3) + 4Expand the right side:s*sqrt(3) = s*sqrt(3) - 4*sqrt(3) + 4Subtract s*sqrt(3) from both sides:0 = -4*sqrt(3) + 4Which simplifies to:4*sqrt(3) = 4Divide both sides by 4:sqrt(3) = 1But sqrt(3) is approximately 1.732, which is not equal to 1. So, this leads to a contradiction. Hmm, that means my assumption that the space diagonal of the cube equals the diameter of the larger sphere is incorrect.Maybe the cube is not inscribed in the larger sphere. Instead, the larger sphere must enclose the cube and all the small spheres. So, the radius of the larger sphere must be the distance from the origin to the farthest point of any small sphere.The farthest points would be the centers of the small spheres plus their radius. So, the distance from the origin to the center of a small sphere is (s/2 - 2)*sqrt(3), and adding the radius 2, we get R = (s/2 - 2)*sqrt(3) + 2.But we also need to relate 's' to the cube. Since each small sphere is tangent to one face of the cube, the distance from the center of the small sphere to that face is 2, so s/2 = (s/2 - 2) + 2, which again is just s/2 = s/2, which doesn't help.Wait, maybe the cube's side length is determined by the placement of the small spheres. Since each small sphere is tangent to one face and three edges, the cube must be large enough such that the small spheres don't overlap.But I'm stuck here. Maybe I should consider the configuration differently. If each small sphere is tangent to one face and three edges, then the cube must have side length such that the distance from the center of the cube to the center of each small sphere is (s/2 - 2)*sqrt(3), and this must be equal to the radius of the larger sphere minus the radius of the small sphere.Wait, no. The radius of the larger sphere must be equal to the distance from the origin to the farthest point of any small sphere. The farthest points are the surfaces of the small spheres away from the origin. So, for the small spheres not at the origin, their centers are at (s/2 - 2, s/2 - 2, s/2 - 2), and their surfaces are 2 units away from their centers in all directions. So, the farthest point from the origin on such a small sphere would be along the line from the origin to the center of the small sphere, extended by 2 units.Therefore, the distance from the origin to the farthest point on the small sphere is (s/2 - 2)*sqrt(3) + 2.Similarly, the sphere at the origin has its surface at a distance of 2 from the origin. So, the radius R of the larger sphere must satisfy R = max[(s/2 - 2)*sqrt(3) + 2, 2]. Since (s/2 - 2)*sqrt(3) + 2 is greater than 2 for s > 4, which it must be because s/2 - 2 must be positive.So, R = (s/2 - 2)*sqrt(3) + 2.Now, we need to find 's'. How?Wait, each small sphere is tangent to three edges. The distance from the center of the small sphere to each edge is 2. The distance from a point (x, y, z) to an edge can be calculated using the formula for the distance from a point to a line in 3D.For example, the distance from (x, y, z) to the edge along the x-axis (which is the line where y=0, z=0) is sqrt(y^2 + z^2). Similarly, the distance to the edge along the y-axis is sqrt(x^2 + z^2), and to the edge along the z-axis is sqrt(x^2 + y^2).Since the small sphere is tangent to these three edges, these distances must be equal to the radius, which is 2. So:sqrt(y^2 + z^2) = 2sqrt(x^2 + z^2) = 2sqrt(x^2 + y^2) = 2And we know that x = s/2 - 2, because the small sphere is tangent to the face x = s/2.So, substituting x = s/2 - 2 into the equations:sqrt((s/2 - 2)^2 + z^2) = 2sqrt((s/2 - 2)^2 + y^2) = 2Squaring both sides:(s/2 - 2)^2 + z^2 = 4(s/2 - 2)^2 + y^2 = 4So, z^2 = 4 - (s/2 - 2)^2Similarly, y^2 = 4 - (s/2 - 2)^2Therefore, y = z = sqrt(4 - (s/2 - 2)^2)But we also have sqrt(x^2 + y^2) = 2, so substituting x and y:sqrt((s/2 - 2)^2 + (sqrt(4 - (s/2 - 2)^2))^2) = 2Simplify inside the sqrt:(s/2 - 2)^2 + (4 - (s/2 - 2)^2) = 4Which simplifies to 4 = 4, which is always true. So, this doesn't help us find 's'.Hmm, maybe I need another approach. Let's consider the cube's side length 's'. The small spheres are placed such that their centers are at (s/2 - 2, s/2 - 2, s/2 - 2) and similar positions for the other eight spheres. So, the cube must be large enough to contain these centers.But the cube is centered at the origin, so the distance from the origin to each face is s/2. The centers of the small spheres are at s/2 - 2 from the origin along each axis. Therefore, s/2 - 2 must be less than or equal to s/2, which it is, as long as s > 4.But we need to find 's' such that the small spheres fit without overlapping. The distance between the centers of two adjacent small spheres should be at least 4 (since each has radius 2). Let's calculate the distance between two centers.Take two centers: (s/2 - 2, s/2 - 2, s/2 - 2) and (s/2 - 2, s/2 - 2, -s/2 + 2). The distance between these two points is sqrt[(0)^2 + (0)^2 + (s - 4)^2] = |s - 4|.Since s > 4, this distance is s - 4. For the spheres not to overlap, s - 4 must be at least 4, so s - 4 >= 4 => s >= 8.Wait, that seems too large. If s = 8, then s/2 = 4, so the centers of the small spheres are at 4 - 2 = 2 units from the origin along each axis. So, the centers are at (2, 2, 2) and similar points. The distance between two centers would be sqrt[(2 - (-2))^2 + (2 - 2)^2 + (2 - 2)^2] = sqrt[16 + 0 + 0] = 4, which is exactly the minimum distance required to prevent overlapping. So, s = 8 is the minimum side length.Therefore, the cube has side length s = 8.Now, the radius R of the larger sphere is (s/2 - 2)*sqrt(3) + 2. Substituting s = 8:R = (4 - 2)*sqrt(3) + 2 = 2*sqrt(3) + 2 = 2 + 2*sqrt(3)Wait, but looking at the answer choices, option B is 2 + 2√3, which would be 2 + 2√3. But wait, let me double-check.Wait, s = 8, so s/2 = 4. The center of each small sphere is at (4 - 2, 4 - 2, 4 - 2) = (2, 2, 2). The distance from the origin to this center is sqrt(2^2 + 2^2 + 2^2) = sqrt(12) = 2*sqrt(3). Then, adding the radius of the small sphere, which is 2, the total radius R is 2*sqrt(3) + 2, which is 2 + 2*sqrt(3). So, that's option B.But wait, let me check if this makes sense. The cube has side length 8, so the space diagonal is 8*sqrt(3). The radius of the larger sphere would be half of that, which is 4*sqrt(3). But according to my previous calculation, it's 2 + 2*sqrt(3), which is approximately 2 + 3.464 = 5.464, while 4*sqrt(3) is approximately 6.928. So, why is there a discrepancy?Ah, because the larger sphere doesn't need to enclose the entire cube, just the small spheres. The cube is just a structure to which the small spheres are tangent. The larger sphere only needs to enclose the small spheres, which are placed near the corners of the cube.So, the radius R is the distance from the origin to the farthest point of any small sphere, which is 2*sqrt(3) + 2, as calculated.But wait, let me think again. The center of each small sphere is at (2, 2, 2), which is 2*sqrt(3) away from the origin. Adding the radius 2, the farthest point is 2*sqrt(3) + 2. So, R = 2 + 2*sqrt(3), which is option B.But wait, the answer choices include option D: 2(sqrt(3) + 1), which is the same as 2 + 2*sqrt(3). So, both B and D are the same? Wait, no, option B is 2 + 2√3, and option D is 2(√3 + 1), which is also 2 + 2√3. So, they are the same. Wait, no, 2(√3 + 1) is 2√3 + 2, which is the same as 2 + 2√3. So, both B and D are the same expression, just written differently. But looking back at the problem, the options are:(A) 4 + 2√3 (B) 2 + 2√3 (C) 2√3 (D) 2(√3 + 1) (E) 6So, options B and D are equivalent. But in the original problem, the answer is supposed to be one of these. Wait, maybe I made a mistake in the calculation.Wait, if s = 8, then the centers of the small spheres are at (2, 2, 2). The distance from the origin is sqrt(2^2 + 2^2 + 2^2) = sqrt(12) = 2*sqrt(3). Adding the radius 2, the total radius R is 2*sqrt(3) + 2, which is 2 + 2*sqrt(3), which is option B. But option D is 2(√3 + 1), which is also 2 + 2√3. So, both B and D are the same. But in the original problem, the answer choices are listed as (A) through (E), with B and D being different expressions but equal in value. So, perhaps the intended answer is D, written as 2(√3 + 1).Wait, but in my calculation, I got R = 2 + 2√3, which is option B. But the problem might have intended the answer as D, which is the same value but written differently. So, perhaps both B and D are correct, but since they are the same, it's a bit confusing.Wait, let me check the calculation again. If s = 8, then the centers are at (2, 2, 2). The distance from the origin is 2√3, and adding the radius 2, R = 2√3 + 2, which is 2 + 2√3, which is option B. So, I think the correct answer is B.But wait, let me think about the cube again. If the cube has side length 8, then the space diagonal is 8√3, so the radius of the sphere enclosing the cube would be 4√3, which is larger than 2 + 2√3. But we don't need to enclose the cube, just the small spheres. So, the radius is indeed 2 + 2√3.Therefore, the answer should be B.

💡First, calculate the total cost of the Lego sets by multiplying the number of sets by the cost per set.Next, since Tonya wants to spend the same amount on each sister, the total amount spent on dolls will be equal to the amount spent on Lego sets.Finally, determine the number of dolls by dividing the total amount allocated for dolls by the cost per doll.

💡First, I'll assume the original value of the baseball card is 100 for simplicity.Let ( x % ) be the percent decrease in the first year. After the first year, the value of the card becomes:[100 times (1 - frac{x}{100})]In the second year, the card decreases by 10%, so the value after the second year is:[100 times (1 - frac{x}{100}) times (1 - 0.10)]Given that the total decrease over two years is 37%, the final value of the card is:[100 times (1 - 0.37) = 63]Setting up the equation:[100 times (1 - frac{x}{100}) times 0.90 = 63]Simplifying:[(1 - frac{x}{100}) times 0.90 = 0.63]Solving for ( x ):[1 - frac{x}{100} = frac{0.63}{0.90} = 0.7][frac{x}{100} = 1 - 0.7 = 0.3][x = 0.3 times 100 = 30]Therefore, the percent decrease in the first year is 30%.

💡First, I'll calculate the total number of texts Sydney sent to each person individually.For Allison:- Monday: 5 texts- Tuesday: 15 texts- Wednesday: 20 textsTotal for Allison = 5 + 15 + 20 = 40 textsFor Brittney:- Monday: 5 texts- Tuesday: 10 texts- Wednesday: 18 textsTotal for Brittney = 5 + 10 + 18 = 33 textsFor Carol:- Monday: 5 texts- Tuesday: 12 texts- Wednesday: 7 textsTotal for Carol = 5 + 12 + 7 = 24 textsFinally, I'll add up the totals for each person to find the overall total number of texts sent to all three friends.Total texts to all three = Total for Allison + Total for Brittney + Total for CarolTotal texts to all three = 40 + 33 + 24 = 97 texts