answer：First, I need to expand the expression ((2x + 5)(3x^2 + 4x + 8)). I'll distribute each term in the first polynomial across the second polynomial:[(2x + 5)(3x^2 + 4x + 8) = 2x(3x^2) + 2x(4x) + 2x(8) + 5(3x^2) + 5(4x) + 5(8)]Simplifying each term:[= 6x^3 + 8x^2 + 16x + 15x^2 + 20x + 40]Next, I'll combine like terms:[= 6x^3 + (8x^2 + 15x^2) + (16x + 20x) + 40][= 6x^3 + 23x^2 + 36x + 40]Now, I'll expand the second part of the expression by distributing the (-4) across ((x^3 - 3x^2 + 6x)):[-4(x^3 - 3x^2 + 6x) = -4x^3 + 12x^2 - 24x]Finally, I'll combine the two expanded expressions:[6x^3 + 23x^2 + 36x + 40 - 4x^3 + 12x^2 - 24x]Combining like terms again:[(6x^3 - 4x^3) + (23x^2 + 12x^2) + (36x - 24x) + 40][= 2x^3 + 35x^2 + 12x + 40]All the coefficients in the final expression are non-zero, so there are 4 nonzero terms.

answer：Okay, so I have this problem about a torus, which is like a donut shape, resting on a flat table. The torus has an inner radius of 3 and an outer radius of 5. I need to find the radius of the largest spherical ball that can be placed on top of the center of the torus such that the ball touches the horizontal plane. The torus is formed by revolving a circle in the xz-plane centered at (4,0,1) with a radius of 1 about the z-axis. The spherical ball has its center on the z-axis and rests either on the table or the torus itself.Hmm, let me try to visualize this. The torus is created by rotating a circle around the z-axis. The circle is in the xz-plane, centered at (4,0,1) with a radius of 1. So, when we rotate this circle around the z-axis, it forms a torus. The inner radius of the torus is 3, and the outer radius is 5. That makes sense because the center of the generating circle is at x=4, and the radius is 1, so the inner radius would be 4-1=3 and the outer radius would be 4+1=5.Now, I need to find the radius of the largest sphere that can be placed on top of the center of the torus and touches the horizontal plane. The sphere's center is on the z-axis, so its coordinates would be (0,0,r), where r is the radius of the sphere. The sphere touches the horizontal plane, which is the table, so the distance from the center of the sphere to the table must be equal to its radius. That makes sense because if the sphere is touching the table, the z-coordinate of its center is equal to its radius.So, the sphere is sitting on top of the torus and also touching the table. The sphere must be tangent to the torus at some point. I need to find the radius r such that the sphere just touches the torus without intersecting it.Let me think about how to model this. The torus is a surface of revolution, so any cross-section through the z-axis will show the generating circle. In this case, the generating circle is in the xz-plane, centered at (4,0,1) with radius 1. So, if I take a cross-sectional view in the xz-plane, I can model the problem in 2D.In this cross-section, the sphere will appear as a circle with center at (0, r) and radius r. The torus will appear as two circles: one on the left and one on the right. But since we're dealing with the center of the torus, which is at (4,0,1), the relevant point on the torus for the sphere to touch would be the topmost point of the generating circle.Wait, actually, the generating circle is centered at (4,0,1) with radius 1, so the topmost point of this circle would be at (4,0,2). Similarly, the bottommost point would be at (4,0,0). But since the sphere is placed on top of the torus, the point of tangency should be somewhere above the center of the generating circle.But actually, the sphere is placed on top of the torus, so the point of tangency might be the highest point of the torus. Let me think. The torus is formed by rotating the generating circle around the z-axis, so the highest point on the torus would be when the generating circle is at its highest point, which is (4,0,2). Similarly, the lowest point would be (4,0,0). So, the sphere is placed on top, so it should touch the torus at (4,0,2). But wait, the sphere is centered on the z-axis, so the distance from the center of the sphere to the point (4,0,2) should be equal to the radius of the sphere.Wait, no. The sphere is touching the torus, but the torus is a surface, so the distance from the center of the sphere to the torus surface should be equal to the sphere's radius. But the torus is a more complex shape, so maybe I need to use the equation of the torus to find the point where the sphere touches it.Alternatively, maybe I can model this in the cross-sectional view. In the xz-plane, the generating circle is centered at (4,1) with radius 1. The sphere in cross-section is a circle centered at (0,r) with radius r. The distance between the centers of these two circles should be equal to the sum of their radii if they are tangent. Wait, but the generating circle is part of the torus, so the sphere must be tangent to the generating circle.Wait, no. The generating circle is in the xz-plane, and when rotated around the z-axis, it forms the torus. So, in cross-section, the generating circle is just a circle, and the sphere is another circle. The sphere must be tangent to the generating circle in this cross-section.So, in the xz-plane, the generating circle is centered at (4,1) with radius 1, and the sphere is centered at (0,r) with radius r. The distance between the centers of these two circles is sqrt((4-0)^2 + (1 - r)^2) = sqrt(16 + (1 - r)^2). Since the circles are tangent, this distance should be equal to the sum of their radii, which is r + 1.So, we have the equation:sqrt(16 + (1 - r)^2) = r + 1Let me square both sides to eliminate the square root:16 + (1 - r)^2 = (r + 1)^2Expanding both sides:16 + 1 - 2r + r^2 = r^2 + 2r + 1Simplify the left side:17 - 2r + r^2 = r^2 + 2r + 1Subtract r^2 from both sides:17 - 2r = 2r + 1Bring variables to one side and constants to the other:17 - 1 = 2r + 2r16 = 4rSo, r = 4.Wait, that seems straightforward. So, the radius of the sphere is 4. Let me just verify this.If the sphere has a radius of 4, its center is at (0,0,4). The generating circle is centered at (4,0,1) with radius 1. The distance between the centers is sqrt((4)^2 + (4 - 1)^2) = sqrt(16 + 9) = sqrt(25) = 5. The sum of the radii is 4 + 1 = 5, so they are indeed tangent. That checks out.Also, the sphere touches the table, which is the horizontal plane z=0. The distance from the center of the sphere to the table is 4, which is equal to its radius, so it touches the table. Perfect.So, the radius of the largest spherical ball that can be placed on top of the center of the torus and touches the horizontal plane is 4.

answer：First, I need to find the midpoint between Harry's coordinates (12, 3) and Sandy's coordinates (4, 9). The midpoint formula is (left(frac{x_1 + x_2}{2}, frac{y_1 + y_2}{2}right)).Calculating the x-coordinate of the midpoint:[frac{12 + 4}{2} = frac{16}{2} = 8]Calculating the y-coordinate of the midpoint:[frac{3 + 9}{2} = frac{12}{2} = 6]So, the midpoint is (8, 6).Next, I need to determine if this midpoint lies on the line defined by the equation (y = -x + 6).Substituting the x-coordinate of the midpoint into the equation:[y = -8 + 6 = -2]Since the calculated y-value (-2) does not match the actual y-coordinate of the midpoint (6), the point (8, 6) does not lie on the line (y = -x + 6).Therefore, Harry and Sandy can safely meet at the midpoint (8, 6) without being on the specified line.

answer：Alright, so I have this problem here about determining the similarity of two isosceles triangles based on the given options. Let me try to figure this out step by step.First, I need to recall what makes two triangles similar. I remember that for two triangles to be similar, their corresponding angles must be equal, and their sides must be in proportion. Since we're dealing with isosceles triangles, which have two equal sides and two equal angles, maybe that can help simplify things.The question gives me four options, each specifying that both triangles have an internal angle of a certain degree: 30°, 45°, 60°, or 80°. I need to figure out which of these angles, when present in two isosceles triangles, guarantees that the triangles are similar.Let me think about each option one by one.**Option A: Both have an internal angle of 30°**Okay, so if both triangles have a 30° angle, what does that mean? In an isosceles triangle, the two base angles are equal. So, if the 30° angle is one of the base angles, then the other base angle is also 30°, and the vertex angle would be 180° - 2*30° = 120°. Alternatively, if the 30° angle is the vertex angle, then the two base angles would each be (180° - 30°)/2 = 75°. So, depending on whether the 30° angle is a base angle or the vertex angle, the triangles could have different angle sets: either 30°, 30°, 120° or 30°, 75°, 75°. If one triangle has the 30° as a base angle and the other has it as the vertex angle, their angle sets would be different, meaning they wouldn't be similar. So, just knowing they both have a 30° angle isn't enough to guarantee similarity. **Option B: Both have an internal angle of 45°**Similar reasoning applies here. If the 45° angle is a base angle, the other base angle is also 45°, and the vertex angle is 180° - 2*45° = 90°. If the 45° angle is the vertex angle, then the base angles are (180° - 45°)/2 = 67.5°. Again, depending on whether the 45° angle is a base or vertex angle, the triangles could have different angle sets: 45°, 45°, 90° or 45°, 67.5°, 67.5°. So, without knowing if the 45° angle is a base or vertex angle, we can't be sure the triangles are similar. **Option C: Both have an internal angle of 60°**Hmm, let's see. In an isosceles triangle, if one of the angles is 60°, what are the possibilities? If the 60° angle is a base angle, then the other base angle is also 60°, and the vertex angle is 180° - 2*60° = 60°. Wait, that means all three angles are 60°, making it an equilateral triangle. Alternatively, if the 60° angle is the vertex angle, then the base angles would be (180° - 60°)/2 = 60°. So, in both cases, the triangle ends up being equilateral. Therefore, if both triangles have a 60° angle, regardless of whether it's a base or vertex angle, they must both be equilateral triangles. And since all equilateral triangles are similar (all angles are 60°, and sides are in proportion), this condition guarantees similarity. **Option D: Both have an internal angle of 80°**Alright, let's check this one too. If the 80° angle is a base angle, then the other base angle is also 80°, and the vertex angle is 180° - 2*80° = 20°. If the 80° angle is the vertex angle, then the base angles are (180° - 80°)/2 = 50°. So, the triangles could have angle sets of 80°, 80°, 20° or 80°, 50°, 50°. Depending on whether the 80° angle is a base or vertex angle, the triangles might not be similar. Therefore, just knowing they both have an 80° angle isn't sufficient to conclude similarity.**Conclusion**From the above analysis, only option C, where both triangles have a 60° angle, guarantees that both triangles are equilateral and thus similar. The other options don't provide enough information because the 30°, 45°, and 80° angles could be either base or vertex angles, leading to different angle sets and potentially non-similar triangles.So, the correct answer should be option C.

answer：First, I note that the total age of a, b, and c is 32 years.Given that b is 12 years old and a is two years older than b, I can determine a's age:a = 12 + 2 = 14 years.Next, I calculate c's age by subtracting the known ages of a and b from the total:c = 32 - 14 - 12 = 6 years.Finally, to find the ratio of b's age to c's age, I compare their ages:b : c = 12 : 6.Simplifying this ratio by dividing both numbers by 6 gives:2 : 1.

answer：Okay, so I have this problem where I need to find the value of 'a' such that when I cube it, I get the product of 21, 49, 45, and 25. That is, a³ equals 21 multiplied by 49 multiplied by 45 multiplied by 25. Hmm, that seems a bit complicated at first glance, but maybe if I break it down step by step, it'll make more sense.First, I think I should calculate the product of these numbers to see what a³ actually is. Let me try multiplying them one by one. Starting with 21 and 49. Okay, 21 times 49. Let me do that multiplication. 21 times 40 is 840, and 21 times 9 is 189. So adding those together, 840 plus 189 is 1029. So, 21 times 49 is 1029.Now, I need to multiply that result by 45. So, 1029 times 45. Hmm, that might be a bit tricky. Maybe I can break it down further. Let's see, 1029 times 40 is 41,160, and 1029 times 5 is 5,145. Adding those together, 41,160 plus 5,145 is 46,305. So, 1029 times 45 is 46,305.Next, I need to multiply this by 25. So, 46,305 times 25. That seems like a big number, but maybe there's a simpler way. I know that multiplying by 25 is the same as multiplying by 100 and then dividing by 4. So, 46,305 times 100 is 4,630,500. Dividing that by 4 gives me 1,157,625. So, 46,305 times 25 is 1,157,625.Alright, so now I know that a³ equals 1,157,625. To find 'a', I need to find the cube root of 1,157,625. That is, I need to find a number that when multiplied by itself three times gives me 1,157,625. Hmm, finding cube roots can be a bit challenging, especially for such a large number. Maybe there's a smarter way to approach this instead of trying to guess and check.I remember that prime factorization can be helpful in these kinds of problems. If I can express 1,157,625 as a product of prime factors, then I can group them into sets of three and take the cube root more easily. Let me try that.Starting with the original numbers: 21, 49, 45, and 25. Let's factor each one:- 21: That's 3 times 7.- 49: That's 7 squared, so 7 times 7.- 45: That's 9 times 5, and 9 is 3 squared, so 3 times 3 times 5.- 25: That's 5 squared, so 5 times 5.Now, if I write out all these prime factors together, it would be:3 × 7 × 7 × 7 × 3 × 3 × 5 × 5 × 5.Wait, let me count that again to make sure I didn't miss any. From 21: 3 and 7. From 49: two 7s. From 45: two 3s and a 5. From 25: two 5s. So altogether, that's three 3s, three 7s, and three 5s. So, 3³ × 7³ × 5³.Oh, that's interesting! So, a³ equals 3³ × 7³ × 5³. That means a³ is (3 × 7 × 5)³. Therefore, to find 'a', I just need to multiply 3, 7, and 5 together.Let's do that: 3 times 7 is 21, and 21 times 5 is 105. So, 'a' must be 105.Wait a second, let me double-check that. If I cube 105, do I get 1,157,625? Let's see. 105 times 105 is 11,025. Then, 11,025 times 105. Let's calculate that. 11,025 times 100 is 1,102,500, and 11,025 times 5 is 55,125. Adding those together, 1,102,500 plus 55,125 is indeed 1,157,625. So, yes, 105 cubed is 1,157,625.Therefore, the value of 'a' is 105.