answer:To calculate the total number of metal rods needed for the fence, we need to first determine the number of metal rods required for each fence panel and then multiply that by the total number of fence panels. Each fence panel consists of: - 3 metal sheets - 2 metal beams Each metal sheet is made of 10 metal rods, so for 3 metal sheets, we need: 3 sheets * 10 rods/sheet = 30 rods Each metal beam is made of 4 metal rods, so for 2 metal beams, we need: 2 beams * 4 rods/beam = 8 rods Now, for each fence panel, we need a total of: 30 rods (for sheets) + 8 rods (for beams) = 38 rods Since the fence is made of 10 fence panels, the total number of metal rods needed for the entire fence is: 10 panels * 38 rods/panel = 380 rods Therefore, the community needs boxed{380} metal rods for the fence.

answer:From the problem, we have frac {100}{2500}= frac {a}{1000}= frac {b}{600}, thus a=40, b=24, Therefore, the line ax+by+8=0 becomes 5x+3y+1=0, The distance from A(1, -1) to the line is frac {|5-3+1|}{ sqrt {25+9}}= frac {3}{ sqrt {34}}, Since the line ax+by+8=0 intersects with the circle centered at A(1, -1) at points B and C, and angle BAC=120°, Therefore, r= frac {6}{ sqrt {34}}, Thus, the equation of circle C is (x-1)^2+(y+1)^2= frac {18}{17}, Hence, the correct choice is boxed{C}. This problem is solved by defining a and b based on stratified sampling, using the formula for the distance from a point to a line to find the distance from A(1, -1) to the line, which gives the radius and thus the conclusion. This question tests knowledge of stratified sampling, the equation of a circle, and the relationship between a line and a circle, and is considered a medium-level question.

answer:**Analysis** This question tests the operation of complex numbers and examines the students' computational ability, which is quite basic. First, simplify the complex number using the rules of complex number operations, then calculate its modulus. **Solution** Given ({{left( 1+sqrt{3}{i} right)}^{2}}z=1-{{{i}}^{3}}), we have (left(1+2 sqrt{3}i+3{i}^{2}right)z=1+i ), Thus, (z= dfrac{1+i}{2 sqrt{3}i-2} ), Which leads to (z= dfrac{left(1+iright)left(2 sqrt{3}i+2right)}{left(2 sqrt{3}i-2right)left(2 sqrt{3}i+2right)}= dfrac{left(2-2 sqrt{3}right)+left(2+2 sqrt{3}right)i}{-16} ), Therefore, (left| z right|=dfrac{sqrt{2}}{4}). Hence, the correct answer is boxed{text{D}}.

answer:1. Given the polynomial: [ f(x)=6left(x^{2}+1right)^{2}+5 g(x) cdotleft(x^{2}+1right)-21 g^{2}(x) ] Let ( y = x^2 + 1 ). Then, [ f(x) = 6y^2 + 5g(x)y - 21g^2(x) ] 2. To factorize ( 6y^2 + 5y - 21 = 0 ), we first find the roots using the quadratic formula where ( a = 6 ), ( b = 5 ), and ( c = -21 ): [ y = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] Substituting the values: [ y = frac{-5 pm sqrt{5^2 - 4(6)(-21)}}{2 cdot 6} ] Simplifying under the square root: [ y = frac{-5 pm sqrt{25 + 504}}{12} = frac{-5 pm sqrt{529}}{12} = frac{-5 pm 23}{12} ] Thus, the roots are: [ y_1 = frac{18}{12} = frac{3}{2}, quad y_2 = frac{-28}{12} = -frac{7}{3} ] 3. Therefore, the polynomial can be factored as: [ 6y^2 + 5y - 21 = 6 left( y - frac{3}{2} right) left( y + frac{7}{3} right) = (2y - 3)(3y + 7) ] 4. Substituting back ( y = x^2 + 1 ): [ f(x) = (2(x^2 + 1) - 3g(x)) cdot (3(x^2 + 1) + 7g(x)) ] This gives: [ f(x) = [2(x^2 + 1) - 3g(x)] cdot [3(x^2 + 1) + 7g(x)] ] 5. Assume that there is an integer ( x_0 ) such that ( f(x_0) = 0 ). Then one of the factors must be zero at ( x_0 ). 6. Case 1: If ( 2(x_0^2 + 1) - 3g(x_0) = 0 ), then: [ 2(x_0^2 + 1) = 3g(x_0) implies x_0^2 + 1 = frac{3}{2}g(x_0) ] Since ( g(x_0) ) is an integer, ( x_0^2 + 1 ) must be an integer, hence ( x_0^2 + 1 ) must be divisible by ( frac{3}{2} ), which is a contradiction because ( x_0^2 + 1 ) is not divisible by 3 for any integer ( x_0 ). 7. Case 2: If ( 3(x_0^2 + 1) + 7g(x_0) = 0 ), then: [ 3(x_0^2 + 1) = -7g(x_0) implies x_0^2 + 1 = -frac{7}{3}g(x_0) ] Again, ( x_0^2 + 1 ) is not divisible by 7 for any integer ( x_0 ) as shown by residue calculations modulo 7. Thus, there is a contradiction in both cases. 8. Conclusion: Hence, the polynomial ( f(x) ) cannot have any integer roots. (blacksquare)