answer:The builder has already installed 8 windows, so there are 14 - 8 = 6 windows left to install. If it takes 8 hours to install each window, then for 6 windows, it will take 6 windows * 8 hours/window = boxed{48} hours to install the rest of the windows.

answer:First, find the prime factorizations of 660 and 924. - 660 = 2^2 cdot 3 cdot 5 cdot 11 - 924 = 2^2 cdot 3 cdot 7 cdot 11 Now, find the GCD by taking the lowest power of all prime factors present in both factorizations. - Common primes are 2, 3, and 11. - The smallest power of 2 present in both factorizations is 2^2. - The smallest power of 3 present in both factorizations is 3^1. - The smallest power of 11 present in both factorizations is 11^1. Thus, the GCD is 2^2 cdot 3^1 cdot 11^1 = 4 cdot 3 cdot 11 = 132. Conclusion: The GCD of 660 and 924 is boxed{132}.

answer:Given the function: [ y = frac{4x+1}{16x^2 + 8x + 3} + frac{1}{sqrt{2}} cdot operatorname{arctg} frac{4x+1}{sqrt{2}} ] We need to find the derivative of (y). 1. Start by differentiating each term individually: [ y' = left( frac{4x+1}{16x^2 + 8x + 3} right)' + left( frac{1}{sqrt{2}} cdot operatorname{arctg} frac{4x+1}{sqrt{2}} right)' ] 2. Use the quotient rule for the first term ( frac{4x+1}{16x^2 + 8x + 3} ): [ left( frac{4x+1}{16x^2 + 8x + 3} right)' = frac{(4x+1)' cdot (16x^2 + 8x + 3) - (4x+1) cdot (16x^2 + 8x + 3)'}{(16x^2 + 8x + 3)^2} ] 3. Calculate the derivatives of the numerator and the denominator: [ (4x+1)' = 4 ] [ (16x^2 + 8x + 3)' = 32x + 8 ] 4. Substitute back into the quotient rule formula: [ left( frac{4x+1}{16x^2 + 8x + 3} right)' = frac{4 cdot (16x^2 + 8x + 3) - (4x+1) cdot (32x + 8)}{(16x^2 + 8x + 3)^2} ] 5. Expand the numerator: [ = frac{4 cdot (16x^2 + 8x + 3) - (4x+1) cdot (32x + 8)}{(16x^2 + 8x + 3)^2} ] [ = frac{64x^2 + 32x + 12 - (4x + 1)(32x + 8)}{(16x^2 + 8x + 3)^2} ] [ = frac{64x^2 + 32x + 12 - (128x^2 + 32x + 32x + 8)}{(16x^2 + 8x + 3)^2} ] 6. Simplify the numerator: [ = frac{64x^2 + 32x + 12 - 128x^2 - 64x - 8}{(16x^2 + 8x + 3)^2} ] [ = frac{-64x^2 - 32x + 4}{(16x^2 + 8x + 3)^2} ] 7. Now, differentiate the second term: (frac{1}{sqrt{2}} cdot operatorname{arctg} frac{4x+1}{sqrt{2}}): [ left( frac{1}{sqrt{2}} cdot operatorname{arctg} frac{4x+1}{sqrt{2}} right)' ] Use the chain rule: [ = frac{1}{sqrt{2}} cdot frac{1}{1+left(frac{4x+1}{sqrt{2}}right)^2} cdot left( frac{4x+1}{sqrt{2}} right)' ] 8. Differentiate the inner function (frac{4x+1}{sqrt{2}}): [ left( frac{4x+1}{sqrt{2}} right)' = frac{4}{sqrt{2}} = frac{4}{sqrt{2}} cdot frac{sqrt{2}}{sqrt{2}} = frac{4sqrt{2}}{2} = 2sqrt{2} ] 9. Substitute back: [ = frac{1}{sqrt{2}} cdot frac{2sqrt{2}}{2 + (4x+1)^2} = frac{4}{16x^2 + 8x + 3} ] 10. Combine both derivatives: [ y' = frac{-64x^2 - 32x + 4}{(16x^2 + 8x + 3)^2} + frac{4}{16x^2 + 8x + 3} ] 11. Simplify the result: [ y' = frac{-64x^2 - 32x + 4 + 4(16x^2 + 8x + 3)}{(16x^2 + 8x + 3)^2} ] [ = frac{-64x^2 - 32x + 4 + 64x^2 + 32x + 12}{(16x^2 + 8x + 3)^2} ] [ = frac{16}{(16x^2 + 8x + 3)^2} ] Conclusion: [ boxed{frac{16}{(16x^2 + 8x + 3)^2}} ]

answer:To solve for angle A in triangle triangle ABC using the given information, we apply the Law of Sines, which states frac{a}{sin A} = frac{b}{sin B}. Given a=4, b=4sqrt{3}, and B=60^{circ}, we can substitute these values into the equation: [ sin A = frac{a sin B}{b} = frac{4 times frac{sqrt{3}}{2}}{4sqrt{3}} = frac{2sqrt{3}}{4sqrt{3}} = frac{1}{2} ] Given sin A = frac{1}{2}, we know that A could theoretically be 30^{circ} or 150^{circ} because the sine function yields a value of frac{1}{2} at both of these angles in the unit circle. However, since a < b, it implies that A < B. Given that B = 60^{circ}, the only possible value for A that is less than B and satisfies sin A = frac{1}{2} is 30^{circ}. Therefore, the correct answer is boxed{text{A: } 30^{circ}}.