answer:The Great Barrier Reef, as the world's largest coral reef system, holds significant ecological and economic value for Australia. Firstly, it serves as a natural resource provider, supplying food through fishing industries and contributing to the development of pharmaceuticals. Secondly, it is known for its immense biodiversity, hosting a wide range of species in a relatively small geographic area, which is vital for the study of marine life and conservation efforts. Lastly, the reef plays a crucial role in marine ecosystems by serving as a source of nitrogen, supporting the growth of various organisms within the food chain. However, the Great Barrier Reef is currently facing numerous threats, including global warming and poor management practices. In response, the Australian Government has allocated significant funding, such as the 40 million dollars, to initiatives aimed at preserving the reef's health and maintaining its ecological balance.

answer:As the bucket is lifted, sand leaks out, causing the weight of the bucket and sand to decrease. To lift the bucket, a force equal to the weight of the bucket and sand must be exerted upward. When the bucket is y feet above the floor of the building, 0.1y pounds of sand has leaked out, and the weight of the sand remaining and the bucket equals (15 - 0.1y) pounds. Therefore, the force required to lift the bucket is F(y) = 15 - 0.1y pounds. The work done in lifting the bucket a small distance dy is F(y) dy = (15 - 0.1y) dy foot-pounds. The work done in lifting the bucket to the top of the building is int_{0}^{50} (15 - 0.1y) dy = 15y - 0.05y^2 bigg|_{0}^{50} = 625 text{ foot-pounds.} Therefore, the work done lifting the bucket to the top of the building is 625 foot-pounds.

answer:To show pi_{m,n} is well-defined, we need to prove that if [a]_n = [b]_n, then pi_{m,n}([a]_n) = pi_{m,n}([b]_n). Assume [a]_n = [b]_n. This implies a equiv b bmod{n}, which means n mid (a-b). Since m mid n, we also have m mid (a-b), leading to a equiv b bmod{m}. Therefore, [a]_m = [b]_m. Thus, pi_{m,n}([a]_n) = [a]_m = [b]_m = pi_{m,n}([b]_n), and the function is well-defined.

answer:To find frac{x}{y}, we can divide the real parts and the imaginary parts separately. Given x = -2 - 5i and y = -1 - 6i, the real part of the fraction is frac{-2}{-1} = 2, and the imaginary part is frac{-5}{-6} = frac{5}{6}. So, frac{x}{y} = 2 + frac{5}{6}i. To express the answer in standard form (a + bi), we can multiply the imaginary part by 6 to get the denominator to be 37: frac{x}{y} = 2 + frac{5}{6}i = 2 + left(frac{5}{6} cdot frac{37}{37}right)i = 2 + frac{37}{37}i - frac{7}{37}i = frac{37}{37} + frac{30}{37}i - frac{7}{37}i = frac{32}{37} - frac{7}{37}i. Therefore, the complex fraction simplified in standard form is frac{32}{37} - frac{7}{37}i.