Aprofessor bases semester grades on four 100-point items: homework, quizzes, a midterm exam, and a final exam. students may choose one of the three schemes, scheme i, scheme ii and scheme iii, where • scheme i: hw 10%, quiz 10%, midterm 30% and finalterm 50%; • scheme ii: hw 10%, quiz 20%, midterm 30% and finalterm 40%; • scheme iii: hw 15%, quiz 15%, midterm 35% and finalterm 35%. 1. find the matrix which gives the link between these three schemes and these four different kind of examinations. 2. using a multiplication between two matrices, determine the most advantageous weighting scheme for a student who earned 97

Submarines control how much they float or sink in the ocean by changing the volume of air and water contained in large ballast tanks. when the tanks are completely full of water, the submarine increases its overall mass and sinks down to the bottom. when the tanks are completely full of air, the submarine reduces its overall mass and floats to the surface. depending on the density of the seawater surrounding the submarine, it will pump seawater in or out of the tanks in order to achieve the same overall density as the sea water and float neutrally in the water. the volume of the submarine never changes. when the tanks are completely full of water, a submarine with a volume of 7.8\times10^3\text{ m}^37.8×10 3 m 3 7, point, 8, times, 10, start superscript, 3, end superscript, space, m, start superscript, 3, end superscript has a total mass of 8\times10^6\text{ kg}8×10 6 kg8, times, 10, start superscript, 6, end superscript, space, k, g. the density of the seawater is 10^3\text{ kg/m}^310 3 kg/m 3 10, start superscript, 3, end superscript, space, k, g, slash, m, start superscript, 3, end superscript. to make that submarine float neutrally, and neither float nor sink in the ocean, what volume of water does that submarine need to subtract from its tanks?