Use projection matrices to find the matrix exponential and particular solution of the given linear system x' = Ax+ f(t), x(a) = xa. 5 1 3 ×-[2]}~-[:] ×0-[:] |x, f(t)= x(0) = 25 -5 2 X'= Find the projection matrix(matrices) for A. 9 10 0 1 (Type exact answers, using radicals and i as needed. Use a comma to separate matrices as needed.) Find the matrix exponential. At = 5 The projection matrix(matrices) is/are 1 + 5t - t 25t 1 - 5t Find the particular solution to the initial value problem. x(t) =

Advanced Math Question

A tank can be filled by 2 water keys together in 3 hours with 3 quarters. The main key would be able to fill it by itself in 4 hours less than the other one. In how many hours can each key fill the tank separately?

A homeowner wants to build, along his driveway, a garden surrounded by a fence. A rectangular garden has been enclosed along a side of a driveway. The enclosure is bordered by a length of fence along the driveway (a long side), another long side of fence, and two short sides of fence. The side along the driveway is labeled "$6 per foot" while the remaining three sides are labeled "$2 dollars per foot." If the garden is to be 1,800 square feet, and the fence along the driveway costs $6 per foot while on the other three sides it costs only $2 per foot, find the dimensions (in ft) that will minimize the cost. length along the driveway  ft length perpendicular to the driveway      ft Find the minimum cost (in dollars). $