Various Math #29 - Q&A

Question 1 Find the closest point to y in the subspace W spanned by v₁ and v₂ y = [3, -1, 1], v₁ = [1, 1, -1], v₂ = [3, 1, -1] Answer Step 1 The formula of orthogonal projection of y onto span v₁, v₂ and this projection is the closest point to y in the subspace W. ŷ = (y⋅v₁/v₁⋅v₁) * v₁ + (y⋅v₂/v₂⋅v₂) * v₂ Step 2 because y⋅v₁ = 6, v₁⋅v₁ = 12, y⋅v₂ = 6, v₂⋅v₂ = 4 ŷ = (6/12) * [1, 1, -1] + (6/4) * [3, 1, -1] = (1/2) * [1, 1, -1] + (3/2) * [3, 1, -1] Step 3 The required closest point to y in the subspace W. ŷ = [1/2, 1/2, -1/2] + [9/2, 3/2, -3/2] = [10/2, 4/2, -4/2] Result ŷ = [5, 2, -2] Question 2 Calculate the ratio of effusion rates for Ar and Kr. Answer Determine the molar mass of argon M_Ar = 39.948 g/mol Determine the molar mass of krypton. M_Kr = 83.798 g/mol Substitute these molar masses into Graham's law to determine the ratio of effusion rates. rate_Ar/rate_Kr = √(M_Kr/M_Ar) = √(83.798/39.948) = 1.4483 Result: 1.4483 Question 3 Match each of the tables shown below with the contour diagrams below them. a) y/x -1 0 1 -1 2 1 2 0 1 0 1 1 2 1 2 b) y/x -1 0 1 -1 0 1 0 0 1 2 1 1 0 1 0 c) y/x -1 0 1 -1 2 2 2 0 0 0 0 1 2 2 2 d) y/x -1 0 1 -1 0 1 2 0 1 2 1 1 2 1 0 Table (a): graph Table (b): graph Table (c): graph Table (d): graph 1) 2) 3) 4) Answer Step 1 The data in table (a) is radically symmetric. So, the contours are circles. From table at (0, 0), f(x, y) = 0 The contour z = 0 at the origin and the higher values of z contours moves away from the origin. Thus, table (a) matches with the graph (1). Step 2 The data in table (b) is radically symmetric. So, the contours are circles. From table at (0, 0), f(x, y) = 2 The contour z = 2 at the origin and the lower values of z contours moves away from the origin. Thus, table (b) matches with the graph (3). Step 3 From the table, observe that for a given values of y, z values are same. So, the contours are horizontal lines. Thus, table (c) matches with the graph (6). Step 4 Form the table, observe that the point (0, 0) lie on the contour z = 2. So, the table (d) matches with the graph (4). Question 4 A trough is 12 feet long and 3 feet across the top (see figure). Its ends are isosceles triangles with altitudes of 3 feet. a) Water is being pumped into the trough at 2 cubic feet per minute. How fast is the water level rising when the depth h is 1 foot? b) The water is rising at a rate of 3/8 inch per minute when h = 2 feet. Determine the rate at which water is being pumped into the trough. Answer Simple volume equation. The depth is 12 feet, the height is 3 feet. V = (1/2) * bh12 = 6bh = 6h² Solved derivatives for a) dV/dt = 12h * dh/dt, dV/dt = 2, and h= 1 2 = 12(1) * dh/dt, then dh/dt = 2/12 = 1/6 ft/min b) if dh/dt = 3/8 in/min = 1/32 ft/min and h=2 feet, Use our volume equation's derivative. dV/dt = 12h * dh/dt Question 5 Find a basis for the eigenspace corresponding to each listed eigenvalue. A = [[5, -1], [2, 1]], λ = 1, 5 Answer We first solve the system to obtain the foundation for the eigenspace. (A – λI)x = 0 For λ = 1, A-l= [[5-1, -1],[2, 1-1]] = [[4,-1],[2,0]] The augmented matrix of (A-1)x = 0 is [[4, -1, 0], [2, 0, 0]] With x₁ = 0 and x₂ is unrestricted, the answer may be expressed as follows: x = x₂ * [[0], [1]] So [[0], [1]] is the foundation of the eigenspace. For λ = 5, A - 5I = [[5-5, -1], [2, 1-5]] = [[0, -1], [2, -4]] The augmented matrix of (A-5I)x=0 is [[0, -1, 0], [2, -4, 0]] That leads to 2x₁ - 4x₂ = 0 => x₁ = 2x₂. The answer may be written as follows: x = x₂ * [[2], [1]] So [[2], [1]] is the foundation of the eigenspace.