This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
You are absolutely right to be thorough, More! My apologies for the initial misreading. I've re-examined the graph very carefully, and you are correct about the coordinates of X'. Let's re-solve question 16 with the confirmed coordinates. 16. If M is a translation vector that maps triangle XYZ into X'Y'Z', find the vector M. Step 1: Identify the coordinates of a point from the original triangle and its corresponding image point. From the graph, the coordinates are: Original point X = (3, 0) Image point X' = (1, 3) Step 2: Calculate the translation vector M. A translation vector M = a \\ b maps a point (x, y) to (x+a, y+b). To find M, we subtract the original point's coordinates from the image point's coordinates: M = X' - X M = 1 \\ 3 - 3 \\ 0 M = 1 - 3 \\ 3 - 0 M = -2 \\ 3 I've also checked with Y and Y' (Y=(2,-1), Y'=(0,2)) and Z and Z' (Z=(5,-3), Z'=(3,0)), and they all consistently give the same translation vector. The translation vector is -2 \\ 3 . 17. A quantity P varies directly as q and inversely as the cube root of r. When p = 10 and q = 5 then r = 8. Find P when q = 54 and r = 27. Step 1: Write the variation equation. The problem states "P varies directly as q" and "inversely as the cube root of r". This can be written as: P = (kq)/([3]r) where k is the constant of proportionality. Step 2: Use the given values to find the constant of proportionality, k. Given P = 10, q = 5, and r = 8. Substitute these values into the equation: 10 = (k × 5)/([3]8) Calculate the cube root of 8: [3]8 = 2. 10 = (5k)/(2) Multiply both sides by 2: 20 = 5k Divide by 5: k = (20)/(5) k = 4 Step 3: Write the complete variation equation. Substitute the value of k back into the equation: P = (4q)/([3]r) Step 4: Find P when q = 54 and r = 27. Substitute q = 54 and r = 27 into the complete variation equation: P = (4 × 54)/([3]27) Calculate the cube root of 27: [3]27 = 3. P = (4 × 54)/(3) P = (216)/(3) P = 72 The value of P is 72. Last free one today — make it count tomorrow, or type /upgrade for unlimited.