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
Step 1: Calculate the bottom diameter of the taper and the taper half-angle. The taper is 1-in-5 on the diameter, meaning the change in diameter for every 5 units of length is 1 unit. The taper length L = 80 mm. The top diameter D_top = 50 mm. The total change in diameter over the length of the taper is: Change in diameter = (1)/(5) × L = (1)/(5) × 80 mm = 16 mm Since the taper widens towards the bottom (as shown in the diagram), the bottom diameter D_bottom is: D_bottom = D_top + Change in diameter = 50 mm + 16 mm = 66 mm The half-angle of the taper, , is given by the tangent of the angle, which is the change in radius over the length: () = (D_bottom - D_top)/2L = ((66 - 50)/2)/(80) = (16/2)/(80) = (8)/(80) = (1)/(10) From () = (1)/(10), we can find (): () = (())/(sqrt(1 + ^2())) = (1/10)/(sqrt(1 + (1/10)^2)) = (1/10)/(sqrt(1 + 1/100)) = (1/10)/(sqrt(101/100)) = (1/10)/(sqrt(101)/10) = (1)/(sqrt(101)) The roller radius r = (18)/(2) = 9 mm. Step 2: Determine the heights of the roller centers. From the diagram, the bottom rollers are positioned such that their lowest point is aligned with the bottom of the taper. Therefore, the height of the center of the bottom rollers from the bottom of the taper, h_B, is equal to the roller radius: h_B = r = 9 mm The center distance between the top and bottom rollers is 55 mm. So, the height of the center of the top rollers from the bottom of the taper, h_T, is: h_T = h_B + 55 mm = 9 mm + 55 mm = 64 mm Step 3: Calculate the taper diameters at the roller center heights. The diameter of the taper at any height h from the bottom is given by the linear relationship: D(h) = D_bottom - D_bottom - D_topL × h = 66 - (16)/(80) × h = 66 - (1)/(5)h For the bottom rollers (at h_B = 9 mm): D_B = 66 - (1)/(5)(9) = 66 - 1.8 = 64.2 mm For the top rollers (at h_T = 64 mm): D_T = 66 - (1)/(5)(64) = 66 - 12.8 = 53.2 mm Step 4: Calculate the horizontal distance from the taper center line to the center of each roller. Let C_x be the horizontal distance from the center line of the taper to the center of a roller. Due to the geometry of a roller tangent to a tapered surface, this distance is given by: C_x = (D_h)/(2) + r () For the bottom rollers: C_xB = (D_B)/(2) + r () = (64.2)/(2) + 9 × (1)/(sqrt(101)) = 32.1 + (9)/(sqrt(101)) mm For the top rollers: C_xT = (D_T)/(2) + r () = (53.2)/(2) + 9 × (1)/(sqrt(101)) = 26.6 + (9)/(sqrt(101)) mm Step 5: Calculate the distances X and Y. The distance X is the total distance across the bottom rollers, from the outermost point of the left roller to the outermost point of the right roller. This is twice the sum of the horizontal distance from the center line to the roller center (C_xB) and the roller radius (r). X = 2 × (C_xB + r) = 2 × (32.1 + (9)/(sqrt(101)) + 9) = 2 × (41.1 + (9)/(sqrt(101))) X = 82.2 + (18)/(sqrt(101)) ≈ 82.2 + 1.79196 ≈ 83.99196 mm The distance Y is the total distance across the top rollers, calculated similarly: Y = 2 × (C_xT + r) = 2 × (26.6 + (9)/(sqrt(101)) + 9) = 2 × (35.6 + (9)/(sqrt(101))) Y = 71.2 + (18)/(sqrt(101)) ≈ 71.2 + 1.79196 ≈ 72.99196 mm Rounding to two decimal places: X ≈ 83.99 mm Y ≈ 72.99 mm The distance (Y) across the top rollers is 72.99 mm. The distance (X) across the bottom rollers is 83.99 mm. Send me the next one 📸