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 capacity of the bucket (frustum). The formula for the volume of a frustum is V = (1)/(3) h (R^2 + Rr + r^2). From the diagram and problem description: Height of bucket, h = 36 cm Larger diameter = 45 cm, so larger radius R = (45)/(2) = 22.5 cm Smaller diameter = 30 cm, so smaller radius r = (30)/(2) = 15 cm Substitute the values into the formula: V_bucket = (1)/(3) (36) ((22.5)^2 + (22.5)(15) + (15)^2) V_bucket = 12 (506.25 + 337.5 + 225) V_bucket = 12 (1068.75) V_bucket = 12825 cm^3 Step 2: Calculate the capacity of the tank (cylinder). The formula for the volume of a cylinder is V = r^2 h. From the problem description: Diameter of tank = 2.1 m, so radius r_tank = (2.1)/(2) = 1.05 m = 105 cm Height of tank = 1.6 m = 160 cm Substitute the values into the formula: V_tank = (105)^2 (160) V_tank = (11025) (160) V_tank = 1764000 cm^3 Step 3: Determine the number of buckets needed to fill the tank. To find the number of buckets, divide the capacity of the tank by the capacity of the bucket. Number of buckets = V_tankV_bucket Number of buckets = 1764000 cm^312825 cm^3 Number of buckets = (1764000)/(12825) Number of buckets ≈ 137.54386 Since you cannot draw a fraction of a bucket, and the tank must be filled, we round up to the next whole number. a) i) The capacity of the bucket: 12825 cm^3 ii) The capacity of the tank: 1764000 cm^3 b) The number of buckets that must be drawn in order to fill the tank: 138 buckets Send me the next one 📸