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: Identify the dimensions of the bucket (frustum). The bucket is a frustum with: Top diameter D = 40 cm, so top radius R = (40)/(2) = 20 cm. Bottom diameter d = 30 cm, so bottom radius r = (30)/(2) = 15 cm. Height h = 30 cm. Step 2: Identify the dimensions of the cylindrical tank. The cylindrical tank has: Radius R_tank = 1.2 m = 1.2 × 100 = 120 cm. Height H_tank = 1.35 m = 1.35 × 100 = 135 cm. It is initially three-fifths full of water. a) (i) Calculate the capacity of the bucket in litres, leaving in your answer. Step 3: Calculate the volume of the bucket (frustum). The formula for the volume of a frustum is V = (1)/(3) h (R^2 + Rr + r^2). V_bucket = (1)/(3) (30 cm) ((20 cm)^2 + (20 cm)(15 cm) + (15 cm)^2) V_bucket = 10 (400 cm^2 + 300 cm^2 + 225 cm^2) V_bucket = 10 (925 cm^2) V_bucket = 9250 cm^3 Step 4: Convert the bucket's volume to litres. Since 1 litre = 1000 cm^3: V_bucket = (9250)/(1000) L V_bucket = 9.25 L The capacity of the bucket is 9.25 L. a) (ii) Calculate the volume of water required to fill the tank in litres. Step 5: Calculate the total volume of the cylindrical tank. The formula for the volume of a cylinder is V = R^2 H. V_tank = (120 cm)^2 (135 cm) V_tank = (14400 cm^2) (135 cm) V_tank = 1944000 cm^3 Step 6: Calculate the volume of water required to fill the tank. The tank is initially three-fifths full, so the remaining volume to fill is two-fifths of the total volume. V_required = (2)/(5) V_tank V_required = (2)/(5) (1944000 cm^3) V_required = 777600 cm^3 Step 7: Convert the required volume to litres. V_required = (777600)/(1000) L V_required = 777.6 L The volume of water required to fill the tank is 777.6 L. b) Calculate the number of buckets that must be drawn to fill the tank. Step 8: Calculate the number of buckets needed. Divide the volume of water required by the capacity of one bucket. Number of buckets = V_requiredV_bucket Number of buckets = 777.6 L9.25 L Number of buckets = (777.6)/(9.25) Number of buckets ≈ 84.06486 Since you cannot use a fraction of a bucket, round up to the nearest whole number. Number of buckets = 85 The number of buckets that must be drawn to fill the tank is 85.