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
b) i) Step 1: Identify the dimensions of the cylinder and the water. The diameter of the cylinder is 10 cm, so the radius r = (10)/(2) = 5 cm. The depth of the water is h = 4 cm. The area of the container in contact with water consists of the circular base and the curved surface area up to the water level. Area in contact with water = Area of base + Curved surface area of water A = r^2 + 2 rh Step 2: Substitute the values into the formula and calculate the area. A = (5 cm)^2 + 2 (5 cm)(4 cm) A = 25 cm^2 + 40 cm^2 A = 65 cm^2 A ≈ 65 × 3.14159265... cm^2 A ≈ 204.2035... cm^2 Rounding to three significant figures: A ≈ 204 cm^2 The area of the container that is in contact with water is 204 cm^2. ii) Step 1: Calculate the initial volume of water in the cylinder. Initial volume of water V_w = r^2 h V_w = (5 cm)^2 (4 cm) V_w = 100 cm^3 Step 2: Calculate the volume of the solid cube. The side of the cube is s = 6 cm. Volume of cube V_c = s^3 V_c = (6 cm)^3 V_c = 216 cm^3 Step 3: Determine the total volume when the cube is immersed. Since the side of the cube (6 cm) is less than the diameter of the cylinder (10 cm), the cube fits inside the cylinder. Also, the height of the cube (6 cm) is greater than the initial water depth (4 cm), but we need to check if the new water level will fully submerge the cube. The total volume occupied by water and the submerged cube is the sum of the initial water volume and the cube's volume. V_total = V_w + V_c V_total = 100 cm^3 + 216 cm^3 Step 4: Calculate the new height of the water level. The new total volume will occupy the cylindrical container with the same base area. Let the new height be h_new. V_total = r^2 h_new 100 + 216 = (5)^2 h_new 100 + 216 = 25 h_new Solve for h_new: h_new = (100 + 216)/(25) h_new = (100)/(25) + (216)/(25) h_new = 4 + (216)/(25) h_new ≈ 4 + (216)/(25 × 3.14159265...) h_new ≈ 4 + (216)/(78.5398...) h_new ≈ 4 + 2.7490... h_new ≈ 6.7490... cm Since h_new ≈ 6.75 cm is greater than the cube's height of 6 cm, the cube is fully submerged, and our calculation is valid. Rounding to three significant figures: h_new ≈ 6.75 cm The new height of the level of water in the cylinder is 6.75 cm. Last free one today — make it count tomorrow, or type /upgrade for unlimited.