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: Convert all dimensions to a consistent unit. The radius is given as 63 cm. The length is given as 3 m. Convert the length to centimeters: L = 3 m = 3 × 100 cm = 300 cm So, r = 63 cm and L = 300 cm. Step 2: Identify the components of the surface area in contact with water. When the semi-cylindrical trough is full, the water is in contact with two parts of the trough: • The curved bottom surface of the semi-cylinder. • The two semi-circular end surfaces. Step 3: Write down the formulas for each component. • The curved surface area of a semi-cylinder is half the curved surface area of a full cylinder: A_curved = (1)/(2) (2 r L) = r L • The area of the two semi-circular ends is equivalent to the area of one full circle: A_ends = 2 × ((1)/(2) r^2) = r^2 Step 4: Calculate the total surface area in contact with water. The total surface area (A) is the sum of the curved surface area and the area of the two semi-circular ends: A = A_curved + A_ends = r L + r^2 Factor out r: A = r (L + r) Substitute the values r = 63 cm and L = 300 cm: A = (63 cm) (300 cm + 63 cm) A = (63) (363) cm^2 A = 22869 cm^2 Using the approximation ≈ 3.14159: A ≈ 22869 × 3.14159 A ≈ 71848.05 cm^2 The surface area of the trough in contact with water when it is full is 71848.05 cm^2. 3 done, 2 left today. You're making progress.