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.
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You're on a roll — Here's the solution to the problem: Assumption: The speed of water flowing through the pipe is 10 cm/s, as indicated in your scratch work. The question text is incomplete regarding this value. a) Calculate the depth of water in the tank at 8:30 a.m. Step 1: Calculate the volume of water flowing into the tank per second. Given the cross-section area of the pipe A_pipe = 6.16 cm^2 and the speed of water v = 10 cm/s. The inflow rate Q_in is: Q_in = A_pipe × v Q_in = 6.16 cm^2 × 10 cm/s Q_in = 61.6 cm^3/s Step 2: Calculate the total time water flows. Water starts flowing at 6:00 a.m. and we need to calculate the depth at 8:30 a.m. Time duration t = 8:30 a.m. - 6:00 a.m. = 2 hours 30 minutes. Convert this time to seconds: t = (2 × 60 minutes + 30 minutes) × 60 seconds/minute t = (120 minutes + 30 minutes) × 60 seconds/minute t = 150 minutes × 60 seconds/minute t = 9000 s Step 3: Calculate the total volume of water in the tank. V_water = Q_in × t V_water = 61.6 cm^3/s × 9000 s V_water = 554400 cm^3 Step 4: Convert the base area of the tank to cm^2. Given the base area of the tank A_tank = 3 m^2. A_tank = 3 m^2 × (100 cm/m)^2 A_tank = 3 × 10000 cm^2 A_tank = 30000 cm^2 Step 5: Calculate the depth of water in the tank. The volume of water in the tank is V_water = A_tank × h, where h is the depth. h = V_waterA_tank h = 554400 cm^330000 cm^2 h = 18.48 cm The depth of water in the tank at 8:30 a.m. is 18.48 cm. b) If the tank is 1.2M high and has a hole at the bottom through which water leaks at the rate of 11.6cm³ per second. Determine the time the tank will be filled. Step 1: Calculate the net inflow rate. Inflow rate $Q_in = 61.6 cm^