Final answer:
The water level in the tank 15 minutes later is approximately 15.16 ft. The mass change in the tank 15 minutes later is approximately 37,337.7 lb.
Step-by-step explanation:
To find the water level and the mass change in the tank 15 min later, we need to consider the flow rate and the dimensions of the tank.
The tank has a diameter of 10 ft and a height of 15 ft. Using the formula for the volume of a cylinder, we can calculate the initial volume of water in the tank:
Volume = π * (radius)^2 * height
Given that the diameter is 10 ft, the radius is 5 ft. Substituting the values into the formula, we get:
Volume = 3.14 * (5 ft)^2 * 15 ft = 1177.5 ft³
Therefore, the initial volume of water in the tank is 1177.5 ft³.
Since the tank is half full, the initial volume of water is half of the total volume:
1177.5 ft³ / 2 = 588.75 ft³
Now, we can calculate the flow rate:
Flow Rate = 300 gpm = (300 gallons / 7.48 gallons/ft³) / 1 min = 40.11 ft³/min
15 minutes later, the additional volume of water in the tank is:
Volume = Flow Rate * Time = 40.11 ft³/min * 15 min = 601.65 ft³
Adding the initial volume and the additional volume, we can find the total volume of water in the tank after 15 minutes:
Total Volume = Initial Volume + Additional Volume = 588.75 ft³ + 601.65 ft³ = 1190.4 ft³
The water level can be calculated by dividing the volume by the area of the base of the tank:
Area of Base = π * (radius)^2 = 3.14 * (5 ft)^2 = 78.5 ft²
Water Level = Total Volume / Area of Base = 1190.4 ft³ / 78.5 ft² = 15.16 ft
Therefore, the water level in the tank 15 minutes later is approximately 15.16 ft.
The mass change in the tank can be calculated using the density of water:
Mass Change = Density * Volume
Given that the density of water is 62.1 lb/ft³, we can calculate the mass change:
Mass Change = 62.1 lb/ft³ * 601.65 ft³ = 37,337.7 lb
Therefore, the mass change in the tank 15 minutes later is approximately 37,337.7 lb.