water is run at a constant rate of 2 ft^3/min` ` to fill a cylindrical tank of radius 2ft and height 6ft. Assuming that the tank is initially empty, make a conjecture about the average weight of the water in the tank over the time period required to fill it. verify the conjecture by integrating. (take weight density of water to be 62.4 lb/ft^3). Round answer to two decimal places.
The cylindrical tank with radius 2 ft and height 6 ft is filled with water at the rate 2 ft^3/min. The total volume of the tank is `pi*r^2*h = pi*4*6 = 24*pi` . The density of water is 62.4 lb/ft^3. As water fills into the tank the mass of water in the tank increases in a linear fashion.The average mass of water in the tank is `(1/2)*24*pi*62.4 = 748.8*pi`
The increase in mass of water in the tank is given by `pi*4*62.4*dx` . Taking the integral for x = 0 to x = 6 gives:
`int_0^6 249.6*pi dx`
The average mass of the tank as `(1497.6*pi)/2` = `748.8*pi` pounds.