Water coming out of a pipe at 3.0 m/s takes 12 minutes to fill a circular tank of radius 0.45 m and height 1.20 m. a) Derive an expression for finding the internal diameter D of a pipe, if you know the speed v at which the water comes out of the pipe, and the time t that it takes to fill a tank of radius r and height h.

To derive an expression for finding the internal diameter D of a pipe, we first need to understand the relationship between the volume of the water coming out of the pipe and the volume of the tank.

The volume V of the water coming out of the pipe can be calculated using the formula for the volume of a cylinder, V = πr²h, where r is the radius of the pipe and h is the height of the water column in the pipe. However, since the water is flowing, we need to consider the speed v at which the water is coming out of the pipe. The height of the water column in the pipe per unit time is v, so the volume of the water coming out of the pipe per unit time is V = πr²v.

The volume V' of the tank is also given by the formula for the volume of a cylinder, V' = πR²H, where R is the radius of the tank and H is the height of the tank.

The time t it takes to fill the tank is the volume of the tank divided by the volume of the water coming out of the pipe per unit time, so t = V'/V = (πR²H)/(πr²v).

We want to find the diameter D of the pipe, which is 2r. Solving the equation for r, we get r = sqrt[(πR²H)/(πvt)]. Therefore, the diameter D of the pipe is D = 2r = 2sqrt[(πR²H)/(πvt)].

So, the expression for finding the internal diameter D of a pipe, if you know the speed v at which the water comes out of the pipe, and the time t that it takes to fill a tank of radius R and height H, is D = 2sqrt[(πR²H)/(πvt)].