PAGE 82 from the textbook might be helpful Torricellis Law d

PAGE 82 from the textbook might be helpful

Torricelli\'s Law describes the relationship between the velocity of fluid leaving a container under the force of gravity and the height of the fluid. Torricelli\'s Law states that the velocity of water exiting through a hole is proportional to Root2gh, where g Is the acceleration due to gravity (9.8m/s2) and h is the depth of water above the hole. Suppose the tank is filled with water with a tap that is 0.019 square meters in size. Prove Torricelli\'s Law by doing the following. Show that the standard gravity differential equation d2h / dt2 =-g leads to the conclusion that an object that falls from a height of h meters will land with a velocity of -Root2gh . (Assume initial velocity is at rest (v0=O)). Use the chain rule (calc I / III concept) to write an expression for the rate of change of the volume with respect to time, dV / dt , in terms of dh / dt. A. Use what you know about the volume and height of a cylinder to create an expression for What geometry formula does represent? Substitute this into your equation from #2 and simplify Now, let\'s consider the rate at Which water flows out of the hole. Use Torricelli\'s law to determine the differential equation fordV / dt. You will now have two different dV / dt Steps 1-4 have derived the differential equation provided in part (b) of textbook. Set steps 3 and 4 equal to each other to create your differential equation. Solve you differential equation and explain what your new equation represents. Suppose you have a cylinder of radius 3 meters and height 8 meters full of water with a 0.019 square meter hole taken from the bottom of the tank. Use your differential equation to determine how long it will take for the tank to empty. Let A(h) be the cross-sectional area of the water in the lank at height h and a the area of the drain hole. The rate at which water is flowing out of the lank at time t can be expressed as the cross-sectional area at height h times the rate at which the height of ihc water is changing. Alternatively, the rate at which water flows out of the hole can be expressed as the area of the hole times the velocity of the draining water. Set these two equal to each other and insert Torricelli\'s law to derive the differential equation A(h)dh / dt = -aRoot2gh

Solution

h(t) can be found by solving the differential equation:

dh/dt= -1/72 (h)^1/2

Separating variables

h^(-1/2)dh = (-1/72)dt

Integrating both sides:

(1/2)h^(1/2) = (-1/72)t + c

h^(1/2) = (-1/36)t + c

h = {(-1/36)t + c}