Question

Suppose water is leaking from a tank through a circular hole of area

*A*_{h}

at its bottom. When water leaks through a hole, friction and
contraction of the stream near the hole reduce the volume of water
leaving the tank per second to
*cA*_{h}

2gh |

, where

* c* (0 <

is an empirical constant.

A tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (Assume the removed apex of the cone is of negligible height and volume.)

(a)

Suppose the tank is 20 feet high and has radius 8 feet and the
circular hole has radius 2 inches. The differential equation
governing the height *h*in feet of water leaking from a tank
after *t* seconds is

dh |

dt |

= −

5 |

6h^{3/2} |

. In this model, friction and contraction of the water at the hole are taken into account with

* c* = 0.6,

and *g* is taken to be

32 ft/s^{2}.

See the figure below.A right-circular conical tank containing water is shown.

- The cone opens upward and the point of the cone at the bottom is labeled: circular hole.
- A dashed line extends horizontally from the circular hole.
- The surface of the water in the tank is labeled:
*A*_{w}and a line segment from the dashed line to*A*_{w}is labeled:*h*. - The radius at the top of the tank is labeled: 8 ft.
- The height of the tank is labeled: 20 ft.

Solve the initial value problem that assumes the tank is initially full.

* h*(

If the tank is initially full, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)

minutes

(b)

Suppose the tank has a vertex angle of 60° and the circular hole
has radius 4 inches. Determine the differential equation governing
the height *h*of water. Use

* c* = 0.6

and

* g* = 32 ft/s

dh |

dt |

=

Solve the initial value problem that assumes the height of the water is initially 10 feet.

* h*(

If the height of the water is initially 10 feet, how long (in minutes) will it take the tank to empty? (Round your answer to two decimal places.)

minutes

Answer #1

The two equations below express conservation of energy and
conservation of mass for water flowing from a circular hole of
radius 2 centimeters at the bottom of a cylindrical tank of radius
20 centimeters. In these equations, Δ m is the mass that leaves the
tank in time Δt, v is the velocity of the water flowing through the
hole, and h is the height of the water in the tank at time t. g is
the accelertion of gravity,...

A jet of water squirts out horizontally from a hole near the
bottom of the tank shown in the figure. If the hole has a diameter
of 5.95 mm, what is the height h of the water level in the tank? x=
.600 m y=1.00 m

Suppose Aaron is pumping water into tank, shaped like an
inverted circular cone, at a rate of 1600ft^3/min. If the altitude
of the cone is 10ft and the radius of the base of the cone is 5ft,
find the rate at which the radius of the liquid is changing when
the height of the liquid is 7ft.

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