   Chapter 6, Problem 5PS

Chapter
Section
Textbook Problem

Torricelli's Law Torricelli’s Law states that water will flow from an opening at the bottom of a tank with the same speed that it would attain falling from the surface of the water to the opening. One of the forms of Torricelli’s Law is A ( h ) d h d t = − k 2 g h where h is the height of the water in the tank, k is the area of the opening at the bottom of the tank, A(h) is the horizontal cross-sectional area at height h, and g is the acceleration due to gravity ( g ≈ 32 feet per second per second). A hemispherical water tank has a radius of 6 feet. When the tank is full, a circular valve with a radius of 1 inch is opened at the bottom, as shown in the figure. How long will it take for the tank to drain completely? To determine

To calculate: The time taken by the tank to drain completely if the radius of hemispherical water tank is 6 feet.

Explanation

Given:

Radius of hemispherical water tank is 6 feet. And radius of valve which is opened is 1 inch, as shown in the provided figure

For the figure refer to the question.

Formula used:

Torricelli’s Law is:

A(h)dhdt=k2gh

Calculation:

According to Torricelli’s Law,

A(h)dhdt=k2gh

where h is the height of the water in the tank, k is the area of the opening at the bottom of the tank, A(h) is the horizontal cross-sectional area at height h, and g is the acceleration due to gravity.

Since, a circular valve of radius of 1 inch is open at the bottom so, the value of k will be

k=π(112)2 1 feet=12 inch

Thus, the center of the hemisphere is at (0,6)

Thus, the equation of tank will be the equation of circle with center (0,6) and radius 6 feet. Then,

x2+(y6)2=36

Now, separate the variables as

x2=36(y6)2=36y2+12y36=12yy2

Thus, the equation of the tank in terms of y will be

x=12yy2

Thus, the area of cross section of tank at a height h will be

A(h)=πx2=π(12hh2)

Now, according to Torricelli’s Law.

A(h)dhdt=k2gh

Substitute k=π(112)2, A(h)=π(12hh2), g=32

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