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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Torricelli's Law A tank similar to the one in Exercise 6 has a height of 20 feet and a radius of 8 feet, and the valve is circular with a radius of 2 inches. The tank is full when the valve is opened. How long will it take for the tank to drain completely?

To determine

To calculate: The time takenby the tank to drain completely if the height of cylindrical tank 20 feet and a radius is 8 feet, and the valve is circular with a radius of 2 inches.

Explanation

Given:

The height of cylindrical tank 20 feet and a radius is 8 feet, and the valve is circular with a radius of 2 inches.

Formula used:

One of the forms of the Torricelli’s Law is:

A(h)dhdt=k2gh

Calculation:

According to the 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 the radius of 2 an inch is open at the bottom so, the value of k will be

As 1 feet=12 inch.

Then, 2 inches=16feet.

k=π(16)2

Now, since the tank is cylindrical so the cross-sectional area of the tank remains constant at all heights.

So, at any height, the cross-sectional area is:

A(h)=π(8)2=64π

According to Torricelli’s Law.

A(h)dhdt=k2gh

Substitute k=π(16)2, A(h)=64π, g=32

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