4. A spherical tank of radius 4 meters is full of water. Use a definite integral to find the work required to pump the water out of the spout that is 2 meters above the sphere. The density of water is 1,000 kg/m³ and acceleration due to gravity is g = 9.8 m/s?. Give the exact value (no decimals).

4. A spherical tank of radius 4 meters is full of water. Use a definite integral to find the work required to pump the water out of the spout that is 2 meters above the sphere. The density of water is 1,000 kg/m³ and acceleration due to gravity is g = 9.8 m/s?. Give the exact value (no decimals).

Mathematics For Machine Technology

8th Edition

ISBN:9781337798310

Author:Peterson, John.

Publisher:Peterson, John.

Chapter65: Achievement Review—section Six

Section: Chapter Questions

Problem 44AR: Solve these prism and cylinder exercises. Where necessary, round the answers to 2 decimal places...

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Question

page 4
4. A spherical tank of radius 4 meters is full of water. Use a definite integral to find the work required to
pump the water out of the spout that is 2 meters above the sphere. The density of water is 1,000 kg/m³ and
acceleration due to gravity is g = 9.8 m/s². Give the exact value (no decimals).
%3D
2 m

Expert Solution

Step 1

We need to find a cross-section of the tank such that the equation we find describes the radius of any slice of the tank. The cross-sectional diagram of the tank is as follows:

The slice with radius $r$ is placed at a depth of $y$ meters from the top. We shall use the Pythagorean theorem to find $r$ in terms of $y$ .

${(4 - y)}^{2} + r^{2} = 4^{2} 16 - 8 y + y^{2} + r^{2} = 16 r^{2} = 8 y - y^{2} r = \sqrt{8 y - y^{2}}$

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