Explanation Given information: A cylindrical tank of radius R and length L placed horizontally filled with oil to height h is redrawn as shown in Figure 1. Proof: Consider A as the area of circular cross section area occupied by oil. The volume V ( h ) of the oil in the tank is calculated as follows: V ( h ) = L ⋅ A ..... (1) A sector A B C made an angle θ and a perpendicular bisector drawn from point A to the line B C as shown in Figure 2. Consider A 1 as the area of the sector A B C and A 2 as the area of the triangle A B C . The area of the sector and area of triangle A B C can be calculated as follows: A 1 = θ 2 R 2 A 2 = 1 2 A D ⋅ B C From Pythagoras theorem the length D C can be calculated as shown below: D C 2 = A C 2 − A D 2 ..... (2) The length A C is the radius R of the cylinder. The length A D can be calculated as follows: A D = R − h Substitute R − h for A D and R for A C in equation (2). D C 2 = R 2 − ( R − h ) 2 D C 2 = R 2 − ( R 2 + h 2 − 2 h R ) D C 2 = 2 h R − h 2 D C = 2 h R − h 2 The relationship between B C and D C is shown below. B C = 2 D C Substitute 2 h R − h 2 for D C in the above equation. B C = 2 2 h R − h 2 Substitute R − h for A D and 2 2 h R − h 2 for B C in A 2 = 1 2 A D ⋅ B C to obtain area A 2 . A 2 = 1 2 ( R − h ) ⋅ ( 2 2 h R − h 2 ) = ( R − h ) ⋅ ( 2 h R − h 2 ) The area A can be calculated as follows: A = A 1 − A 2 Substitute θ 2 R 2 for A 1 and R 2 ( R − h ) for A 2 in the above equation as below To determine (b) To show: The expression for the rate of change of volume with respect to h is d V d h = 2 L h ( 2 R − h ) . To determine (c) To find: The rate of change of height h if the value of h is 0.5 .

Loose-leaf Version For Calculus: Early Transcendentals

4th Edition

ISBN: 9781319055912

Author: Jon Rogawski, Colin Adams, Robert Franzosa

Publisher: W. H. Freeman

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Question

Chapter 3.10, Problem 45E

To determine

(a)

To show:

That the volume V(h) of the oil in the tank is L(R2cos−1(1−hR)−(R−h)2hR−h2).

To determine

(b)

To show:

The expression for the rate of change of volume with respect to h is dVdh=2Lh(2R−h).

To determine

(c)

To find:

The rate of change of height h if the value of h is 0.5.

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Chapter 3.10, Problem 44E

Chapter 3, Problem 1CRE

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Chapter 3 Solutions

Loose-leaf Version For Calculus: Early Transcendentals

Ch. 3.1 - Prob. 1PQ Ch. 3.1 - Prob. 2PQ Ch. 3.1 - Prob. 3PQ Ch. 3.1 - Prob. 4PQ Ch. 3.1 - Prob. 5PQ Ch. 3.1 - Prob. 6PQ Ch. 3.1 - Prob. 7PQ Ch. 3.1 - Prob. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E

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Prob. 93E Ch. 3.9 - Prob. 1PQ Ch. 3.9 - Prob. 2PQ Ch. 3.9 - Prob. 3PQ Ch. 3.9 - Prob. 4PQ Ch. 3.9 - Prob. 5PQ Ch. 3.9 - Prob. 1E Ch. 3.9 - Prob. 2E Ch. 3.9 - Prob. 3E Ch. 3.9 - Prob. 4E Ch. 3.9 - Prob. 5E Ch. 3.9 - Prob. 6E Ch. 3.9 - Prob. 7E Ch. 3.9 - Prob. 8E Ch. 3.9 - Prob. 9E Ch. 3.9 - Prob. 10E Ch. 3.9 - Prob. 11E Ch. 3.9 - Prob. 12E Ch. 3.9 - Prob. 13E Ch. 3.9 - Prob. 14E Ch. 3.9 - Prob. 15E Ch. 3.9 - Prob. 16E Ch. 3.9 - Prob. 17E Ch. 3.9 - Prob. 18E Ch. 3.9 - Prob. 19E Ch. 3.9 - Prob. 20E Ch. 3.9 - Prob. 21E Ch. 3.9 - Prob. 22E Ch. 3.9 - Prob. 23E Ch. 3.9 - Prob. 24E Ch. 3.9 - Prob. 25E Ch. 3.9 - Prob. 26E Ch. 3.9 - Prob. 27E Ch. 3.9 - Prob. 28E Ch. 3.9 - Prob. 29E Ch. 3.9 - Prob. 30E Ch. 3.9 - Prob. 31E Ch. 3.9 - Prob. 32E Ch. 3.9 - Prob. 33E Ch. 3.9 - Prob. 34E Ch. 3.9 - Prob. 35E Ch. 3.9 - Prob. 36E Ch. 3.9 - Prob. 37E Ch. 3.9 - Prob. 38E Ch. 3.9 - Prob. 39E Ch. 3.9 - Prob. 40E Ch. 3.9 - Prob. 41E Ch. 3.9 - Prob. 42E Ch. 3.9 - Prob. 43E Ch. 3.9 - Prob. 44E Ch. 3.9 - Prob. 45E Ch. 3.9 - Prob. 46E Ch. 3.9 - Prob. 47E Ch. 3.9 - Prob. 48E Ch. 3.9 - Prob. 49E Ch. 3.9 - Prob. 50E Ch. 3.9 - Prob. 51E Ch. 3.9 - Prob. 52E Ch. 3.9 - Prob. 53E Ch. 3.9 - Prob. 54E Ch. 3.9 - Prob. 55E Ch. 3.9 - Prob. 56E Ch. 3.9 - Prob. 57E Ch. 3.9 - Prob. 58E Ch. 3.9 - Prob. 59E Ch. 3.9 - Prob. 60E Ch. 3.9 - Prob. 61E Ch. 3.9 - Prob. 62E Ch. 3.9 - Prob. 63E Ch. 3.9 - Prob. 64E Ch. 3.9 - Prob. 65E Ch. 3.9 - Prob. 66E Ch. 3.9 - Prob. 67E Ch. 3.9 - Prob. 68E Ch. 3.9 - Prob. 69E Ch. 3.9 - Prob. 70E Ch. 3.9 - Prob. 71E Ch. 3.9 - Prob. 72E Ch. 3.9 - Prob. 73E Ch. 3.9 - Prob. 74E Ch. 3.9 - Prob. 75E Ch. 3.9 - Prob. 76E Ch. 3.9 - Prob. 77E Ch. 3.9 - Prob. 78E Ch. 3.9 - Prob. 79E Ch. 3.9 - Prob. 80E Ch. 3.9 - Prob. 81E Ch. 3.9 - Prob. 82E Ch. 3.9 - Prob. 83E Ch. 3.9 - Prob. 84E Ch. 3.9 - Prob. 85E Ch. 3.9 - Prob. 86E Ch. 3.9 - Prob. 87E Ch. 3.9 - Prob. 88E Ch. 3.10 - Prob. 1PQ Ch. 3.10 - Prob. 2PQ Ch. 3.10 - Prob. 3PQ Ch. 3.10 - Prob. 4PQ Ch. 3.10 - Prob. 5PQ Ch. 3.10 - Prob. 6PQ Ch. 3.10 - Prob. 1E Ch. 3.10 - Prob. 2E Ch. 3.10 - Prob. 3E Ch. 3.10 - Prob. 4E Ch. 3.10 - Prob. 5E Ch. 3.10 - Prob. 6E Ch. 3.10 - Prob. 7E Ch. 3.10 - Prob. 8E Ch. 3.10 - Prob. 9E Ch. 3.10 - Prob. 10E Ch. 3.10 - Prob. 11E Ch. 3.10 - Prob. 12E Ch. 3.10 - Prob. 13E Ch. 3.10 - Prob. 14E Ch. 3.10 - Prob. 15E Ch. 3.10 - Prob. 16E Ch. 3.10 - Prob. 17E Ch. 3.10 - Prob. 18E Ch. 3.10 - Prob. 19E Ch. 3.10 - Prob. 20E Ch. 3.10 - Prob. 21E Ch. 3.10 - Prob. 22E Ch. 3.10 - Prob. 23E Ch. 3.10 - Prob. 24E Ch. 3.10 - Prob. 25E Ch. 3.10 - Prob. 26E Ch. 3.10 - Prob. 27E Ch. 3.10 - Prob. 28E Ch. 3.10 - Prob. 29E Ch. 3.10 - Prob. 30E Ch. 3.10 - Prob. 31E Ch. 3.10 - Prob. 32E Ch. 3.10 - Prob. 33E Ch. 3.10 - Prob. 34E Ch. 3.10 - Prob. 35E Ch. 3.10 - Prob. 36E Ch. 3.10 - Prob. 37E Ch. 3.10 - Prob. 38E Ch. 3.10 - Prob. 39E Ch. 3.10 - Prob. 40E Ch. 3.10 - Prob. 41E Ch. 3.10 - Prob. 42E Ch. 3.10 - Prob. 43E Ch. 3.10 - Prob. 44E Ch. 3.10 - Prob. 45E Ch. 3 - Prob. 1CRE Ch. 3 - Prob. 2CRE Ch. 3 - Prob. 3CRE Ch. 3 - Prob. 4CRE Ch. 3 - Prob. 5CRE Ch. 3 - Prob. 6CRE Ch. 3 - Prob. 7CRE Ch. 3 - Prob. 8CRE Ch. 3 - Prob. 9CRE Ch. 3 - Prob. 10CRE Ch. 3 - Prob. 11CRE Ch. 3 - Prob. 12CRE Ch. 3 - Prob. 13CRE Ch. 3 - Prob. 14CRE Ch. 3 - Prob. 15CRE Ch. 3 - Prob. 16CRE Ch. 3 - Prob. 17CRE Ch. 3 - Prob. 18CRE Ch. 3 - Prob. 19CRE Ch. 3 - Prob. 20CRE Ch. 3 - Prob. 21CRE Ch. 3 - Prob. 22CRE Ch. 3 - Prob. 23CRE Ch. 3 - Prob. 24CRE Ch. 3 - Prob. 25CRE Ch. 3 - Prob. 26CRE Ch. 3 - Prob. 27CRE Ch. 3 - Prob. 28CRE Ch. 3 - Prob. 29CRE Ch. 3 - Prob. 30CRE Ch. 3 - Prob. 31CRE Ch. 3 - Prob. 32CRE Ch. 3 - Prob. 33CRE Ch. 3 - Prob. 34CRE Ch. 3 - Prob. 35CRE Ch. 3 - Prob. 36CRE Ch. 3 - Prob. 37CRE Ch. 3 - Prob. 38CRE Ch. 3 - Prob. 39CRE Ch. 3 - Prob. 40CRE Ch. 3 - Prob. 41CRE Ch. 3 - Prob. 42CRE Ch. 3 - Prob. 43CRE Ch. 3 - Prob. 44CRE Ch. 3 - Prob. 45CRE Ch. 3 - Prob. 46CRE Ch. 3 - Prob. 47CRE Ch. 3 - Prob. 48CRE Ch. 3 - Prob. 49CRE Ch. 3 - Prob. 50CRE Ch. 3 - Prob. 51CRE Ch. 3 - Prob. 52CRE Ch. 3 - Prob. 53CRE Ch. 3 - Prob. 54CRE Ch. 3 - Prob. 55CRE Ch. 3 - Prob. 56CRE Ch. 3 - 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(a) To show: That the volume V ( h ) of the oil in the tank is L ( R 2 cos − 1 ( 1 − h R ) − ( R − h ) 2 h R − h 2 ) .

Solution Summary: The author illustrates how a cylindrical tank of radius R and length L placed horizontally filled with oil is redrawn as shown in Figure 1.

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(a)

To show:

That the volume V(h) of the oil in the tank is L(R2cos−1(1−hR)−(R−h)2hR−h2).

(b)

To show:

The expression for the rate of change of volume with respect to h is dVdh=2Lh(2R−h).

(c)

To find:

The rate of change of height h if the value of h is 0.5.

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Chapter 3 Solutions

(a) To show: That the volume V ( h ) of the oil in the tank is L ( R 2 cos − 1 ( 1 − h R ) − ( R − h ) 2 h R − h 2 ) .

Solution Summary: The author illustrates how a cylindrical tank of radius R and length L placed horizontally filled with oil is redrawn as shown in Figure 1.

Videos

(a) To show: That the volume V(h) of the oil in the tank is L(R2cos−1(1−hR)−(R−h)2hR−h2).

(b) To show: The expression for the rate of change of volume with respect to h is dVdh=2Lh(2R−h).

(c) To find: The rate of change of height h if the value of h is 0.5.

Want to see the full answer?

Chapter 3 Solutions

(a)

To show:

That the volume V(h) of the oil in the tank is L(R2cos−1(1−hR)−(R−h)2hR−h2).

(b)

To show:

The expression for the rate of change of volume with respect to h is dVdh=2Lh(2R−h).

(c)

To find:

The rate of change of height h if the value of h is 0.5.