Explanation Given information: The height of the conical tank is 3 m . Radius of the conical tank at the top is 2 m . The rate of the water flow is 2 m 3 / min . Formula Used: The volume of a cone is: V = 1 3 π r 2 h Where, V is the volume of the cone, r is the radius of the cone at top and h is the height of the cone. Chain Rule: If f ( x ) is defined as a composite function such that, f ( x ) = ( g ∘ h ) ( x ) = g [ h ( x ) ] Then the derivative of f ( x ) is: f ′ ( x ) = g ′ [ h ( x ) ] ⋅ h ′ ( x ) Calculation: Consider, the volume of the conical tank and the radius at the top be V and r respectively at time t and the height of the conical tank when the water level is 2 m is h at time t . Therefore, the rate of the water flow is d V d t . The rate of rising of water level is d h d t . According to the question and figure, D E = r A D = h = 2 m A C = 3 m B C = 2 m d V d t = 2 m 3 / min From the figure, in Δ A D E and Δ A C B , ∠ A D E = ∠ A C B = 90 ° Since, B C and D E are the radii of the cone therefore they are parallel to each other. Hence, ∠ A E D and ∠ A B C are corresponding angles. ∠ A E D = ∠ A B C Thus, by AA similarity criterion, Δ A D E ∼ Δ A C B And A E A B = A D A C = D E C B Take the second and third factor of the above expression. A D A C = D E C B ...... (1) Substitute, h for A D , 3 m for A C , r for D E and 2 m for C B in the equation (1). h 3 m = r 2 m ( 3 m ) × r = ( 2 m ) × h r = ( 2 m ) × h 3 m r = 2 h 3 To find the volume of the cone, use the formula V = 1 3 π r 2 h

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 3.9, Problem 13E

To determine

To calculate:

The rate of rising of water level when it is 2 m.

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Chapter 3.9, Problem 12E

Chapter 3.9, Problem 14E

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Chapter 3 Solutions

Calculus - Standalone book

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. 1E Ch. 3.1 - Prob. 2E Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E

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