Explanation Explanation Given : The volume of right circular cone of radius r and height h is V = π 3 r 2 h . r = h = 12 Formula Used: Power rule of derivative: d ( x n ) d x = n x n − 1 Change in value of function z = f ( x , y ) for a given change in variable x is given as: Δ z = f x ( Δ x ) Calculations: The volume of right circular cone of radius r and height h is: V ( r , h ) = π 3 r 2 h . Change in value of V for a given change in r is given as: Δ V = V r ( r 0 , h 0 ) ( r − r 0 ) = V r ( r 0 , h 0 ) Δ r Taking derivative of V with respect to r , we get: V r ( r , h ) = ∂ ( π 3 r 2 h ) ∂ r Since we are differentiating with respect to r , we can treat h as constant. Hence, we get: V r ( r , h ) = π 3 h ∂ ( r 2 ) ∂ r Applying power rule of derivative: d ( x n ) d x = n x n − 1 , we get: V r ( r , h ) = π 3 h ( 2 ) r 2 − 1 V r ( r , h ) = π 3 h ( 2 ) r 1 V r ( r , h ) = 2 π 3 r h Given that r 0 = h 0 = 12 V r ( r 0 , h 0 ) = 2 π 3 ( 12 ) ( 12 ) V

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

3rd Edition

ISBN: 9781319019846

Author: Jon Rogawski, Colin Adams

Publisher: W. H. Freeman

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Question

Chapter 14.3, Problem 49E

To determine

To determine:

The volume of right circular cone of radius r and height h is $V = \frac{π}{3} r^{2} h$ . Suppose $r = h = 12$ . What leads to greater increase in V, a 1 cm increase in r or 1cm increase in h? Argue using partial derivatives.

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Chapter 14.3, Problem 48E

Chapter 14.3, Problem 50E

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Chapter 14 Solutions

Loose-leaf Version for Calculus: Early Transcendentals Combo 3e & WebAssign for Calculus: Early Transcendentals 3e (Life of Edition)

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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The volume of right circular cone of radius r and height h is V = π 3 r 2 h . Suppose r = h = 12 . What leads to greater increase in V , a 1 cm increase in r or 1cm increase in h ? Argue using partial derivatives.

To determine: The volume of right circular cone of radius r and height h is V=π3r2h. Suppose r=h=12. What leads to greater increase in V, a 1 cm increase in r or 1cm increase in h? Argue using partial derivatives.

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Chapter 14 Solutions

To determine:

The volume of right circular cone of radius r and height h is $V = \frac{π}{3} r^{2} h$ . Suppose $r = h = 12$ . What leads to greater increase in V, a 1 cm increase in r or 1cm increase in h? Argue using partial derivatives.