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15.3Ex47-48

15.3Ex47-48 - S E C T I O N 15.3 Partial Derivatives(ET...

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SECTION 15.3 Partial Derivatives (ET Section 14.3) 647 Hence, V P T = nR P T = nR V . 47. The volume of a right-circular cone of radius r and height h is V = π 3 r 2 h . Calculate V r and V h . SOLUTION We obtain the following derivatives: V r = r ³ 3 r 2 h ´ = h 3 r r 2 = h 3 · 2 r = 2 hr 3 V h = h ³ 3 r 2 h ´ = 3 r 2 48. A right-circular cone has r = h = 12 cm. What leads to a greater increase in V , a 1-cm increase in r or 1-cm increase in h ? Argue using partial derivatives. SOLUTION The partial derivatives V r and V h can be estimated for small values of 1 r and 1 h by V r V ( r + 1 r , h ) V ( r , h ) 1 r and V h V ( r , h + 1 h ) V ( r , h ) 1 h Hence, V ( r + 1 r , h ) V ( r , h ) V r 1 r V ( r , h + 1 h ) V ( r , h ) V h 1 h Therefore an increase 1 r = 1cmin r leads to an increase of V r
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