Pre-Calc Homework Solutions 203

Pre-Calc Homework Solutions 203 - a was da 1 in 5 1 1 2 ft...

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67. Let C 5 circumference, r 5 radius, S 5 surface area, and V 5 volume. (a) Since C 5 2 p r , we have } d d C r } 5 2 p and so dC 5 2 p dr . Therefore, ) } d C C } ) 5 ) } 2 2 p p d r r } ) 5 ) } d r r } ) , } 0 1 . 0 4 c c m m } 5 0.04 The calculated radius will be within approximately 4% of the correct radius. (b) Since S 5 4 p r 2 , we have } d d S r } 5 8 p r and so dS 5 8 p r dr . Therefore, ) } d S S } ) 5 ) } 8 4 p p r r d 2 r } ) 5 ) } 2 r dr } ) # 2(0.04) 5 0.08. The calculated surface area will be within approximately 8% of the correct surface area. (c) Since V 5 } 4 3 } p r 3 , we have } d d V r } 5 4 p r 2 and so dV 5 4 p r 2 dr . Therefore ) } d V V } ) 5 ) ) 5 ) } 3 r dr } ) # 3(0.04) 5 0.12. The calculated volume will be within approximately 12% of the correct volume. 68. By similar triangles, we have } a 6 } 5 } a 1 h 20 } , which gives ah 5 6 a 1 120, or h 5 6 1 120 a 2 1 . The height of the lamp post is approximately 6 1 120(15) 2 1 5 14 ft. The estimated error in measuring
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Unformatted text preview: a was ) da ) # 1 in. 5 } 1 1 2 } ft. Since } d d h a } 5 2 120 a 2 2 , we have ) dh ) 5 ) 2 120 a 2 2 da ) # 120(15) 2 2 1 } 1 1 2 } 2 5 } 4 2 5 } ft, so the estimated possible error is 6} 4 2 5 } ft or 6} 1 8 5 } in. 69. } d d y x } 5 2 sin x cos x 2 3. Since sin x and cos x are both between 1 and 2 1, the value of 2 sin x cos x is never greater than 2. Therefore, } d d y x } # 2 2 3 5 2 1 for all values of x . Since } d d y x } is always negative, the function decreases on every interval. 4 p r 2 dr } } 4 3 } p r 3 Chapter 4 Review 203...
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