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Pre-Calc Homework Solutions 179

Pre-Calc Homework Solutions 179 - Since the height is 3 8...

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14. Step 1: s 5 length of kite string x 5 horizontal distance from Inge to kite Step 2: At the instant in question, } d d x t } 5 25 ft/sec and s 5 500 ft Step 3: We want to find } d d s t } . Step 4: x 2 1 300 2 5 s 2 Step 5: 2 x } d d x t } 5 2 s } d d s t } or x } d d x t } 5 s } d d s t } Step 6: At the instant in question, since x 2 1 300 2 5 s 2 , we have x 5 ˇ s 2 w 2 w 3 w 0 w 0 w 2 w 5 ˇ 5 w 0 w 0 w 2 w 2 w 3 w 0 w 0 w 2 w 5 400. Thus (400)(25) 5 (500) } d d s t } , so } d d s t } 5 20 ft/sec. Inge must let the string out at the rate of 20 ft/sec. 15. Step 1: The cylinder shown represents the shape of the hole. r 5 radius of cylinder V 5 volume of cylinder Step 2: At the instant in question, } d d r t } 5 } 0. 3 00 m 1 in in. } 5 } 30 1 00 } in./min and (since the diameter is 3.800 in.), r 5 1.900 in. Step 3: We want to find } d d V t } . Step 4: V 5 p r 2 (6) 5 6 p r 2 Step 5: } d d V t } 5 12 p r } d d r t } Step 6: } d d V t } 5 12 p (1.900) 1 } 30 1 00 } 2 5 } 2 1 5 9 0 p 0 } 5 0.0076 p < 0.0239 in 3 /min The volume is increasing at the rate of approximately 0.0239 in 3 /min. 16. Step 1: r 5 base radius of cone h 5 height of cone V 5 volume of cone Step 2: At the instant in question, h 5 4 m and } d d V t } 5 10 m 3 /min. Step 3: We want to find } d d h t } and } d d r t } . Step 4:
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Unformatted text preview: Since the height is } 3 8 } of the base diameter, we have h 5 } 3 8 } (2 r ) or r 5 } 4 3 } h . We also have V 5 } 1 3 } p r 2 h 5 } 1 3 } p 1 } 4 3 } h 2 2 h 5 } 16 2 p 7 h 3 } . We will use the equations V 5 } 16 2 p 7 h 3 } and r 5 } 4 3 } h . Steps 5 and 6: (a) } d d V t } 5 } 16 p 9 h 2 } } d d h t } 10 5 } 16 p 9 (4) 2 } } d d h t } } d d h t } 5 } 12 4 8 5 p } m/min 5 } 1 3 1 2 2 p 5 } cm/min The height is changing at the rate of } 1 3 1 2 2 p 5 } < 11.19 cm/min. (b) Using the results from Step 4 and part (a), we have } d d r t } 5 } 4 3 } } d d h t } 5 } 4 3 } 1 } 1 3 1 2 2 p 5 } 2 5 } 3 8 7 p 5 } cm/min. The radius is changing at the rate of } 3 8 7 p 5 } < 14.92 cm/min. h r 6 in. r Inge Kite 300 ft x s Section 4.6 179...
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