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No.33 The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l＝1m and w＝h＝2m, and l and w are increasing at a rate of 2 m/s while h is decreasing at a rate of 3 m/s. At that instant find the rates at which the following quantities are changing. (a) The volume (b) The surface area (c) The length of a diagonal (a)Volume V l w h dV dt l w h w l h h l w t t t And l 1 m w h 2 m l t w t 2 m/s h t 2 m/s 3 m/s dV 2 2 2 2 1 2 3 1 2 dt 8 4 6 6 m 3 /s (b)Surface t S 2l w lh wh 2 l w w l l h h l w h h w dS dt t t t t t 22 2 2 1 2 2 3 1 2 2 3 2 24 2 4 3 4 6 2 5 10 m 2 /s 1 (c)Length of diagonal D l 2 w2 h 2 l 2 w 2 h 2 2 1 1 1 2 2 2 2 2 2 2 2 2 2 2 dD 1 2 dl dw 1 1 2l dt 2 l w h 2w dt 2 l w h 2h dh l w h dt 2 dt 1 l 2 w 2 h 2 2 l dl w dw h dh dt dt dt 1 12 2 2 2 2 2 1 2 2 2 2 3 13 2 4 6 0 m/s Length of diagonal do not changing. No.36 If a sound with frequency fs is produced by a source traveling along a line with speed vo along the same line from the opposite direction toward the source, then the c vo frequency of the sound heard by the observer is f o f s , where c is the speed c vs of the sound, about 332m/s.(This is the Doppler effect.) Suppose that, at a particular moment, you are in a train traveling at 34 m/s and accelerating at 1.2 m/s. A train is approaching you from the opposite direction on the other track at 40 m/s, accelerating at 1.4 m/s, and sounds its whistle, which has a frequency of 460 Hz. At that instant, what is the perceived frequency that you hear and how fast is it changing? c vo (a) f o fs c vs 332 40 332 34 460 576.6 The perceived frequency is 576.6 Hz. c vo (b) Since f o f s , in this case c and f s are constants. c vs dvo dv c v s s c v o dt df o dt fs dt 2 c v s dvo dv c v s s c vo dt dt fs c v s 2 and dvo 2 1.2 m/s dt df o 1.2 332 40 1.4 332 34 460 dt 332 40 2 350 .4 512 .4 460 4.66 Hz/s 85264 No.39 If z f x y , show that z z 0 . x then u 1 設 x y u x z f x y f u z x z u z z y z u z u x u y u u x y Chain rule u z z z z 0 x y u u u y x 1 dv s 2 1.4 m/s dt