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Unformatted text preview: 178 Section 4.6
13. Step 1: x distance from wall to base of ladder y height of top of ladder A area of triangle formed by the ladder, wall, and ground angle between the ladder and the ground Step 2:
dx x dt dy y dt
2 10. continued (c) s
ds dt x2 y2
1 z2 2x
dx dt 2 x 2 y2
dz z dt z2 2y dy dt 2z dz dy At the instant in question, x Step 3:
(2)(1) 22 0 29 12 ft and dx dt 5 ft/sec. x ds dt y 2 z 2 (4)(1) (3)( 2) 42 32 0 m/sec We want to find Step 4, 5, and 6: (a) x 2
dx 2x dt dy dA d , , and . dt dt dt The rate of change of the diagonal length is 0 m/sec. 11. Step 1: s (diagonal) distance from antenna to airplane x horizontal distance from antenna to airplane Step 2: At the instant in question, s 10 mi and
ds dt dx . dt y2 169
dy 2y dt 0 To evaluate, note that, at the instant in question, y 169 x2 169 122 5. 300 mph. Step 3: We want to find Step 4: x 2 49 Step 5:
dx dt dx dt 1 2 s2 10 102 49 49 dy Then 2(12)(5) 2(5) 0 dt dy dy 12 ft/sec or 12 ft/sec dt dt The top of the ladder is sliding down the wall at the s2 49 rate of 12 ft/sec. (Note that the downward rate of motion is positive.) s 2 or x 2s ds dt s s2 3000 51 ds 49 dt (b) A
dA dt Step 6: (300) mph 420.08 mph 1 xy 2 1 dy x 2 dt dA dt y dx dt 119 ft/sec. 2 Using the results from step 2 and from part (a), we have
1 [(12)( 12) 2 The speed of the airplane is about 420.08 mph. 12. Step 1: h height (or depth) of the water in the trough V volume of water in the trough Step 2: At the instant in question, Step 3:
dh We want to find . dt dV dt (5)(5)] The area of the triangle is changing at the rate of 59.5 ft2/sec.
y x
2 2.5 ft3/min and h 2 ft. (c) tan sec
d dt x dy dt y x2 dx dt Since tan
4 3 Step 4: The width of the top surface of the water is h, so we have V
1 4 (h) h (15), or V 2 3 dh dt dh dt 1 ft/min 16 1 ft/min. 16 cos 12 and so sec2 13 5 , we have for 0 12 1
12 2 13 2 169 . 144 10h 2 Combining this result with the results from step 2 and from part (a), we have
d dt 169 d 144 dt (12)( 12) (5)(5) , so 122 Step 5:
dV dt 20h 1 radian/sec. The angle is changing at the rate 1 radian/sec. Step 6: 2.5
dh dt of 20(2) 0.0625 The water level is increasing at the rate of ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Spring '08 term at University of Florida.
 Spring '08
 GERMAN

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