Pre-Calc Homework Solutions 434

Pre-Calc Homework Solutions 434 - 434 Chapter 10 Review 46....

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Unformatted text preview: 434 Chapter 10 Review 46. If g( ) 2f ( ), then (g( )2 g ( )2) 2 g( ) sin 4[2 f ( ) sin ( f ( )2 f ( )2)], so the area generated by g is 4 times that of f. 47. (a) Let r 1.75 0.06 . 2 50. 2 3 0 a3 sin 0 d 2 3 a 3 cos 0 4 3 a , and 3 a2 d 4 3 a 3 a2. So 4a . By symmetry, x 3 dr (b) Since d b , this is just Equation 4 for the 2 y length of the curve. 80 a2 4a 0, . 3 0, so the centroid is (c) Using NINT, 0 1.75 741.420 cm 0.06 2 2 0.06 2 d 2 s Chapter 10 Review Exercises (pp. 569572) 1. (a) 3 3, 4 4 2, 17, 32 172 3 1 6 2 evaluates to 7.414 m. b 2 r 2 1/2 5 1313 9 8, 12 20 (d) r 2 since b 2 1/2 2 b 2 r 2 r 1 r (b) 2. (a) (b) 3. (a) (b) (b) 322 2, 4 1 8 2 5 2 2(4) 10 10, 725 1, 6, 25 5 8 1 is a very small quantity squared. 2( 3), 2 2 (e) L La 741.420 cm (from part (c)), 80 0 1.75 0.03 2 0.06 2 2 80 4. (a) 5(2), 5( 5) d 236 741.416 cm 5. 10 2 25 2 29 3 2 1.75 0 6 radians below the negative x-axis: , 1 2 48. (a) Use the approximation, La, from 47(e). If the reel has made n complete turns, then the angle is 2 n. So from the integral, La n r0 b bL r02 [assuming counterclockwise]. 6. 7. 2 8. 5 dy dx 3 1 , 2 1 42 12 1 (3/5)2 dy/dt dx/dt 3 (4/5)2 2 bn 2 1 2 r0n. Solving for n gives 1 . 4, 1 3 4 , 5 5 8 17 , 2 17 (b) The take up reel slows down as time progresses. (c) Since L is proportional to time, the formula in part (a) shows that n will grow roughly as the square root of time. 2 49. 3 2 0 3, 4 9. (a) (1/2)sec t tan t (1/2)sec2 t 3 2 sin t dy dx 3 2 3 2 For t a (1 2 3 a 3 2 ;x ,y 1, and 3 2 . So the 3 cos ) cos (cos 3 cos2 15 sin 8 1 cos3 4 3 d 3 cos3 cos 2 tangent line is y cos4 ) d cos2 sin y (b) d 2y dx 2 3 2 1 x or 0 x 1 . 4 dy /dt dx/dt cos t (1/2)sec2 t 1 equals . 3 4 3/t 2 2/t 3 5 ,y 4 2 3 a 3 sin 3 15 8 dy dx 2 cos3 t, sin 0 which for t 10. (a) dy dx dy/dt dx/dt 5 3 a , and 2 2 0 a2(1 cos )2 d cos2 ) d 1 sin 2 4 2 0 a2 a2 2 (1 0 2 cos 2 sin 3 2 3 a2. (b) 5a ,0 . 6 3 t 2 1 dy For t 2: x , and 2 dx 1 So the tangent line is y 3 x 2 13 y 3x . 4 d 2y dx 2 dy dx dy /dt dx/dt 3/2 2/t 3 3. 5 or 4 So x 5 3 a 2 3 a2 5a . 6 3t 3 , which for t 4 2 equals 6. By symmetry, y 0, so the centroid is ...
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