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Unformatted text preview: Mathematics Department, UCLA T. Richthammer fall 08, sheet 5 Oct 24, 2008 Homework assignments: Math 31A Calculus, Sec. 1 075. You inflate a large balloon by filling it with water at a constant rate of 0 . 005 cubic meter per second. Determine the growth rate of the diameter of the ball, in the following situations: (a) when the diameter is 0 . 1 meter (b) when the diameter is 0 . 5 meter (c) when the volume is 0 . 5 cubic meter (d) after 20 seconds. (Hint: The volume of a ball with radius r is V ( r ) = 4 3 r 3 π . Use a calculator!) Answer: Volume V cubic meter, diameter d meter, time t seconds. V = 4 3 r 3 π , d = 2 r , so V = d 3 π/ 6. V ( t ) = 0 . 005. We have V ( t ) = d ( t ) 2 d ( t ) π/ 2, so d ( t ) = . 01 πd ( t ) 2 . (a) d = 1 /π ≈ . 32 (b) d = 1 / (25 π ) ≈ . 013 (c) d = (6 V/π ) 1 / 3 ≈ . 98, so d = 0 . 0033 (d) V = 0 . 1, so d ≈ . 58, so d = 0 . 0096 076. A car travels down a highway at a speed of 50 mph. You are standing 0 . 1 miles from the highway. How fast is the distance between you and the car increasing or decreasing in the following situations (use a calculator): (a) when the car passes in front of you (b) when the car has a distance of 0 . 2 miles to you (c) an hour after the car has passed you Answer: Let P be the point, where the car passes in front of you, let x be the distance in miles of the car to that point, t the time in hours, counted from the moment when the car passes you, and l the distance from you to the car. We have l 2 = 0 . 01 + x 2 and x ( t ) = 50. So l = xx /l . (a) x = 0, so l = 0 (b) l = 0 . 2, so x ≈ . 17, so l ≈ 43 . 3 (c) x = 50, so l ≈ 50 . 0001, so l = ≈ 49 . 9999 077. A particle moves counterclockwise around the ellipse 9 x 2 + 16 y 2 = 25. (a) Determine the sign of x ( t ) and y ( t ) in every quadrant. (Explain your answer!) (b) At what rate is the x-coordinate changing when the particle passes through (1 , 1) if its y-coordinate is increasing at a rate of 18 m/s ? (c) At what rate is the y-coordinate changing when the particle passes through (0 , 5 / 4)? Answer: (a) First quadrant ( x, y > 0): x is decreasing, y is increasing, so x ( t ) < 0, y ( t ) > 0. Similarly for the other quadrants: Second qu.: x ( t ) < , y ( t ) < 0. Third qu.: x ( t ) > , y ( t ) < 0. Fourth qu.: x ( t ) > , y ( t ) > 0. (b) 18 xx + 32 yy = 0, so x = 16 yy 9 x = 32 m/s . (c) y = 9 xx 16 y = 0. 078. The hour hand of a clock is 3 in. long, and it is broken so that it keeps on showing 12 o’clock. The minute hand is 4 in. long and works fine. Show that the angle θ between the two hands and the distance d between their tips are related by d 2 = (3- 4 cos θ ) 2 +(4 sin θ ) 2 . How fast is the distance d changing at a quarter past the full hour? Answer: Let A be the tip of the hour hand, B the tip of the minute hand, O the center of the watch and P the point on the line OA such that P B is perpendicular to OA (so that P B is the height of triangle OBA ). The length of P B is 4 sin θ and the length of...
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This note was uploaded on 05/20/2010 for the course MATH 31A Math 31A taught by Professor Thomasrichthammer during the Fall '08 term at UCLA.
- Fall '08