Report

25. A police car traveling south toward Sioux Falls at 160 km/h pursues a truck traveling east away from Sioux Falls, Iowa, at 140 km/h. At time t = 0, the police car is 20 km north and the truck is 30 km east of Sioux Falls. Calculate the rate at which the distance between the vehicles is changing: dl l 20 30 1300 2 dl dt dy dx 160 & 140 dt dt y 20 160t & x 30 140t l 2 30 140t 20 160t 2 2l 2 dt 27.735 km/h t 0 @ t 1/12 h, l 42.1966 km 112.963 km/h t 1/12 2 dl dl 2 30 140t 140 2 20 160t 160 l 4200 19600t 3200 25, 600t dt dt dl 1000 45, 200t dt l 30. A particle moves counterclockwise around the ellipse with equation 9x2 + 16y2 = 25. (a) In which of the four quadrants is dx/dt > 0? Explain. (b) Find a relation between dx/dt and dy/dt. (c) At what rate is the x-coordinate changing when the particle passes the point (1, 1) if its y-coordinate is increasing at a rate of 6 m/s? (d) Find dy/dt when the particle is at the top and bottom of the ellipse. dx x w/ respect to t dt dx is positive in quadrants III & IV dt x is moving to the right . 9 x 2 16 y 2 25 18 x dx dy 32 y 0 dt dt dx dy 32 y 0 dt dt dy dx 1,1 & 6 18 32 6 0 dt dt dx 32 6 32 m/s dt 18 3 18 x dy 0 dt 41. A roller coaster has the shape of the graph. Show that when the roller coaster passes the point (x, f (x)), the vertical velocity of the roller coaster is equal to f '(x) times its horizontal velocity. dy dx y f x f ' x dt dt In other words, vertical velocity equals dx f ' x or f ' x horizontal velocity. dt y f x