1.3,7,9, 11, 13, 17, 19,
3.9 EXERCISES 1. If V is the volume of a cube with edge length x and the cube expands as time passes, find dV/di in terms of dx/dt. 3 cm/s. How fast is the x-coordinate of the point changing at that instant? 2. (a) If A is the area of a circle with radius r and the circle expands as time passes, find dA/dt in terms of dr/dt. 13-16 (b) Suppose oil spills from a ruptured tanker and spreads in (a) What quantities are given in the problem? a circular pattern. If the radius of the oil spill increases at b) What is the unknown? a constant rate of I m/s, how fast is the area of the spill c) Draw a picture of the situation for any time t. increasing when the radius is 30 m? d) Write an equation that relates the quantities. e) Finish solving the problem. 3. Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the 13. A plane flying horizontally at an altitude of 1 mi and a speed of 500 mi/h passes directly over a radar station. Find the rate at square is 16 cm?? which the distance from the plane to the station is increasing 4. The length of a rectangle is increasing at a rate of 8 cm/s and when it is 2 mi away from the station. its width is increasing at a rate of 3 cm/s. When the length is 14. If a snowball melts so that its surface area decreases at a rate of 20 cm and the width is 10 cm, how fast is the area of the Icm / min, find the rate at which the diameter decreases when rectangle increasing? the diameter is 10 cm. 5. A cylindrical tank with radius 5 m is being filled with water 15. A street light is mounted at the top of a 15-ft-tall pole. A man at a rate of 3 m'/ min. How fast is the height of the water 6 ft tall walks away from the pole with a speed of 5 ft/s along a increasing? straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole? 6. The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 80 mm? 16. At noon, ship A is 150 km west of ship B. Ship A is sailing east at 35 km /h and ship B is sailing north at 25 km / h. How fast is 7. The radius of a spherical ball is increasing at a rate of the distance between the ships changing at 4:00 PM? 2 cm /min. At what rate is the surface area of the ball increasing when the radius is 8 cm? 17. Two cars start moving from the same point. One travels south 8. The area of a triangle with sides of lengths a and b and at 60 mi / h and the other travels west at 25 mi/ h. At what rate contained angle 0 is is the distance between the cars increasing two hours later? A = zab sin 0 18. A spotlight on the ground shines on a wall 12 m away. If a man (a) If a = 2 cm, b = 3 cm, and 0 increases at a rate of 2 m tall walks from the spotlight toward the building at a speed 0.2 rad/ min, how fast is the area increasing when of 1.6 m/s, how fast is the length of his shadow on the build- 0 = TT/3? ing decreasing when he is 4 m from the building? (b) If a = 2 cm, b increases at a rate of 1.5 cm /min, and 0 19. A man starts walking north at 4 ft/s from a point P. Five min- increases at a rate of 0.2 rad / min, how fast is the area utes later a woman starts walking south at 5 ft/s from a point increasing when b = 3 cm and 0 = 17 /3? 500 ft due east of P. At what rate are the people moving apart (c) If a increases at a rate of 2.5 cm / min, b increases at a rate 15 min after the woman starts walking? of 1.5 cm / min, and 0 increases at a rate of 0.2 rad / min, 20. A baseball diamond is a square with side 90 ft. A batter hits how fast is the area increasing when a = 2 cm, b = 3 cm, ball and runs toward first base with a speed of 24 ft/s. and 0 = 17/3? a) At what rate is his distance from second base decreasing 9. Suppose y = v2x + 1, where x and y are functions of t. when he is halfway to first base? (a) If dx/dt = 3, find dy/dt when x = 4. (b) At what rate is his distance from third base increasing at (b) If dy/dt = 5, find dx/dt when x = 12. the same moment? 10. Suppose 4x2 + 9y2 = 36, where x and y are functions of t. (a) If dy/dt = 3, find dx/di when x = 2 and y = 3 5. (b) If dx/dt = 3, find dy /dt when x = -2 and y = 3 5. 1. If x2 + y2 + z? = 9, dx/di = 5, and dy/dt = 4, find dz/dt when (x, y, z) = (2, 2, 1). 90 ft 2. A particle is moving along a hyperbola xy = 8. As it reaches the point (4, 2), the y-coordinate is decreasing at a rate of
