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Related Rates Worksheet 1. If x + y = x2, and dx/dt = 2, find dy/dt when x = 1. 2. If xy = 12, and dx/dt = 3, find dy/dt when y = 2. 3. If x2 + 3xy + y2 = 11, and dy/dt = 2, find dx/dt when y = 1, and x = 2. 4. If xy + z2 = z, and dx/dt = -1, dz/dt = 3, find dy/dt when x = 2 and z = 4 5. Air is being pumped into a spherical balloon so that its volume increases at a rate of 100cm 3/s. How fast is the radius of the balloon increasing when the diameter is 50cm? 6. A ladder 10m long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1m/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6m from the wall? 7. A water tank has the shape of an inverted cone with base radius 2m and height 4m. If water is being pumped into the tank at a rate if 2m3/min, find the rate at which the water level is rising when the tank is 3m deep. 8. Car A is traveling west at 50km/h and car B is traveling north at 60km/h. Both cars are headed for the intersection of the two roads. At what rate are the two cars approaching each other when car A is 300m and car B is 400m from the intersection? 9. Sven, who is 1.6 m tall, walks away from the base of a 4.5 m high lamppost at a rate of 1.2 m/s. At what rate is the length of his shadow increasing when he is 6m from the lamppost? 10. One leg of a right triangle is always 12 cm long, and the other leg is increasing at a rate of 1 cm/s. Find the rate of change of the hypotenuse (h), when it is 20 cm long. 11. A spherical snowball is melting in such a way that its volume is decreasing at a rate of 1cm3/min. At what rate is the diameter decreasing when the diameter is 10cm? 12. If a snowball melts so that its surface area decreases at a rate of 1cm2/min, find the rate at which the diameter decreases when the diameter is 10cm. 13. A street light is at the top of a 15ft pole. A man 6ft tall walks away from the pole with a speed of 5ft/s along a straight path. How fast is his shadow lengthening when he is 40ft from the pole? 14. A plane flying horizontally at an altitude of 1km and a speed of 500km/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when the plane is 2km away from the station. 15. Two cars start at the same point. One travels south at 60km/h and the other travels west at 25km/h. At what rate is the distance between them increasing two hours later? 16. At noon, ship A is 100km west of ship B. Ship A is sailing south at 35km/h and ship B is sailing north at 25km/h. How fast is the distance between them changing at 4:00PM? 17. The altitude of a triangle is increasing at a rate of 1cm/min while the area of the triangle is increasing at a rate of 2cm2/min. At what rate is the base of the triangle changing when the altitude is 10cm and the area is 100cm2. 18. Water is leaking out of an inverted conical tank at a rate of 10 000cm3/min at the same time that water is being pumped into the tank at a constant rate. The tank has a height of 6m and a diameter at the top is 4m. If the water level is rising at a rate of 20cm/min when the height of the water is 2m, find the rate at which water is being pumped into the tank. 19. The radius of a right circular cone is decreasing at a rate of 1.5 cm/s and the height is increasing at the rate of 5 cm/s. At what rate is the volume changing when the height is 12 cm and the radius is 2 cm? 20. At high noon, boat A is exactly 30km due West of boat B. Boat A is heading South at a speed of 50 km/h and boat B is heading North with a speed of 40 km/h. Determine at what rate the distance between the boats is increasing at 2:30PM. 21. Gravel is being dumped from a conveyer belt at a rate of 30m3/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 10m high? 22. A pebble is thrown into water and causes a circular ripple to spread outward at a rate of 2m/s. Find the rate of change of the area of the circle in terms of , 3 seconds after the pebble falls in the water. 23. The sides of a cube are increasing at a rate of 1cm/s. How fast is the diagonal of the cube changing when the side length is 1cm? 24. The radius of a cylinder is decreasing at a rate of 1cm/min. The height remains the same at 20cm. How fast is the volume changing when the radius is 12cm? 25. At what rate is the volume of a box changing if the width of the box is increasing at a rate of 3cm/s, the length is increasing at a rate of 2cm/s and the height is decreasing at a rate of 1cm/s, when the height is 4cm, the width is 2cm and the volume is 40cm3. Related Rates Worksheet Solutions 1. 2 2. –1 16 3. 7 27 4. 2 1 5. cm/s 25 3 6. m/s 4 7. 0.28 m/min 8. 78km/h 9. 0.662m/s 10. 0.8cm/s 1 11. cm/min 50 1 12. cm/min 20 13. 3.3ft/s 14. 433.01km/h 15. 65km/h 16. 55.4km/h 17. –1.6cm/min 18. 2.89E5cm3/min 19. decreasing at 54.45cm3/s 20. 89.2km/h 21. 0.382m/min 22. 24m2/s 23. 3 cm/s 24. -480cm3/min 25. 66cm3/s