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Calculus Name: __________________________ Related Rates Packet (SOLUTIONS) – PART I Date: 12/6 Solutions to Related Rates Exercises (#1—4, 7) d y 3x 5 dt dy dx 3 0 dt dt 1. Equation: y 3 x 5 a. Given that dx / dt 2 , find dy / dt when x = 1. b. Given that dy / dt 1 , find dx / dt when x = 0. dy dx 3 0 dt dt dx b) 1 3 dt dx 1 dt 3 dy dx 3 0 dt dt a) dy 32 0 6 dt Note: The derivatives themselves do not depend on the x or y coordinates… x=1 and x=0 were extraneous pieces of information! 2. A 6m ladder is against a wall. If its bottom is pushed toward the wall at a constant rate of 1m , 2 s how fast is the top of the ladder sliding when it (the top of the ladder) reaches 5m up the wall? dx 1 dy m s Find: ?? m s when y = 5. dt 2 dt Triangle Relationship Needed: Pythagorean Theorem x 2 y 2 36 Take the derivative: d x 2 y 2 36 dt dx dy 2x 2y 0 dt dt Plug in the “moment” y = 5….need to figure out x at this time too! x 2 52 36 2 2 x y 36 x 11 2x dx dy 2y 0 dt dt 2 11 1 dy 25 0 2 dt Conclusion: The top of the ladder is sliding up at a rate of the wall. Document1 dy 11 ms dt 10 11 m s when the ladder is 5 meters up 10 Calculus Name: __________________________ Related Rates Packet (SOLUTIONS) – PART I Date: 12/6 3. A winch on a dock is 20ft above the water level. It is attached to a boat at a point at water level and reels the boat in at 2 ft sec . How fast is the boat moving when the rope the rope is 30 ft long? 20.05ft long? dx 2 ft s dt dr Find: ?? ft s dt when r = 30 and when r = 20.05 Triangle Relationship Needed: Pythagorean Thm. x 2 202 r 2 Take the derivative: d dx dr x2 202 r 2 2x 0 2r dt dt dt Plug in the “moment” r = 30….need to figure out x at this time too! x 2 202 302 x 500 2x dx 60 dx dr dx ft s 2.683 ft s 2 500 0 2r 0 2 30 2 dt dt dt dt 500 Conclusion: The boat is moving toward the dock at a rate of 2.683 ft s when rope is 30 ft long. Plug in the “moment” r = 20.05….need to figure out x at this time too! x 2 202 20.052 x 2.0025 dx dr dx 2 20.0025 0 2r 0 2 20.05 2 dt dt dt dx 60 ft s 28.337 ft s dt 20.0025 2x Conclusion: The boat is moving toward the dock at a rate of 28.337 ft s when rope is 20.05 ft long. Document1 Calculus Name: __________________________ Related Rates Packet (SOLUTIONS) – PART I Date: 12/6 3 4. Water is flowing into a cone (whose height is 16cm and radius is 4cm) at a rate of 2 cm min . How fast is the water level rising when it is 10 cm deep? dV 2 cm3 min dt Find: dh ?? cm min when h = 10 dt Volume Relationship Needed: 1 V r 2h Uh-oh! We have 2 variables, but we only want “h” 3 4 r 16 h Similar Triangles Relationship Needed: 16 4 1 r h h r 4 Now substitute: 2 V Take the derivative: d 1 V h3 dt 48 1 1 h h 3 4 V 1 h3 48 dV 1 dh 3h2 dt 48 dt Plug in the “moment” when h = 10: dh .32 1 .102 cm min 2 dh dt 2 3(10) 48 dt Conclusion: The water level is rising at a rate of .102 cm min when the water is 10 cm deep. Document1 Calculus Name: __________________________ Related Rates Packet (SOLUTIONS) – PART I Date: 12/6 7. A stone dropped into a still pond sends out a circular ripple whose radius increases at a constant rate of 3 ft/s. How rapidly is the area enclosed by the ripple increasing at the end of 10 s? We know: r dr 3 ft s dt We want: dA ?? ft when t = 10 s dt Geometry relationship: A r 2 Take the derivative and use the moment (when t = 10, the radius will be 30 ft): d dA dr A r 2 2r dt dt dt dA 2 30 3 180 ft s dt Conclusion: The area enclosed by the ripple is increasing at the rate of 180 ft seconds. Document1 s at the end of 10