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Related Rates of Change The Big Idea: Given 2 or more variables whose measures change with respect to time, 1. Write an equation that relates the variables. 2. Use derivatives with respect to time to see how their rates of change are related. Ex. 1 A circular puddle of water is expanding at the rate of 16π in2 per minute. At what rate is the radius growing when r = 4 in.? Ex. 2. If the radius of an expanding sphere is growing at an the rate of 0.1 cm/sec, at what rate is the volume growing at the moment the radius is 10 cm? 4 V r3 3 dV dr 4 r 2 dt dt dV cm 2 4 10cm 0.1 dt sec dV cm3 40 dt sec The sphere is growing at a rate of 40 cm3 / sec . Steps for Related Rates Problems: 1. When necessary, draw a picture (sketch). 2. Identify known information. 3. Identify what you are looking for. 4. Write an equation to relate the variables. 5. Differentiate both sides with respect to t. 6. Plug in known quantities & solve. Water is draining from a cylindrical tank of radius 3 at 3000 cm3/second. How fast is the water level dropping? dV cm3 3000 dt sec V r 2 h, with r 3 V 9 h dV dh 9 dt dt cm3 dh 3000 9 sec dt dh Find dt (We need a formula to relate V and h. ) dh 1000 cm3 sec dt 3 The water level is dropping at 1000/3π cm/sec. Hot Air Balloon Problem: h h tan 500 d 1 dh 2 sec dt 500 dt 2 1 dh 2 0.14 500 dt 500ft The balloon is rising at a rate of 140 feet per minute. Truck Problem: Truck A travels east at 40 mi/hr. Truck B travels north at 30 mi/hr. How fast is the distance between the trucks changing 6 minutes later? r t x d y z 1 1 dz dx dy 40 2 x 10 42 y 30 10 2z 3 dt dt dt 32 42 z 2 dz 4 40 3 30 2 5 9 16 z dt 2 dz 25 z 250 5 dz dt 50 5 z dt 2 2 B 2 z y dy 30 dt A x dx 40 dt miles 50 hour A 14 foot ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/s, how fast will the bottom be moving away from the wall when the top is 6 ft above the ground? dy 2 dt x 2 y 2 L2 x2 62 142 x 4 10 y dL 14 dt 0 L x dx dt 2x 2 4 10 dx dy dL 2y 2L dt dt dt dx 2 6 2 2 14 0 dt dx 3 dt 10 The ladder is moving away at a rate of How fast is the area changing? 3 10 10 A 14 foot ladder is leaning against a wall. If the top of the ladder slips down the wall at a rate of 2 ft/s, let’s now find how fast the area of the triangle is changing when the top is 6 ft above the ground? dy 2 dt y dL 14 dt 0 L x dx 3 dt 10 A man 6 ft tall is walking at a rate of 2 ft/s toward a street light 16 ft tall. At what rate is the length of his shadow changing? 6 x 16 x y 16 6 x dx dt y dy 2 dt 6x 6y 16x 6y 10x dy dx 6 10 dt dt dx 6 2 10 dt dx 6 dt 5 The length of his shadow is reducing at a rate of 6/5 ft/s. A boat whose deck is 10 ft below the level of a dock, is being drawn in by means of a rope attached to a pulley on the dock. When the boat is 24 ft away and approaching the dock at ½ ft/sec, how fast is the rope being pulled in? 2 2 2 dx 1 x y R dt 2 2 2 24 10 R2 x R 26 dy y dx dy dR R 0 2x 2y 2R dR dt dt dt dt dt dR 1 2 24 2 10 0 2 26 dt 2 dR 6 dt 13 The rope is being pulled in at a rate of 6/13 ft/sec. A pebble is dropped into a still pool and sends out a circular ripple whose radius increases at a constant rate of 4 ft/s. How fast is the area of the region enclosed by the ripple increasing at the end of 8 seconds. A r 2 At t = 8, r = (8)(4) = 32 dr 4 dt dA dt dA dr 2r dt dt dA 2 32 4 dt dA 256 dt The area is increasing at a rate of 256 ft2/sec. A spherical container is deflated such that its radius decreases at a constant rate of 10 cm/min. At what rate must air be removed when the radius is 5 cm? 4 3 V r 3 dr 10 dt dV dt dV 2 dr 4r dt dt dV 2 4 5 10 1000 dt Air must be removed at a rate of 1000 ft3/min. Sand pours into a conical pile whose height is always one half its diameter. If the height increases at a constant rate of 4 ft/min, at what rate is sand pouring from the chute when the pile is 15 ft high? 1 1 2 h d V r h 2 3 1 1 3 h 2r dh V h 4 2 3 dt hr dV 2 dh h dt dt dV dt dV 2 15 4 dt dV 900 dt The sand is pouring from the chute at a rate of 900 ft3/min. Liquid is pouring out of a cone shaped filter at a rate of 3 cubic inches per minute. Assume that the height of the cone is 12 inches and the radius of the base of the cone is 3 inches. How rapidly is the depth of the liquid in the filter decreasing when the level is 6 inches deep? 3 1 2 V r h 3 2 r 12 h dV 3 dt 1 1 V h h 3 4 1 V h3 48 dV 3 2 dh h dt 48 dt 3 2 dh 3 6 48 dt 4 dh 3 dt r h 3 12 1 r h 4 The depth of the liquid is decreasing at a rate of 4 in/sec. 3 If xy 2 20 and x is decreasing at the rate of 3 units per second, the rate at which y is changing when y = 2 is nearest to: a. –0.6 u/s b. –0.2 u/s xy 2 20 x 2 20 x5 2 c. 0.2 u/s d. 0.6 u/s e. 1.0 u/s dx 2 dy dt y 2y dt x 0 dy 2 3 2 2 2 5 0 dt dy 20 12 dt dy 3 dt 5 A particle moves along a curve x 2 y 2 at time t 0. dx dy at that time? 8 when x 1, what is the value of If dt dt x2y 2 2 -1 y 2 y 2 dx dy 2 2 x dt y dt x 0 dx 2 2 1 2 8 1 dt 0 dx 4 8 dt dx 2 dt