2.6 Related Rates Today we will stretch even farther! 2.6 Related Rates Suppose x and y are differentiable functions of t related by the equation y x 3 2 dy dx Find when x 1, given 2 when x 1. dt dt dy dx y x 3 2x 0 dt dt dy 2 1 2 4 dt 2 2.6 Related Rates A plane is flying over a radar tracking station. If s is decreasing at a rate of 400 mph when s 10 mi, what is the speed of the plane? Geometric Model dx/dt? Solve on Board s 2 x 2 36 ds dx 2s 2 x 0 dt dt dx 2 10 400 dt 2x x? 10 6 x 2 2 x8 2 ds 400 dt s 10 dx 500 mph dt speed 500 mph dx ? dt Given h t =50t 2 , (h in feet and t in seconds), find the rate of change in the of elevation of the camera at 10 seconds after lift-off. h t =50t x 20002 50002 You cannot put in values for variables until you have differentiated! d ? dt x Variable Constant Geometric Model h tan 2000 Implicit Differentiation d 1 dh dt 2000 dt 1 d 1 dh 2 cos dt 2000 dt d co s 2 d h dt 2000 dt sec 2 h 2 h 10 =5000 Variable cos 2000 20002 50002 dh 100 10 1000 dt t 10 2 radians per second 29 In an engine, a 7” connecting rod is fastened to a crankshaft of radius 3”. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when . 3 Constants 3,7 Variables ,x Crankshaft = 3” Connecting Rod = 7” Piston = x” Given Rate: 200 Revolutions per minute d 200(2 ) 400 d dt 400 dt In an engine, a 7” connecting rod is fastened to a crankshaft of radius 3”. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when 3 . Connecting Rod = 7” Crankshaft = 3” dx Find: when = dt 3 Piston = x” Do not substitute before you differentiate!! Equation: Find an equation that relates and x. Law of Cosines: b a c 2ac cos 2 2 2 7 2 32 x 2 (2)(3) x cos d 400 dt Law of Cosines In an engine, a 7” connecting rod is fastened to a crankshaft of radius 3”. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when 3 . Connecting Rod = 7” Crankshaft = 3” Piston = x” Implicit Differentiation: 49 9 x 6 x cos 2 Solution 49 9 x 6 x cos 2 dx dx d 0 0 2 x( ) 6( cos x( sin ) ) dt dt dt dx dx d 0 2 x( ) 6 cos ( ) 6 x sin ( ) dt dt dt You could divide every term by 2. dx dx d 6 cos ( ) 2 x( ) 6 x sin ( ) d dt dt dt 6 x sin ( ) dx d dx d t (6 cos 2 x) 6 x sin ( ) dt dt dt 6cos 2 x In an engine, a 7” connecting rod is fastened to a crankshaft of radius 3”. The crankshaft rotates counterclockwise at a constant rate of 200 revolutions per minute. Find the velocity of the piston when Crankshaft = 3” Connecting Rod = 7” 3 . Piston = x” d 3x sin ( ) dx dt dt 3cos x Before we substitute, we need to find a value for x. 49 9 x 6 x cos 2 3 1 40 x 6 x( ) 2 2 40 x 3x 0 x 3x 40 2 0 ( x 8)( x 5) 2 x8 d 3x sin ( ) dx dt dt 3cos x 3 (3)(8) sin (400 ) (24)( )(400 ) dx 2 3 3 / 2 8 dt 3(cos ) (8) 3 in. 4018 Length is decreasing min . Diagram 2.6 Related Rates HW 2.6/1-9odd,31-34,41-44,52 Related Rates 1. Diagram 2. Geometric Model 3. Differentiate d? 4. Solve for dt 5. Substitute 6. Answer Question 9.4 Law of Cosines Objective To use the Law of Cosines to find unknown parts of a . The Law of Cosines SAS & SSS In ABC , c 2 a 2 b2 2ab cos C OPP 2 ADJ1 ADJ 2 2 ADJ1 ADJ 2 cos 2 2 Problem