2.5 Related Rates 1A) Equation: x 4 y 3 Given that Given) dx dy 1 , find when x 2 dt dt dx 1; dt Find) dy ; dt When) x2 Equation) x 4 y 3 Work: We will need to take the derivative with respect to time and solve for the “find”. x 4y 3 d d x 4y 3 dt dt dx dy 4 0 dt dt 4dy dx dt dt dy 1 dx dy *Note that x did not appear, so did not depend on When. dt 4 dt dt dy 1 1 1 dt 4 4 2B) Given that Given) Work: 2 2 2 dy dx 3, Find when x, y 2 , 2 . dt dt dy 3; dt Find) x2 y 2 2x d 2 d x y2 2x dt dt 2x 2 dx dy dx 2y 2 dt dt dt y dy dx xdx dt dt dt dx ; dt 2 2 2 2 , 2 When) x, y Equation) x2 y 2 2x y dy dx 1 x dt dt y dy dx 1 x dt dt 2 dx 2 3 dt 2 2 2 2 2 2 2 3 dx dt 2 2 dx 1 3 dt 3) 3 dx dt Oil from a ruptured tanker spreads in a circular pattern whose radius increases at a constant rate of 2 ft/sec. How fast is the area of the spill increasing when the radius of the spill is 60 ft? r Given) dr ft 2 ; dt sec Work: dA dr 2r dt dt 2 60 ft 2 dA ft 2 240 dt sec ft sec Find) dA ; dt When) r 60 ft Equation) A r 2 Example: 4) A baseball diamond is a square whose sides are 90ft long. Suppose that a player running from second base to third base has a speed of 30 ft/sec at the instant when he is 20 ft from third base. At what rate is the players distance from home plate changing at that instant? W X Y 90ft Work: Given: dx ft 30 dt sec Find: dy dt When: x 20 ft Equation: x 2 902 y 2 d 2 d x 902 y 2 dt dt 2x dx dy 0 2y dt dt x dx dy y dt dt x 2 90 2 y 2 20 2 90 2 y 2 20 ft ft dy 30 sec dt 10 85 ft 400 8100 y 2 *Note you have to solve the original equation for Y. 8500 y 2 8500 y 10 85 y y 10 85 Thus: 60 ft dy 85 sec dt dy ft 6.51 dt sec ***Note: Do not substitute a number in for a variable if the variable is changing. Wait until you have isolated your “find” after you have taken the derivative. 5) A camera 3000 ft away from the launching pad of a rocket is photographing the takeoff. If the rocket is rising vertically at 880 ft/sec when it is 4000 ft above the launch pad, how fast must the camera angle change at that instant to keep the camera aimed at the rocket? h θ 3000 ft Work: dh ft d 880 Find: dt sec dt d d h tan dt dt 3000 Given: sec2 When: h 4000 ft d 1 dh 1 dt 3000 dt d 1 1 dh 2 dt 3000 sec dt d 1 1 880 dt 3000 5 2 3 d 1 9 880 dt 3000 25 d 66 rad d rad 180 6.05 .1056 dt 625 sec dt sec rad sec Equation: tan h 3000 5) Alternate method for solving #5 Work: Given: dh ft 880 dt sec tan d dt d dt Find: h 3000 arctan 1 h 1 3000 2 1 dh 3000 dt 2 1 880 3000 1 4000 1 3000 d dt When: h 4000 ft h 3000 1 1 88 16 300 1 9 1 1 3 1 66 rad 22 22 25 75 25 25 625 sec 9 Equation: tan h 3000 Example: 6) As shown in the figure above, water is draining from a conical tank with height 12 feet and diameter 8 feet into a cylindrical tank that is changing at the rate of (h-12) feet per minute. The volume V of a cone with radius r and height h is 1 V r 2h 3 Work: Given: A) dh ft h 12 dt min Find: dv dt Equation: 1 V r 2h 3 Write an expression for the volume of water in the conical tank as a function of h. 4 * using similar triangles we find what r is equal to. 1 V r 2h 3 12 2 1 1 V h h 3 3 1 V h3 27 B) When: h 3 ft r 4 h 12 1 r h 3 At what rate is the volume of water in the conical tank changing when h=3? Indicate units of measure. d 1 dh V 3h 2 dt 27 dt dv 1 ft 3 3 ft 2 9 dt 27 min dv ft 3 9 dt min C) Let y be the depth, in feet, of the water in the cylindrical tank. At what rate is y changing when h=3 ? Indicate units of measure. Given: dv ft 3 9 dt min Find: dy dt When: h 3 ft Equation: V Area of Base Height V 400 ft 2 y dv dy 1 dv dy 400 ft 2 2 dt dt 400 ft dt dt 9 ft dy 1 9 ft 3 dy 2 400 min dt 400 ft min dt Angular Rate of Change: In a free-fall experiment, an object is dropped from a height of 256 feet. A camera on the ground 500 feet from the point of impact records the fall of the object. S 256ft θ 500ft A) Find the position function giving the height of the object at time t, assuming the object is released at time t = 0. At what time will the object reach ground level? s t 16t 2 V0t S0 s t 16t 2 0 t 256 s t 16t 2 256 (* Position function) When will the object reach the ground? 0 16t 2 256 16t 2 256 t 2 16 t 4 t 4 sec B) Find the rates of change of the angle of elevation of camera when t = 1 and t = 2. Find d dt When t 1sec; t 2sec Equation: tan s 500 s 500 16t 2 256 arctan 500 d 1 1 32t 2 dt 16t 2 256 500 1 500 arctan When t = 1: d dt 1 16 12 256 1 500 2 1 32 1 0.0520 500 When t = 2: d dt 1 16 2 256 1 500 2 2 1 32 2 0.1115511183 500