Aplicações de Derivadas: Problemas Práticos

142 APPLICATIONS OF DERIVATIVES PROBLEMS 1 A stone dropped into a pond sends out a series of con centric ripples If the radius r of the outer ripple in creases steadily at the rate of 6 fts find the rate at which the area of disturbed water is increasing a when 13 r 10 ft and b when r 20 ft 2 A large spherical snowball is melting at the rate of 27T ft3h At the moment when it is 30 inches in diameter determine a how fast the radius is changing and b how fast the surface area is changing 14 3 Sand is being poured onto a conical pile at the constant rate of 50 ft3min Frictional forces in the sand are such that the height of the pile is always equal to the radius of its base How fast is the height of the pile increas ing when the sand is 5 ft deep 15 4 A girl 5 ft tall is running at the rate of 12 fts and passes under a street light 20 ft above the ground Find how rapidly the tip of her shadow is moving when she is a 20 ft past the street light and b 50 ft past the street light 5 In Problem 4 find how rapidly the length of the girls 16 shadow is increasing at each of the stated moments 6 A light is at the top of a pole 80 ft high A ball is dropped from the same height from a point 20 ft away from the light Find how fast the shadow of the ball is moving along the ground a 1 second later b 2 seconds later Assume that the ball falls s 1612 feet in t seconds 7 A woman raises a bucket of cement to a platform 40 ft 17 above her head by means of a rope 80 ft long that passes over a pulley on the platform If she holds her end of the rope firmly at head level and walks away at 5 fts how fast is the bucket rising when she is 30 ft away from the spot directly below the pulley 18 8 A boy is flying a kite at a height of 80 ft and the wind is blowing the kite horizontally away from the boy at the rate of 20 fts How fast is the boy paying out string when the kite is 100 ft away from him 9 A boat is being pulled in to a dock by means of a rope with one end tied to the bow of the boat and the other end passing through a ring attached to the dock at a point 5 ft higher than the bow of the boat If the rope is being pulled in at the rate of 4 fts how fast is the boat moving through the water when 13 ft of rope are out 10 A trough is 10 ft long and has a cross section in the shape of an equilateral triangle 2 ft on each side If wa ter is being pumped in at the rate of 20 ft3min how fast is the water level rising when the water is 1 ft deep 1 1 A spherical meteorite enters the earths atmosphere and bums up at a rate proportional to its surface area Show 19 that its radius decreases at a constant rate 12 A point moves around the circle x2 y2 a2 in such a way that the xcomponent of its velocity is given by dxldt y Find dyldt and decide whether the direc tion of the motion is clockwise or counterclockwise A car moving at 60 mih along a straight road passes under a weather balloon rising vertically at 20 mih If the balloon is 1 mi up when the car is directly beneath it how fast is the distance between the car and the bal loon increasing 1 minute later Most gases obey Boyles law If a sample of the gas is held at a constant temperature while being compressed by a piston in a cylinder then its pressure p and vol ume V are related by the equation pV c where c is a constant Find dpldt in terms of p and dVdt At a certain moment a sample of gas obeying Boyles law Problem 14 occupies a volume of 1000 in3 at a pressure of LO lbin2 If this gas is being compressed isothermally at the rate of 12 in3min find the rate at which the pressure is increasing at the instant when the volume is 600 in3 A ladder 20 ft long is leaning against a wall 12 ft high with its top projecting over the wall Its bottom is be ing pulled away from the wall at the constant rate of 5 ftmin Find how rapidly the top of the ladder is ap proaching the ground a when 5 ft of the ladder pro jects over the wall b when the top of the ladder reaches the top of the wall A conical party hat made of cardboard has a radius of 4 in and a height of 12 in When filled with beer it leaks at the rate of 4 in3min At what rate is the level of beer falling a when the beer is 6 in deep b when the hat is half empty A hemispherical bowl of radius 8 in is being filled with water at a constant rate If the water level is rising at the rate of t ins at the instant when the water is 6 in deep find how fast the water is flowing in a by using the fact that a segment of a sphere has vol ume V 7Th2 a 4 where a is the radius of the sphere and h is the height of the segment b by using the fact that if V is the volume of the wa ter at time t then dV 7Tr2 dh dt dt where r is the radius of the surface and h is the depth Water is being poured into a hemispherical bowl of ra dius 3 in at the rate of 1 in3s How fast is the water level rising when the water is 1 in deep