Using Implicit Differentiation to Solve Related Rates Problems

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Questions

• If a cylindrical tank with radius 5 meters is being filled with water at a rate of 3 cubic meters per minute, how fast is the height of the water increasing?
• If the radius of a sphere is increasing at a rate of 4 cm per second, how fast is the volume increasing when the diameter is 80 cm?
• If #y=x^3+2x# and #dx/dt=5#, how do you find #dy/dt# when #x=2# ?
• If #x^2+y^2=25# and #dy/dt=6#, how do you find #dx/dt# when #y=4# ?
• How do you find the rate at which water is pumped into an inverted conical tank that has a height of 6m and a diameter of 4m if water is leaking out at the rate of #10,000(cm)^3/min# and the water level is rising #20 (cm)/min#?
• How much salt is in the tank after t minutes, if a tank contains 1000 liters of brine with 15kg of dissolved salt and pure water enters the tank at 10 liters/min?
• At what approximate rate (in cubic meters per minute) is the volume of a sphere changing at the instant when the surface area is 5 square meters and the radius is increasing at the rate of 1/3 meters per minute?
• At what approximate rate (in cubic meters per minute) is the volume of a sphere changing at the instant when the surface area is 4 square meters and the radius is increasing at the rate of 1/6 meters per minute?
• What is the rate of change of the width (in ft/sec) when the height is 10 feet, if the height is decreasing at that moment at the rate of 1 ft/sec.A rectangle has both a changing height and a changing width, but the height and width change so that the area of the rectangle is always 60 square feet?
• What is the total amount of water supplied per hour inside of a circle of radius 8 if a sprinkler distributes water in a circular pattern, supplying water to a depth of #e^-r# feet per hour at a distance of r feet from the sprinkler?
• Water is leaking out of an inverted conical tank at a rate of 10,000 cm3/min at the same time water is being pumped into the tank at a constant rate If the tank has a height of 6m and the diameter at the top is 4 m and if the water level is rising at a rate of 20 cm/min when the height of the water is 2m, how do you find the rate at which the water is being pumped into the tank?
• At what rate, in cm/s, is the radius of the circle increasing when the radius is 5 cm if oil is poured on a flat surface, and it spreads out forming a circle and the area of this circle is increasing at a constant rate of 5 cm2/s?
• How fast is the water level rising when the water is 3 cm deep (at its deepest point) if water is poured into a conical container at the rate of 10 cm3/sec. the cone points directly down, and it has a height of 25 cm and a base radius of 15 cm?
• How do you find the rate at which the volume of a cone changes with the radius is 40 inches and the height is 40 inches, where the radius of a right circular cone is increasing at a rate of 3 inches per second and its height is decreasing at a rate of 2inches per second?
• What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building if an 8ft fence runs parallel to a tall building at the distance of 4ft from the building?
• How do you find the rate at which a batter's distance from second base decreases when he is halfway to first base if the baseball diamond is a square with side 90 ft and a batter hits the ball and runs toward first base with a speed of 24 ft/s?
• How do you find the amount of sugar in the tank after t minutes if a tank contains 1640 liters of pure water and a solution that contains 0.09 kg of sugar per liter enters a tank at the rate 5 l/min the solution is mixed and drains from the tank at the same rate?
• How do you determine how much salt is in the tank when it is full if a 30-gallon tank initially contains 15 gallons of salt water containing 5 pounds of salt and suppose salt water containing 1 pound of salt per gallon is pumped into the top of the tank at the rate of 2 gallons per minute, while a well-mixed solution leaves the bottom of the tank at a rate of 1 gallon per minute?
• How do you determine what percentage of chlorine is in the pool after 1 hour if a pool whose volume is 10000 gallons contains water the is 0.01%chlorine and starting at t=0 city water containing .001% chlorine is pumped into the pool at a rate of 5 gal/min while the pool water flows out at the same rate?
• How fast is the radius changing when diameter of the snowball is 10 cm given a spherical snowball with an outer layer of ice melts so that the volume of the snowball decreases at a rate of 2cm per 3min?
• How do you find the rate at which water is being pumped into the tank in cubic centimeters per minute if water is leaking out of an inverted conical tank at a rate of 12500 cubic cm/min at the same time that water is being pumped into the tank at a constant rate, and the tank has 6m height and the the diameter at the top is 6.5m and if the water level is rising at a rate of 20 cm/min when the height of the water is 1.0m?
• How do you find y'' by implicit differentiation for #4x^3 + 3y^3 = 6#?
• How fast is the radius of the basketball increasing when the radius is 16 cm if air is being pumped into a basketball at a rate of 100 cm3/sec?
• Water is being drained from a cone-shaped reservoir 10 ft. in diameter and 10 ft. deep at a constant rate of 3 ft3/min. How fast is the water level falling when the depth of the water is 6 ft?
• The angle of elevation of the sun is decreasing by 1/4 radians per hour. How fast is the shadow cast by a building of height 50 meters lengthening, when the angle of elevation of the sun is #pi/4#?
• A can of soda measures 2.5 inches in diameter and 5 inches in height. If a full-can of soda gets spilled at a rate of 4 in^3/sec, how is the level of soda changing at the moment when the can of soda is half-full?
• The hands of a clock in some tower are approximately 2m and 1.5m in length. How fast is the distance between the tips of the hands changing at 9:00?
• A farmer wishes to enclose a rectangular field of area 450 ft using an existing wall as one of the sides. The cost of the fence for the other 3 sides is $3 per foot. How do you find the dimensions that minimize the cost of the fence?
• Each side of a square is increasing at a rate of 6 cm/s. At what rate is the area of the square increasing when the area of the square is 16 cm^2?
• The altitude of a triangle is increasing at a rate of 1.5 cm/min while the area of the triangle is increasing at a rate of 5 square cm/min. At what rate is the base of the triangle changing when the altitude is 9 cm and the area is 81 square cm?
• Bases are located on the field 90 feet away from one another Jimmy is running at a speed of 10 ft/sec from second to third base. When Jimmy is halfway to third base, how quickly is the distance between him and home plate decreasing?
• The radius of a sphere is increasing at a rate of 4 mm/s. How fast is the volume increasing when the diameter is 40 mm?
• A spherical balloon is being inflated at the rate of 12 cubic feet per second. What is the radius of the balloon when its surface area is increasing at a rate of 8 feet square feet per second? Volume= (4/3)(pi)r^3 Area= 4(pi)r^2?
• A cube of ice is melting and the volume is decreasing at a rate of 3 cubic m/s. How fast is the height decreasing when the cube is 6 inches in height?
• A plane flying horizontally at an altitude of 1 mi and speed of 500mi/hr passes directly over a radar station. How do you find the rate at which the distance from the plane to the station is increasing when it is 2 miles away from the station?
• Two cars start moving from the same point. One travels south at 60mi/h and the other travels west at 25mi/h. At what rate is the distance between the cars increasing two hours later?
• A conical cup of radius 5 cm and height 15 cm is leaking water at the rate of 2 cm^3/min. What rate is the level of water decreasing when the water is 3 cm deep?
• Two sides of a triangle are 6 m and 7 m in length and the angle between them is increasing at a rate of 0.07 rad/s. How do you find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is pi/3?
• A street light is at the top of a 16 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 35 ft from the base of the pole?
• A balloon rises at the rate of 8 ft/sec from a point on the ground 60 ft from the observer. How do you find the rate of change of the angle of elevation when the balloon is 25 ft above the ground?
• The radius of a spherical balloon is increasing at a rate of 2 centimeters per minute. How fast is the volume changing when the radius is 14 centimeters?
• The volume of a cube is increasing at the rate of 20 cubic centimeters per second. How fast, in square centimeters per second, is the surface area of the cube increasing at the instant when each edge of the cube is 10 centimeters long?
• A swimming pool is 25 ft wide, 40 ft long, 3 ft deep at one end and 9 ft deep at the other end. If water is pumped into the pool at the rate of 10 cubic feet/min, how fast is the water level rising when it is 4 ft deep at the deep end?
• The radius of a spherical balloon is increasing by 5 cm/sec. At what rate is air being blown into the balloon at the moment when the radius is 13 cm?
• A girl of height 120 cm is walking towards a light on the ground at a speed of 0.6 m/s. Her shadow is being cast on a wall behind her that is 5 m from the light. How is the size of her shadow changing when she is 1.5 m from the light?
• You throw a rock into a pond and watch the circular ripple travel out in all directions along the surface. If the ripple travels at 1.4 m/s, what is the approximate rate that the circumference is increasing when the diameter of the circular ripple is 6m?
• A beacon on a lighthouse is one mile from shore, and revolves at 10PI Radians per minute. What is the speed with which the light sweeps across the straight shore as it lights the sand 2 miles from the lighthouse?
• The radius r of a sphere is increasing at a constant rate of 0.04 centimeters per second.At the time when the radius of the sphere is 10 centimeters, what is the rate of increase of its volume?
• Question #20dff
• A feeding trough full of water is 5 ft long and its ends are isosceles triangles having a base and height of 3 ft. Water leaks out of the tank at a rate of 5 (ft)^3/min. How fast is the water level falling when the water in the tank is 6 in. deep?
• If a hose filling up a cylindrical pool with a radius of 5 ft at 28 cubic feet per minute, how fast is the depth of the pool water increasing?
• A water tank has the shape of an upright cylinder. The cylinder is filling at a rate of 1.5 m^3 / minute. If the tank has a radius of 2 m, at what rate is the water level increasing when the water is 3.2 m deep?
• The sun is shining and a spherical snowball of volume 340 ft3 is melting at a rate of 17 cubic feet per hour. As it melts, it remains spherical. At what rate is the radius changing after 7 hours?
• A cylinder gets taller at a rate of 3 inches per second, but the radius shrinks at a rate of 1 inch per second. How fast is the volume of the cylinder changing when the height is 20 inches and the radius is 10 inches?
• Anna is 6 ft. tall. She is walking away from a street light that is 24 ft tall at a rate of 4 ft/sec. How fast is the length of her shadow changing?
• A conical tank is 8m high. The radius at the top is 2m. At what rate is water running out if the depth is 3m and is decreasing at the rate of 0.4 m/min?
• A spherical balloon is expanding at the rate of 60pi cubic inches per second. How fast is the surface area of the balloon expanding when the radius of the balloon is 4 in?
• A point is moving along the curve #y=sqrt(x)# in such a way that its x coordinate id increasing at the rate of 2 units per minute. At what rate is its slope changing (a) when x=1? (b) when x=4?
• A street light is mounted at the top of a 15ft tall pole. A man 6ft tall walks away from the pole with a speed of 5ft/sec along a straight path. How fast is the tip of his shadow moving when he is 40ft from the pole?
• A street light is at the top of a 12 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 ft from the base of the pole?
• A person 1.8m tall is walking away from a lamppost 6m high at the rate 1.3m/s. At what rate is the end of the person's shadow moving away from the lamppost?
• Andy is 6 feet tall and is walking away from a street light that is 30 feet above ground at a rate of 2 feet per second. How fast is his shadow increasing in length?
• A plane flying with a constant speed of 14 km/min passes over a ground radar station at an altitude of 13 km and climbs at an angle of 20 degrees. At what rate, in km/min is the distance from the plane to the radar station increasing 5 minutes later?
• Water is pouring into a cylindrical bowl of height 10 ft. and radius 3 ft, at a rate of #5" ft"^3/"min"#. At what rate does the level of the water rise?
• A man standing on a wharf is hauling in a rope attached to a boat, at the rate of 4 ft/sec. if his hands are 9 ft. above the point of attachment, what is the rate at which the boat is approaching the wharf when it is 12 ft away?
• A six foot tall person is walking away from a 14 foot tall lamp post at 3 feet per second. When the person is 10 feet from the lamp post, how fast is the tip of the shadow moving away from the lamp post?
• If the rate at which water vapor condenses onto a spherical raindrop is proportional to the surface area of the raindrop, show that the radius of the raindrop will increase at a constant rate?
• A plane flying with a constant speed of 4 km/min passes over a ground radar station at an altitude of 13 km and climbs at an angle of 40 degrees. At what rate is the distance from the plane to the radar station increasing 4 minutes later?
• A plane flying horizontally at an altitude of 1 mi and a speed of 540 mi/h passes directly over a radar station. How do you find the rate at which the distance from the plane to the station is increasing when it is 5 mi away from the station?
• A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 50 ft from the pole?
• A ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes How fast is a rider rising when the rider is 16 m above ground level?
• Question #acc14
• Determine how fast the length of an edge of a cube is changing at the moment when the length of the edge is #5 cm# and the volume of the cube is decreasing at a rate of #100 (cm^3)/sec#?
• A piston is connected by a rod of #14 cm# to a crankshaft at a point #5 cm# away from the axis of rotation. Determine how fast the crankshaft is rotating when the piston is 11 cm away from the axis of rotation and is moving toward it at 1200 cm/s?
• Question #ec74b
• Question #c977a
• Water leaking onto a floor forms a circular pool. The radius of the pool increases at a rate of 4 cm/min. How fast is the area of the pool increasing when the radius is 5 cm?
• Oil spilling from a ruptured tanker spreads in a circle on the surface of the ocean. The area of the spill increases at a rate of 9π m²/min. How fast is the radius of the spill increasing when the radius is 10 m?
• A conical paper cup is 10 cm tall with a radius of 10 cm. The cup is being filled with water so that the water level rises at a rate of 2 cm/sec. At what rate is water being poured into the cup when the water level is 8 cm?
• A spherical balloon is inflated so that its radius (r) increases at a rate of 2/r cm/sec. How fast is the volume of the balloon increasing when the radius is 4 cm?
• A 7 ft tall person is walking away from a 20 ft tall lamppost at a rate of 5 ft/sec. Assume the scenario can be modeled with right triangles. At what rate is the length of the person's shadow changing when the person is 16 ft from the lamppost?
• A hypothetical square shrinks so that the length of its diagonals are changing at a rate of −8 m/min. At what rate is the area of the square changing when the diagonals are 5 m each?
• A hypothetical square shrinks at a rate of 2 m²/min. At what rate are the diagonals of the square changing when the diagonals are 7 m each?
• Water leaking onto a floor forms a circular pool. The area of the pool increases at a rate of 25π cm²/min. How fast is the radius of the pool increasing when the radius is 6 cm?
• A perfect cube shaped ice cube melts so that the length of its sides are decreasing at a rate of 2 mm/sec. Assume that the block retains its cube shape as it melts. At what rate is the volume of the ice cube changing when the sides are 2 mm each?
• A spherical snowball melts so that its radius decreases at a rate of 4 in/sec. At what rate is the volume of the snowball changing when the radius is 8 in?
• A hypothetical cube shrinks at a rate of 8 m³/min. At what rate are the sides of the cube changing when the sides are 3 m each?
• A hypothetical square grows at a rate of 16 m²/min. How fast are the sides of the square increasing when the sides are 15 m each?
• A hypothetical cube grows so that the length of its sides are increasing at a rate of 4 m/min. How fast is the volume of the cube increasing when the sides are 7 m each?
• A hypothetical square grows so that the length of its diagonals are increasing at a rate of 4 m/min. How fast is the area of the square increasing when the diagonals are 14 m each?
• A conical paper cup is 10 cm tall with a radius of 30 cm. The cup is being filled with water so that the water level rises at a rate of 2 cm/sec. At what rate is water being poured into the cup when the water level is 9 cm?
• Question #c8c35
• CALCULUS RELATED RATE PROBLEM. PLEASE HELP??
• Question #f65d3
• If the rate of change in #x# is #"3 s"^(-1)#, and #(dy)/(dx) = 5#, what is the rate of change in #y#? Is #y# changing faster than #x# or vice versa?
• Assume the bottom of a 16 ft ladder is pulled out at a rate of 3 ft/s. How do you find the rate at the top of the ladder when it is 10 ft from the ground?
• A square is inscribed in a circle. How fast is the area of the square changing when the area of the circle is increasing at the rate of 1 inch squared per minute?
• A collapsible spherical tank is being relieved of air at the rate of 2 cubic inches per minute. At what rate is the radius of the tank changing when the surface area is 12 square inches?
• Calculus Word Problem?
• Water is being pumped into a vertical cylinder of radius 5 meters and height 20 meters at a rate of 3 meters/min. How fast is the water level rising when the cylinder is half full?
• Calculus Word Problem?
• If a snowball melts so that its surface area decreases at a rate of 3 cm2/min, how do you find the rate at which the diameter decreases when the diameter is 12 cm?
• A right cylindrical water tank with a diameter of 3 feet and a height of 6 feet is being drained. At what rate is the volume of the water in the tank changing when the water level of the tank is dropping at a rate of 4 inches per minute?
• Calculus, Optimization: Ship A is 60 miles south of ship B and is sailing North at a rate of 21 mph. If ship B is sailing west at a rate of 22 mph, in how many hours will the distance between the two ships be minimized?
• Question #9a6d1
• Question #8e483
• Two boats leave the port at the same time with one boat traveling north at 15 knots per hour and the other boat traveling west at 12 knots per hour. How fast is the distance between the boats changing after 2 hours?
• Help with a related rates problem?
• Question #cd6e9
• Question #31517
• Question #a249f
• Related rates problem?