STEM Lessons for College Students

related rates sphere application of derivatives differentiation calculus optimization inverted

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#1 Circles Oil is dropping from a punctured tank onto a circular oil pool on the ground with radius r. Find the following and state their meanings and units.
a) dr/dt
c) How fast is the radius changing when the radius is 2 cm and the area is changing at 20p cm2/min

#2 Cones:A conical pit is being filled with concrete. The pit has a base radius of 10 cm, and a depth of 30 cm. The pit is being filled so that the height of water rises at 2 cm/sec. Find the following in terms of V and h, (include meaning & units).
a) dh/dt
c) How fast is pit being filled at the point in time that the water level is 4cm.

#3 Spheres A sphere being inflated at 10ft^3/sec How fast is the radius increasing once the radiums is 4ft ?
#4 Ladder: A ladder 13 ft long is sliding down a wall. At the moment the base is 12ft from the wall, the base of the ladder is moving at 5 ft/sec. How fast is the top of the ladder moving at this moment?

#5 Boat and Winch: A winch 20 feet above sea level is used to reel in a rope connected to a boat at 2 ft/sec. How fast is the boat moving when the rope is 45 feet in length.

Ice Cube: An ice cube melts uniformly at a rate of 27cm3/sec. If the cube retains its perfect form while it is melting, at what rate are the sides shrinking when each side is 3 cm long ?

#Inverted conical Tank: water runs out of a conical tank at 7ft^3/min. The tank has a base of 10ft and a depth of 12 ft. How fast is the water level rising when the water is 5ft deep

#7 Trapezoidal Trough:The ends of a horizontal water trough is an isoceles trapezoidal prism. The length is 8 feet and the lower base is 4 feet. The upper base is 10 feet and the depth of the trough is 2 feet. If the water level is rising at 3 feet/min, when the depth of the water is 1 foot, how fast is water being poured into the trough?

#9 Hot air Balloon:A hot air balloon is rising vertically from a level platform. A range finder 400 feet from the lift point is tracking the hot air balloon. At the moment the angle of elevation of the range finder is 60 degrees , the angle of elevation is increasing at .10 degrees/minute. How fast is the balloon rising at that moment?

Consider a sphere of radius 10cm.
If the radius changes 0.1cm (a very small amount) how much does the volume change?
Water is draining from a cylindrical tank at 3 liters/second. How fast is the surface dropping?
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?

A particle moves along the curve C ú »” Ä B fi As it reaches $
the pointa b #fl $ fl the y-coordinate is increasing at a rate of
4 cm/sec. How fast is the x-coordinate of the point changing
at this instant?
3. Suppose where is a constant and when

  1. Two cars start moving from the same point. One travels south at
    60 m/hr and the other travels west at 25 m/hr. At what rate is
    the distance between the cars increasing two hours later?
  2. A plane flying horizontally at an altitude of 1 mile and a speed
    of 500 mph passes over a radar station. Find the rate at which
    the distance from the plane to the station is increasing when it
    is 2 miles away from the station.
  3. A conical water tower has a height of 12 ft and a radius of 3 ft.
    Water is pumped into the tank at a rate of 4 ft /min. How fast is $
    the water level rising when the water level is 6 ft

A man walks along a straight path at a speed of 4 ft/s.
A searchlight is located on the ground 20 ft from the path and is kept focused on the man.
§At what rate is the searchlight rotating
when the man is 15 ft from the point on
the path closest to the searchlight?
A police cruiser, approaching a right-angled intersection from the north, is chasing a speeding
car that has turned the corner and is now moving straight east. When the cruiser is 0.6
mi north of the intersection and the car is 0.8 mi to the east, the police determine with radar
that the distance between them and the car is increasing at 20 mph. If the cruiser is moving
at 60 mph at the instant of measurement, what is the speed of the car?
The voltage V (volts), current I (amperes),
and resistance R (ohms) of an electric circuit like the one shown
here are related by the equation V IR. Suppose that V is
increasing at the rate of 1 volt sec while I is decreasing at the
rate of 13 ampsec. Let t denote time in sec.
A trough is 15 ft long and 4 ft across the top
as shown in the figure. Its ends are isosceles triangles with
height 3 ft. Water runs into the trough at the rate of 2.5 ft3min.
How fast is the water level rising when it is 2 ft deep?
Sliding Ladder A 13-ft ladder is leaning against a house (see
figure) when its base starts to slide away. By the time the base is
12 ft from the house, the base is moving at the rate of 5 ft sec.

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